3.310 \(\int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx\)
Optimal. Leaf size=399 \[ -\frac {3 d^4 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^5}+\frac {3 d^4 \text {Li}_5\left (-e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \text {Li}_5\left (e^{2 i (a+b x)}\right )}{2 b^5}+\frac {6 i d^3 (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}-\frac {3 i d^3 (c+d x) \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b} \]
[Out]
2*I*d*(d*x+c)^3/b^2+1/2*(d*x+c)^4/b-2*(d*x+c)^4*arctanh(exp(2*I*(b*x+a)))/b-6*d^2*(d*x+c)^2*ln(1+exp(2*I*(b*x+
a)))/b^3+6*I*d^3*(d*x+c)*polylog(2,-exp(2*I*(b*x+a)))/b^4-2*I*d*(d*x+c)^3*polylog(2,exp(2*I*(b*x+a)))/b^2+3*I*
d^3*(d*x+c)*polylog(4,exp(2*I*(b*x+a)))/b^4-3*d^4*polylog(3,-exp(2*I*(b*x+a)))/b^5-3*d^2*(d*x+c)^2*polylog(3,-
exp(2*I*(b*x+a)))/b^3+3*d^2*(d*x+c)^2*polylog(3,exp(2*I*(b*x+a)))/b^3+2*I*d*(d*x+c)^3*polylog(2,-exp(2*I*(b*x+
a)))/b^2-3*I*d^3*(d*x+c)*polylog(4,-exp(2*I*(b*x+a)))/b^4+3/2*d^4*polylog(5,-exp(2*I*(b*x+a)))/b^5-3/2*d^4*pol
ylog(5,exp(2*I*(b*x+a)))/b^5-2*d*(d*x+c)^3*tan(b*x+a)/b^2+1/2*(d*x+c)^4*tan(b*x+a)^2/b
________________________________________________________________________________________
Rubi [A] time = 0.97, antiderivative size = 399, normalized size of antiderivative = 1.00,
number of steps used = 25, number of rules used = 16, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used
= {2620, 14, 4420, 6741, 12, 6742, 2551, 4183, 2531, 6609, 2282, 6589, 3720, 3719, 2190, 32}
\[ \frac {6 i d^3 (c+d x) \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac {3 i d^3 (c+d x) \text {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \text {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^2 (c+d x)^2 \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x)^3 \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}+\frac {3 d^4 \text {PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \text {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^3,x]
[Out]
((2*I)*d*(c + d*x)^3)/b^2 + (c + d*x)^4/(2*b) - (2*(c + d*x)^4*ArcTanh[E^((2*I)*(a + b*x))])/b - (6*d^2*(c + d
*x)^2*Log[1 + E^((2*I)*(a + b*x))])/b^3 + ((6*I)*d^3*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))])/b^4 + ((2*I)*
d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 - ((2*I)*d*(c + d*x)^3*PolyLog[2, E^((2*I)*(a + b*x))])/b^
2 - (3*d^4*PolyLog[3, -E^((2*I)*(a + b*x))])/b^5 - (3*d^2*(c + d*x)^2*PolyLog[3, -E^((2*I)*(a + b*x))])/b^3 +
(3*d^2*(c + d*x)^2*PolyLog[3, E^((2*I)*(a + b*x))])/b^3 - ((3*I)*d^3*(c + d*x)*PolyLog[4, -E^((2*I)*(a + b*x))
])/b^4 + ((3*I)*d^3*(c + d*x)*PolyLog[4, E^((2*I)*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, -E^((2*I)*(a + b*x))])/
(2*b^5) - (3*d^4*PolyLog[5, E^((2*I)*(a + b*x))])/(2*b^5) - (2*d*(c + d*x)^3*Tan[a + b*x])/b^2 + ((c + d*x)^4*
Tan[a + b*x]^2)/(2*b)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rule 2190
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2282
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 2531
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 2551
Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Rule 3719
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]
Rule 3720
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]
Rule 4183
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]
Rule 4420
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Modul
e[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u
, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
Rule 6589
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rule 6609
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]
Rule 6741
Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {align*} \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx &=\frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-(4 d) \int (c+d x)^3 \left (\frac {\log (\tan (a+b x))}{b}+\frac {\tan ^2(a+b x)}{2 b}\right ) \, dx\\ &=\frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-(4 d) \int \frac {(c+d x)^3 \left (2 \log (\tan (a+b x))+\tan ^2(a+b x)\right )}{2 b} \, dx\\ &=\frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(2 d) \int (c+d x)^3 \left (2 \log (\tan (a+b x))+\tan ^2(a+b x)\right ) \, dx}{b}\\ &=\frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(2 d) \int \left (2 (c+d x)^3 \log (\tan (a+b x))+(c+d x)^3 \tan ^2(a+b x)\right ) \, dx}{b}\\ &=\frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(2 d) \int (c+d x)^3 \tan ^2(a+b x) \, dx}{b}-\frac {(4 d) \int (c+d x)^3 \log (\tan (a+b x)) \, dx}{b}\\ &=-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\int 2 b (c+d x)^4 \csc (2 a+2 b x) \, dx}{b}+\frac {(2 d) \int (c+d x)^3 \, dx}{b}+\frac {\left (6 d^2\right ) \int (c+d x)^2 \tan (a+b x) \, dx}{b^2}\\ &=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+2 \int (c+d x)^4 \csc (2 a+2 b x) \, dx-\frac {\left (12 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}} \, dx}{b^2}\\ &=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(4 d) \int (c+d x)^3 \log \left (1-e^{i (2 a+2 b x)}\right ) \, dx}{b}+\frac {(4 d) \int (c+d x)^3 \log \left (1+e^{i (2 a+2 b x)}\right ) \, dx}{b}+\frac {\left (12 d^3\right ) \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {\left (6 i d^2\right ) \int (c+d x)^2 \text {Li}_2\left (-e^{i (2 a+2 b x)}\right ) \, dx}{b^2}+\frac {\left (6 i d^2\right ) \int (c+d x)^2 \text {Li}_2\left (e^{i (2 a+2 b x)}\right ) \, dx}{b^2}-\frac {\left (6 i d^4\right ) \int \text {Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^4}\\ &=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\left (6 d^3\right ) \int (c+d x) \text {Li}_3\left (-e^{i (2 a+2 b x)}\right ) \, dx}{b^3}-\frac {\left (6 d^3\right ) \int (c+d x) \text {Li}_3\left (e^{i (2 a+2 b x)}\right ) \, dx}{b^3}-\frac {\left (3 d^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^5}\\ &=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\left (3 i d^4\right ) \int \text {Li}_4\left (-e^{i (2 a+2 b x)}\right ) \, dx}{b^4}-\frac {\left (3 i d^4\right ) \int \text {Li}_4\left (e^{i (2 a+2 b x)}\right ) \, dx}{b^4}\\ &=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\left (3 d^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_4(-x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{2 b^5}-\frac {\left (3 d^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_4(x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{2 b^5}\\ &=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \text {Li}_5\left (-e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \text {Li}_5\left (e^{2 i (a+b x)}\right )}{2 b^5}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] time = 7.47, size = 2090, normalized size = 5.24 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^3,x]
[Out]
-((c^2*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x
))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^
((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^
((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^3) - (c*d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E
^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*L
og[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLo
g[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*P
olyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/
E^((2*I)*a)))/b^4 - (d^4*E^(I*a)*Csc[a]*((2*b^5*x^5)/E^((2*I)*a) + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 - E^
((-I)*(a + b*x))] + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 + E^((-I)*(a + b*x))] - (20*(-1 + E^((2*I)*a))*(b^3
*x^3*PolyLog[2, -E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, -E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, -E^((-
I)*(a + b*x))] + (6*I)*PolyLog[5, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog
[2, E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, E^((-I)*(a + b*x))]
+ (6*I)*PolyLog[5, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(10*b^5) + (x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*
c*d^3*x^3 + d^4*x^4)*Csc[a]*Sec[a])/5 - ((I/2)*c^2*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^(
(-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*Poly
Log[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - ((I/2)*d^4*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*
Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*
I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^5*E^(I*a)) - (I/2)*c*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - (
(4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^
((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E
^((2*I)*a)))*Sec[a] + (d^4*((-4*I)*x^5 - (10*(1 + E^((2*I)*a))*x^4*Log[1 + E^((-2*I)*(a + b*x))])/b + (5*(1 +
E^((2*I)*a))*((-4*I)*b^3*x^3*PolyLog[2, -E^((-2*I)*(a + b*x))] - 6*b^2*x^2*PolyLog[3, -E^((-2*I)*(a + b*x))] +
(6*I)*b*x*PolyLog[4, -E^((-2*I)*(a + b*x))] + 3*PolyLog[5, -E^((-2*I)*(a + b*x))]))/b^5)*Sec[a])/(20*E^(I*a))
+ ((c + d*x)^4*Sec[a + b*x]^2)/(2*b) - (c^4*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a
]))/(b*(Cos[a]^2 + Sin[a]^2)) - (6*c^2*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a])
)/(b^3*(Cos[a]^2 + Sin[a]^2)) + (c^4*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(
b*(Cos[a]^2 + Sin[a]^2)) - (2*c^3*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Co
t[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*
Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]])
)]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (6*c*d^3*Csc[a]*((b^2*x^2)/E^(I*
ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a
]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a
]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^4*Sqrt[Csc[a]^2*(Cos[a]
^2 + Sin[a]^2)]) - (2*Sec[a]*Sec[a + b*x]*(c^3*d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*
x^3*Sin[b*x]))/b^2 - (2*c^3*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) -
Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[
b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[
a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])
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fricas [C] time = 1.05, size = 3308, normalized size = 8.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x, algorithm="fricas")
[Out]
1/2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4 - 24*d^4*cos(b*x + a)^2*polyl
og(5, cos(b*x + a) + I*sin(b*x + a)) - 24*d^4*cos(b*x + a)^2*polylog(5, cos(b*x + a) - I*sin(b*x + a)) + 24*d^
4*cos(b*x + a)^2*polylog(5, I*cos(b*x + a) + sin(b*x + a)) + 24*d^4*cos(b*x + a)^2*polylog(5, I*cos(b*x + a) -
sin(b*x + a)) + 24*d^4*cos(b*x + a)^2*polylog(5, -I*cos(b*x + a) + sin(b*x + a)) + 24*d^4*cos(b*x + a)^2*poly
log(5, -I*cos(b*x + a) - sin(b*x + a)) - 24*d^4*cos(b*x + a)^2*polylog(5, -cos(b*x + a) + I*sin(b*x + a)) - 24
*d^4*cos(b*x + a)^2*polylog(5, -cos(b*x + a) - I*sin(b*x + a)) + (-4*I*b^3*d^4*x^3 - 12*I*b^3*c*d^3*x^2 - 12*I
*b^3*c^2*d^2*x - 4*I*b^3*c^3*d)*cos(b*x + a)^2*dilog(cos(b*x + a) + I*sin(b*x + a)) + (4*I*b^3*d^4*x^3 + 12*I*
b^3*c*d^3*x^2 + 12*I*b^3*c^2*d^2*x + 4*I*b^3*c^3*d)*cos(b*x + a)^2*dilog(cos(b*x + a) - I*sin(b*x + a)) + (-4*
I*b^3*d^4*x^3 - 12*I*b^3*c*d^3*x^2 - 4*I*b^3*c^3*d - 12*I*b*c*d^3 - 12*I*(b^3*c^2*d^2 + b*d^4)*x)*cos(b*x + a)
^2*dilog(I*cos(b*x + a) + sin(b*x + a)) + (4*I*b^3*d^4*x^3 + 12*I*b^3*c*d^3*x^2 + 4*I*b^3*c^3*d + 12*I*b*c*d^3
+ 12*I*(b^3*c^2*d^2 + b*d^4)*x)*cos(b*x + a)^2*dilog(I*cos(b*x + a) - sin(b*x + a)) + (4*I*b^3*d^4*x^3 + 12*I
*b^3*c*d^3*x^2 + 4*I*b^3*c^3*d + 12*I*b*c*d^3 + 12*I*(b^3*c^2*d^2 + b*d^4)*x)*cos(b*x + a)^2*dilog(-I*cos(b*x
+ a) + sin(b*x + a)) + (-4*I*b^3*d^4*x^3 - 12*I*b^3*c*d^3*x^2 - 4*I*b^3*c^3*d - 12*I*b*c*d^3 - 12*I*(b^3*c^2*d
^2 + b*d^4)*x)*cos(b*x + a)^2*dilog(-I*cos(b*x + a) - sin(b*x + a)) + (4*I*b^3*d^4*x^3 + 12*I*b^3*c*d^3*x^2 +
12*I*b^3*c^2*d^2*x + 4*I*b^3*c^3*d)*cos(b*x + a)^2*dilog(-cos(b*x + a) + I*sin(b*x + a)) + (-4*I*b^3*d^4*x^3 -
12*I*b^3*c*d^3*x^2 - 12*I*b^3*c^2*d^2*x - 4*I*b^3*c^3*d)*cos(b*x + a)^2*dilog(-cos(b*x + a) - I*sin(b*x + a))
+ (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4)*cos(b*x + a)^2*log(cos(b*x +
a) + I*sin(b*x + a) + 1) - (b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 + 1)*b^2*c^2*d^2 - 4*(a^3 + 3*a)*b*c*d^3 + (a^4 +
6*a^2)*d^4)*cos(b*x + a)^2*log(cos(b*x + a) + I*sin(b*x + a) + I) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^
2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4)*cos(b*x + a)^2*log(cos(b*x + a) - I*sin(b*x + a) + 1) - (b^4*c^4 - 4*a*b^
3*c^3*d + 6*(a^2 + 1)*b^2*c^2*d^2 - 4*(a^3 + 3*a)*b*c*d^3 + (a^4 + 6*a^2)*d^4)*cos(b*x + a)^2*log(cos(b*x + a)
- I*sin(b*x + a) + I) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3 + 3*a)*b*
c*d^3 - (a^4 + 6*a^2)*d^4 + 6*(b^4*c^2*d^2 + b^2*d^4)*x^2 + 4*(b^4*c^3*d + 3*b^2*c*d^3)*x)*cos(b*x + a)^2*log(
I*cos(b*x + a) + sin(b*x + a) + 1) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a
^3 + 3*a)*b*c*d^3 - (a^4 + 6*a^2)*d^4 + 6*(b^4*c^2*d^2 + b^2*d^4)*x^2 + 4*(b^4*c^3*d + 3*b^2*c*d^3)*x)*cos(b*x
+ a)^2*log(I*cos(b*x + a) - sin(b*x + a) + 1) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^
2*d^2 + 4*(a^3 + 3*a)*b*c*d^3 - (a^4 + 6*a^2)*d^4 + 6*(b^4*c^2*d^2 + b^2*d^4)*x^2 + 4*(b^4*c^3*d + 3*b^2*c*d^3
)*x)*cos(b*x + a)^2*log(-I*cos(b*x + a) + sin(b*x + a) + 1) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d -
6*a^2*b^2*c^2*d^2 + 4*(a^3 + 3*a)*b*c*d^3 - (a^4 + 6*a^2)*d^4 + 6*(b^4*c^2*d^2 + b^2*d^4)*x^2 + 4*(b^4*c^3*d
+ 3*b^2*c*d^3)*x)*cos(b*x + a)^2*log(-I*cos(b*x + a) - sin(b*x + a) + 1) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^
2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*cos(b*x + a)^2*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) + (b^4*c
^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*cos(b*x + a)^2*log(-1/2*cos(b*x + a) - 1/2*I
*sin(b*x + a) + 1/2) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + 4*a*b^3*c^3*d - 6*
a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*cos(b*x + a)^2*log(-cos(b*x + a) + I*sin(b*x + a) + 1) - (b^4*c^4 -
4*a*b^3*c^3*d + 6*(a^2 + 1)*b^2*c^2*d^2 - 4*(a^3 + 3*a)*b*c*d^3 + (a^4 + 6*a^2)*d^4)*cos(b*x + a)^2*log(-cos(
b*x + a) + I*sin(b*x + a) + I) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + 4*a*b^3*
c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*cos(b*x + a)^2*log(-cos(b*x + a) - I*sin(b*x + a) + 1) -
(b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 + 1)*b^2*c^2*d^2 - 4*(a^3 + 3*a)*b*c*d^3 + (a^4 + 6*a^2)*d^4)*cos(b*x + a)^2
*log(-cos(b*x + a) - I*sin(b*x + a) + I) + (24*I*b*d^4*x + 24*I*b*c*d^3)*cos(b*x + a)^2*polylog(4, cos(b*x + a
) + I*sin(b*x + a)) + (-24*I*b*d^4*x - 24*I*b*c*d^3)*cos(b*x + a)^2*polylog(4, cos(b*x + a) - I*sin(b*x + a))
+ (24*I*b*d^4*x + 24*I*b*c*d^3)*cos(b*x + a)^2*polylog(4, I*cos(b*x + a) + sin(b*x + a)) + (-24*I*b*d^4*x - 24
*I*b*c*d^3)*cos(b*x + a)^2*polylog(4, I*cos(b*x + a) - sin(b*x + a)) + (-24*I*b*d^4*x - 24*I*b*c*d^3)*cos(b*x
+ a)^2*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) + (24*I*b*d^4*x + 24*I*b*c*d^3)*cos(b*x + a)^2*polylog(4, -I
*cos(b*x + a) - sin(b*x + a)) + (-24*I*b*d^4*x - 24*I*b*c*d^3)*cos(b*x + a)^2*polylog(4, -cos(b*x + a) + I*sin
(b*x + a)) + (24*I*b*d^4*x + 24*I*b*c*d^3)*cos(b*x + a)^2*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) + 12*(b^2
*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(b*x + a)^2*polylog(3, cos(b*x + a) + I*sin(b*x + a)) + 12*(b^2*d^4
*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(b*x + a)^2*polylog(3, cos(b*x + a) - I*sin(b*x + a)) - 12*(b^2*d^4*x^2
+ 2*b^2*c*d^3*x + b^2*c^2*d^2 + d^4)*cos(b*x + a)^2*polylog(3, I*cos(b*x + a) + sin(b*x + a)) - 12*(b^2*d^4*x
^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 + d^4)*cos(b*x + a)^2*polylog(3, I*cos(b*x + a) - sin(b*x + a)) - 12*(b^2*d^4
*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 + d^4)*cos(b*x + a)^2*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 12*(b^2*
d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 + d^4)*cos(b*x + a)^2*polylog(3, -I*cos(b*x + a) - sin(b*x + a)) + 12*(b
^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(b*x + a)^2*polylog(3, -cos(b*x + a) + I*sin(b*x + a)) + 12*(b^2*
d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(b*x + a)^2*polylog(3, -cos(b*x + a) - I*sin(b*x + a)) - 4*(b^3*d^4*
x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d)*cos(b*x + a)*sin(b*x + a))/(b^5*cos(b*x + a)^2)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{4} \csc \left (b x + a\right ) \sec \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x, algorithm="giac")
[Out]
integrate((d*x + c)^4*csc(b*x + a)*sec(b*x + a)^3, x)
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maple [B] time = 0.25, size = 1729, normalized size = 4.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x)
[Out]
3/2*d^4*polylog(5,-exp(2*I*(b*x+a)))/b^5-3*d^4*polylog(3,-exp(2*I*(b*x+a)))/b^5+1/b^5*d^4*a^4*ln(exp(I*(b*x+a)
)-1)+12/b^3*c^2*d^2*polylog(3,-exp(I*(b*x+a)))+12/b^3*c^2*d^2*polylog(3,exp(I*(b*x+a)))-1/b^5*d^4*a^4*ln(1-exp
(I*(b*x+a)))+12/b^3*d^4*polylog(3,exp(I*(b*x+a)))*x^2+12/b^3*d^4*polylog(3,-exp(I*(b*x+a)))*x^2-6/b^3*d^2*c^2*
ln(1+exp(2*I*(b*x+a)))-6/b^3*d^4*ln(1+exp(2*I*(b*x+a)))*x^2+4*I/b^2*d^4*x^3-8*I/b^5*d^4*a^3-3/b^3*c^2*d^2*poly
log(3,-exp(2*I*(b*x+a)))-3/b^3*d^4*polylog(3,-exp(2*I*(b*x+a)))*x^2+24*I/b^3*d^3*c*a*x-24*d^4*polylog(5,-exp(I
*(b*x+a)))/b^5-24*d^4*polylog(5,exp(I*(b*x+a)))/b^5-12/b^3*d^3*c*ln(1+exp(2*I*(b*x+a)))*x+12*I/b^2*d^3*c*x^2+1
2*I/b^4*d^3*c*a^2+6*I/b^4*d^4*polylog(2,-exp(2*I*(b*x+a)))*x-12*I/b^4*d^4*a^2*x+6*I/b^4*d^3*c*polylog(2,-exp(2
*I*(b*x+a)))+2*(b*d^4*x^4*exp(2*I*(b*x+a))+4*b*c*d^3*x^3*exp(2*I*(b*x+a))+6*b*c^2*d^2*x^2*exp(2*I*(b*x+a))+4*b
*c^3*d*x*exp(2*I*(b*x+a))-2*I*d^4*x^3*exp(2*I*(b*x+a))+b*c^4*exp(2*I*(b*x+a))-6*I*c*d^3*x^2*exp(2*I*(b*x+a))-6
*I*c^2*d^2*x*exp(2*I*(b*x+a))-2*I*d^4*x^3-2*I*c^3*d*exp(2*I*(b*x+a))-6*I*c*d^3*x^2-6*I*c^2*d^2*x-2*I*c^3*d)/b^
2/(1+exp(2*I*(b*x+a)))^2-4/b*c*d^3*ln(1+exp(2*I*(b*x+a)))*x^3+6*I/b^2*c*d^3*polylog(2,-exp(2*I*(b*x+a)))*x^2+6
*I/b^2*c^2*d^2*polylog(2,-exp(2*I*(b*x+a)))*x-1/b*c^4*ln(1+exp(2*I*(b*x+a)))+1/b*c^4*ln(exp(I*(b*x+a))+1)+1/b*
c^4*ln(exp(I*(b*x+a))-1)+12/b^5*d^4*a^2*ln(exp(I*(b*x+a)))+12/b^3*d^2*c^2*ln(exp(I*(b*x+a)))-1/b*d^4*ln(1+exp(
2*I*(b*x+a)))*x^4+4/b*c^3*d*ln(exp(I*(b*x+a))+1)*x+4/b*c^3*d*ln(1-exp(I*(b*x+a)))*x+4/b^2*c^3*d*ln(1-exp(I*(b*
x+a)))*a+6/b*c^2*d^2*ln(exp(I*(b*x+a))+1)*x^2+24/b^3*c*d^3*polylog(3,-exp(I*(b*x+a)))*x-6/b^3*c^2*d^2*a^2*ln(1
-exp(I*(b*x+a)))+6/b*c^2*d^2*ln(1-exp(I*(b*x+a)))*x^2+24/b^3*c*d^3*polylog(3,exp(I*(b*x+a)))*x+24*I/b^4*c*d^3*
polylog(4,-exp(I*(b*x+a)))+24*I/b^4*c*d^3*polylog(4,exp(I*(b*x+a)))-4*I/b^2*d^4*polylog(2,exp(I*(b*x+a)))*x^3+
24*I/b^4*d^4*polylog(4,exp(I*(b*x+a)))*x-4*I/b^2*d^4*polylog(2,-exp(I*(b*x+a)))*x^3+24*I/b^4*d^4*polylog(4,-ex
p(I*(b*x+a)))*x-4*I/b^2*c^3*d*polylog(2,-exp(I*(b*x+a)))-4*I/b^2*c^3*d*polylog(2,exp(I*(b*x+a)))-4/b^4*c*d^3*a
^3*ln(exp(I*(b*x+a))-1)+6/b^3*c^2*d^2*a^2*ln(exp(I*(b*x+a))-1)-4/b^2*c^3*d*a*ln(exp(I*(b*x+a))-1)+1/b*d^4*ln(1
-exp(I*(b*x+a)))*x^4+1/b*d^4*ln(exp(I*(b*x+a))+1)*x^4-12*I/b^2*c*d^3*polylog(2,-exp(I*(b*x+a)))*x^2-12*I/b^2*c
^2*d^2*polylog(2,-exp(I*(b*x+a)))*x-12*I/b^2*c^2*d^2*polylog(2,exp(I*(b*x+a)))*x-12*I/b^2*c*d^3*polylog(2,exp(
I*(b*x+a)))*x^2+4/b*c*d^3*ln(exp(I*(b*x+a))+1)*x^3+4/b*c*d^3*ln(1-exp(I*(b*x+a)))*x^3+4/b^4*c*d^3*ln(1-exp(I*(
b*x+a)))*a^3-24/b^4*d^3*c*a*ln(exp(I*(b*x+a)))-4/b*c^3*d*ln(1+exp(2*I*(b*x+a)))*x-6/b^3*c*d^3*polylog(3,-exp(2
*I*(b*x+a)))*x-6/b*c^2*d^2*ln(1+exp(2*I*(b*x+a)))*x^2-3*I/b^4*d^4*polylog(4,-exp(2*I*(b*x+a)))*x+2*I/b^2*d^4*p
olylog(2,-exp(2*I*(b*x+a)))*x^3-3*I/b^4*c*d^3*polylog(4,-exp(2*I*(b*x+a)))+2*I/b^2*c^3*d*polylog(2,-exp(2*I*(b
*x+a)))
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maxima [B] time = 6.05, size = 8770, normalized size = 21.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x, algorithm="maxima")
[Out]
-1/2*(c^4*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2)) - 4*a*c^3*d*(1/(sin(b*x + a
)^2 - 1) + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b + 6*a^2*c^2*d^2*(1/(sin(b*x + a)^2 - 1) + log(sin(
b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b^2 - 4*a^3*c*d^3*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - l
og(sin(b*x + a)^2))/b^3 + a^4*d^4*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b^4
+ 2*(24*b^3*c^3*d - 72*a*b^2*c^2*d^2 + 72*a^2*b*c*d^3 - 24*a^3*d^4 + (12*(b*x + a)^4*d^4 + 36*b^2*c^2*d^2 - 7
2*a*b*c*d^3 + 36*a^2*d^4 + 32*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 36*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(
b*x + a)^2 + 24*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x + a) + 4*(3*(b*x +
a)^4*d^4 + 9*b^2*c^2*d^2 - 18*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b
*c*d^3 + (a^2 + 1)*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*
(b*x + a))*cos(4*b*x + 4*a) + 8*(3*(b*x + a)^4*d^4 + 9*b^2*c^2*d^2 - 18*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a
*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2
+ 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (12*I*(b*x + a)^4*d^4 + 36*I*b^2*c^2*d
^2 - 72*I*a*b*c*d^3 + 36*I*a^2*d^4 + (32*I*b*c*d^3 - 32*I*a*d^4)*(b*x + a)^3 + (36*I*b^2*c^2*d^2 - 72*I*a*b*c*
d^3 + (36*I*a^2 + 36*I)*d^4)*(b*x + a)^2 + (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + (72*I*a^2 + 72*I)*b*c*d^3 +
(-24*I*a^3 - 72*I*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + (24*I*(b*x + a)^4*d^4 + 72*I*b^2*c^2*d^2 - 144*I*a*b*c
*d^3 + 72*I*a^2*d^4 + (64*I*b*c*d^3 - 64*I*a*d^4)*(b*x + a)^3 + (72*I*b^2*c^2*d^2 - 144*I*a*b*c*d^3 + (72*I*a^
2 + 72*I)*d^4)*(b*x + a)^2 + (48*I*b^3*c^3*d - 144*I*a*b^2*c^2*d^2 + (144*I*a^2 + 144*I)*b*c*d^3 + (-48*I*a^3
- 144*I*a)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - (6*(b*x + a)^4*
d^4 + 24*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 36*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 24*(b^3*c^3*d
- 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a) + 6*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3
+ 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d
^4)*(b*x + a))*cos(4*b*x + 4*a) + 12*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a
*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*cos(2*b
*x + 2*a) - (-6*I*(b*x + a)^4*d^4 + (-24*I*b*c*d^3 + 24*I*a*d^4)*(b*x + a)^3 + (-36*I*b^2*c^2*d^2 + 72*I*a*b*c
*d^3 - 36*I*a^2*d^4)*(b*x + a)^2 + (-24*I*b^3*c^3*d + 72*I*a*b^2*c^2*d^2 - 72*I*a^2*b*c*d^3 + 24*I*a^3*d^4)*(b
*x + a))*sin(4*b*x + 4*a) - (-12*I*(b*x + a)^4*d^4 + (-48*I*b*c*d^3 + 48*I*a*d^4)*(b*x + a)^3 + (-72*I*b^2*c^2
*d^2 + 144*I*a*b*c*d^3 - 72*I*a^2*d^4)*(b*x + a)^2 + (-48*I*b^3*c^3*d + 144*I*a*b^2*c^2*d^2 - 144*I*a^2*b*c*d^
3 + 48*I*a^3*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + (6*(b*x + a)^4*d^4 +
24*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 36*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 24*(b^3*c^3*d - 3*a*
b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a) + 6*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b
^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b
*x + a))*cos(4*b*x + 4*a) + 12*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d
^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*cos(2*b*x + 2
*a) + (6*I*(b*x + a)^4*d^4 + (24*I*b*c*d^3 - 24*I*a*d^4)*(b*x + a)^3 + (36*I*b^2*c^2*d^2 - 72*I*a*b*c*d^3 + 36
*I*a^2*d^4)*(b*x + a)^2 + (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 - 24*I*a^3*d^4)*(b*x + a))*s
in(4*b*x + 4*a) + (12*I*(b*x + a)^4*d^4 + (48*I*b*c*d^3 - 48*I*a*d^4)*(b*x + a)^3 + (72*I*b^2*c^2*d^2 - 144*I*
a*b*c*d^3 + 72*I*a^2*d^4)*(b*x + a)^2 + (48*I*b^3*c^3*d - 144*I*a*b^2*c^2*d^2 + 144*I*a^2*b*c*d^3 - 48*I*a^3*d
^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 24*((b*x + a)^3*d^4 + 3*(b*c*d^3 -
a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (12*I*(b*x + a)^4*
d^4 + 24*b^3*c^3*d - 72*a*b^2*c^2*d^2 + 72*a^2*b*c*d^3 - 24*a^3*d^4 + (48*I*b*c*d^3 - 24*(2*I*a + 1)*d^4)*(b*x
+ a)^3 + (72*I*b^2*c^2*d^2 - 72*(2*I*a + 1)*b*c*d^3 + (72*I*a^2 + 72*a)*d^4)*(b*x + a)^2 + (48*I*b^3*c^3*d -
72*(2*I*a + 1)*b^2*c^2*d^2 + (144*I*a^2 + 144*a)*b*c*d^3 + (-48*I*a^3 - 72*a^2)*d^4)*(b*x + a))*cos(2*b*x + 2*
a) - (12*b^3*c^3*d - 36*a*b^2*c^2*d^2 + 24*(b*x + a)^3*d^4 + 36*(a^2 + 1)*b*c*d^3 - 12*(a^3 + 3*a)*d^4 + 48*(b
*c*d^3 - a*d^4)*(b*x + a)^2 + 36*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a) + 12*(b^3*c^3*d - 3*a*b
^2*c^2*d^2 + 2*(b*x + a)^3*d^4 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(
b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 24*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 2*(
b*x + a)^3*d^4 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*
a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-12*I*b^3*c^3*d + 36*I*a*b^2*c^2*d^2 - 24*I*(b*x + a
)^3*d^4 + (-36*I*a^2 - 36*I)*b*c*d^3 + (12*I*a^3 + 36*I*a)*d^4 + (-48*I*b*c*d^3 + 48*I*a*d^4)*(b*x + a)^2 + (-
36*I*b^2*c^2*d^2 + 72*I*a*b*c*d^3 + (-36*I*a^2 - 36*I)*d^4)*(b*x + a))*sin(4*b*x + 4*a) - (-24*I*b^3*c^3*d + 7
2*I*a*b^2*c^2*d^2 - 48*I*(b*x + a)^3*d^4 + (-72*I*a^2 - 72*I)*b*c*d^3 + (24*I*a^3 + 72*I*a)*d^4 + (-96*I*b*c*d
^3 + 96*I*a*d^4)*(b*x + a)^2 + (-72*I*b^2*c^2*d^2 + 144*I*a*b*c*d^3 + (-72*I*a^2 - 72*I)*d^4)*(b*x + a))*sin(2
*b*x + 2*a))*dilog(-e^(2*I*b*x + 2*I*a)) + (24*b^3*c^3*d - 72*a*b^2*c^2*d^2 + 72*a^2*b*c*d^3 + 24*(b*x + a)^3*
d^4 - 24*a^3*d^4 + 72*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 72*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + 24*
(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3
*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 48*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b
*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)
*(b*x + a))*cos(2*b*x + 2*a) + (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 + 24*I*(b*x + a)^3*d^4
- 24*I*a^3*d^4 + (72*I*b*c*d^3 - 72*I*a*d^4)*(b*x + a)^2 + (72*I*b^2*c^2*d^2 - 144*I*a*b*c*d^3 + 72*I*a^2*d^4)
*(b*x + a))*sin(4*b*x + 4*a) + (48*I*b^3*c^3*d - 144*I*a*b^2*c^2*d^2 + 144*I*a^2*b*c*d^3 + 48*I*(b*x + a)^3*d^
4 - 48*I*a^3*d^4 + (144*I*b*c*d^3 - 144*I*a*d^4)*(b*x + a)^2 + (144*I*b^2*c^2*d^2 - 288*I*a*b*c*d^3 + 144*I*a^
2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*a)) + (24*b^3*c^3*d - 72*a*b^2*c^2*d^2 + 72*a^2*b*c*d^
3 + 24*(b*x + a)^3*d^4 - 24*a^3*d^4 + 72*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 72*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d
^4)*(b*x + a) + 24*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d
^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 48*(b^3*c^3*d - 3*a*b^
2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a
*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 + 24
*I*(b*x + a)^3*d^4 - 24*I*a^3*d^4 + (72*I*b*c*d^3 - 72*I*a*d^4)*(b*x + a)^2 + (72*I*b^2*c^2*d^2 - 144*I*a*b*c*
d^3 + 72*I*a^2*d^4)*(b*x + a))*sin(4*b*x + 4*a) + (48*I*b^3*c^3*d - 144*I*a*b^2*c^2*d^2 + 144*I*a^2*b*c*d^3 +
48*I*(b*x + a)^3*d^4 - 48*I*a^3*d^4 + (144*I*b*c*d^3 - 144*I*a*d^4)*(b*x + a)^2 + (144*I*b^2*c^2*d^2 - 288*I*a
*b*c*d^3 + 144*I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) + (-6*I*(b*x + a)^4*d^4 - 18*I*b
^2*c^2*d^2 + 36*I*a*b*c*d^3 - 18*I*a^2*d^4 + (-16*I*b*c*d^3 + 16*I*a*d^4)*(b*x + a)^3 + (-18*I*b^2*c^2*d^2 + 3
6*I*a*b*c*d^3 + (-18*I*a^2 - 18*I)*d^4)*(b*x + a)^2 + (-12*I*b^3*c^3*d + 36*I*a*b^2*c^2*d^2 + (-36*I*a^2 - 36*
I)*b*c*d^3 + (12*I*a^3 + 36*I*a)*d^4)*(b*x + a) + (-6*I*(b*x + a)^4*d^4 - 18*I*b^2*c^2*d^2 + 36*I*a*b*c*d^3 -
18*I*a^2*d^4 + (-16*I*b*c*d^3 + 16*I*a*d^4)*(b*x + a)^3 + (-18*I*b^2*c^2*d^2 + 36*I*a*b*c*d^3 + (-18*I*a^2 - 1
8*I)*d^4)*(b*x + a)^2 + (-12*I*b^3*c^3*d + 36*I*a*b^2*c^2*d^2 + (-36*I*a^2 - 36*I)*b*c*d^3 + (12*I*a^3 + 36*I*
a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (-12*I*(b*x + a)^4*d^4 - 36*I*b^2*c^2*d^2 + 72*I*a*b*c*d^3 - 36*I*a^2*d^
4 + (-32*I*b*c*d^3 + 32*I*a*d^4)*(b*x + a)^3 + (-36*I*b^2*c^2*d^2 + 72*I*a*b*c*d^3 + (-36*I*a^2 - 36*I)*d^4)*(
b*x + a)^2 + (-24*I*b^3*c^3*d + 72*I*a*b^2*c^2*d^2 + (-72*I*a^2 - 72*I)*b*c*d^3 + (24*I*a^3 + 72*I*a)*d^4)*(b*
x + a))*cos(2*b*x + 2*a) + 2*(3*(b*x + a)^4*d^4 + 9*b^2*c^2*d^2 - 18*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a*d^
4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 +
3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 4*(3*(b*x + a)^4*d^4 + 9*b^2*c^2*d^2 - 18
*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x
+ a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*
log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + (3*I*(b*x + a)^4*d^4 + (12*I*b*c*d^3 -
12*I*a*d^4)*(b*x + a)^3 + (18*I*b^2*c^2*d^2 - 36*I*a*b*c*d^3 + 18*I*a^2*d^4)*(b*x + a)^2 + (12*I*b^3*c^3*d -
36*I*a*b^2*c^2*d^2 + 36*I*a^2*b*c*d^3 - 12*I*a^3*d^4)*(b*x + a) + (3*I*(b*x + a)^4*d^4 + (12*I*b*c*d^3 - 12*I*
a*d^4)*(b*x + a)^3 + (18*I*b^2*c^2*d^2 - 36*I*a*b*c*d^3 + 18*I*a^2*d^4)*(b*x + a)^2 + (12*I*b^3*c^3*d - 36*I*a
*b^2*c^2*d^2 + 36*I*a^2*b*c*d^3 - 12*I*a^3*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (6*I*(b*x + a)^4*d^4 + (24*I*b*c
*d^3 - 24*I*a*d^4)*(b*x + a)^3 + (36*I*b^2*c^2*d^2 - 72*I*a*b*c*d^3 + 36*I*a^2*d^4)*(b*x + a)^2 + (24*I*b^3*c^
3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 - 24*I*a^3*d^4)*(b*x + a))*cos(2*b*x + 2*a) - 3*((b*x + a)^4*d^4 +
4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^
2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 6*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b
*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d
^3 - a^3*d^4)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (3*I*(b
*x + a)^4*d^4 + (12*I*b*c*d^3 - 12*I*a*d^4)*(b*x + a)^3 + (18*I*b^2*c^2*d^2 - 36*I*a*b*c*d^3 + 18*I*a^2*d^4)*(
b*x + a)^2 + (12*I*b^3*c^3*d - 36*I*a*b^2*c^2*d^2 + 36*I*a^2*b*c*d^3 - 12*I*a^3*d^4)*(b*x + a) + (3*I*(b*x + a
)^4*d^4 + (12*I*b*c*d^3 - 12*I*a*d^4)*(b*x + a)^3 + (18*I*b^2*c^2*d^2 - 36*I*a*b*c*d^3 + 18*I*a^2*d^4)*(b*x +
a)^2 + (12*I*b^3*c^3*d - 36*I*a*b^2*c^2*d^2 + 36*I*a^2*b*c*d^3 - 12*I*a^3*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (
6*I*(b*x + a)^4*d^4 + (24*I*b*c*d^3 - 24*I*a*d^4)*(b*x + a)^3 + (36*I*b^2*c^2*d^2 - 72*I*a*b*c*d^3 + 36*I*a^2*
d^4)*(b*x + a)^2 + (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 - 24*I*a^3*d^4)*(b*x + a))*cos(2*b*
x + 2*a) - 3*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x
+ a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 6*((b*x + a)
^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d
- 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2
- 2*cos(b*x + a) + 1) + (18*I*d^4*cos(4*b*x + 4*a) + 36*I*d^4*cos(2*b*x + 2*a) - 18*d^4*sin(4*b*x + 4*a) - 36*
d^4*sin(2*b*x + 2*a) + 18*I*d^4)*polylog(5, -e^(2*I*b*x + 2*I*a)) + (-144*I*d^4*cos(4*b*x + 4*a) - 288*I*d^4*c
os(2*b*x + 2*a) + 144*d^4*sin(4*b*x + 4*a) + 288*d^4*sin(2*b*x + 2*a) - 144*I*d^4)*polylog(5, -e^(I*b*x + I*a)
) + (-144*I*d^4*cos(4*b*x + 4*a) - 288*I*d^4*cos(2*b*x + 2*a) + 144*d^4*sin(4*b*x + 4*a) + 288*d^4*sin(2*b*x +
2*a) - 144*I*d^4)*polylog(5, e^(I*b*x + I*a)) + (24*b*c*d^3 + 36*(b*x + a)*d^4 - 24*a*d^4 + 12*(2*b*c*d^3 + 3
*(b*x + a)*d^4 - 2*a*d^4)*cos(4*b*x + 4*a) + 24*(2*b*c*d^3 + 3*(b*x + a)*d^4 - 2*a*d^4)*cos(2*b*x + 2*a) + (24
*I*b*c*d^3 + 36*I*(b*x + a)*d^4 - 24*I*a*d^4)*sin(4*b*x + 4*a) + (48*I*b*c*d^3 + 72*I*(b*x + a)*d^4 - 48*I*a*d
^4)*sin(2*b*x + 2*a))*polylog(4, -e^(2*I*b*x + 2*I*a)) - (144*b*c*d^3 + 144*(b*x + a)*d^4 - 144*a*d^4 + 144*(b
*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) + 288*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*a) - (-
144*I*b*c*d^3 - 144*I*(b*x + a)*d^4 + 144*I*a*d^4)*sin(4*b*x + 4*a) - (-288*I*b*c*d^3 - 288*I*(b*x + a)*d^4 +
288*I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, -e^(I*b*x + I*a)) - (144*b*c*d^3 + 144*(b*x + a)*d^4 - 144*a*d^4 + 1
44*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) + 288*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*a)
- (-144*I*b*c*d^3 - 144*I*(b*x + a)*d^4 + 144*I*a*d^4)*sin(4*b*x + 4*a) - (-288*I*b*c*d^3 - 288*I*(b*x + a)*d
^4 + 288*I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, e^(I*b*x + I*a)) + (-18*I*b^2*c^2*d^2 + 36*I*a*b*c*d^3 - 36*I*(
b*x + a)^2*d^4 + (-18*I*a^2 - 18*I)*d^4 + (-48*I*b*c*d^3 + 48*I*a*d^4)*(b*x + a) + (-18*I*b^2*c^2*d^2 + 36*I*a
*b*c*d^3 - 36*I*(b*x + a)^2*d^4 + (-18*I*a^2 - 18*I)*d^4 + (-48*I*b*c*d^3 + 48*I*a*d^4)*(b*x + a))*cos(4*b*x +
4*a) + (-36*I*b^2*c^2*d^2 + 72*I*a*b*c*d^3 - 72*I*(b*x + a)^2*d^4 + (-36*I*a^2 - 36*I)*d^4 + (-96*I*b*c*d^3 +
96*I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) + 6*(3*b^2*c^2*d^2 - 6*a*b*c*d^3 + 6*(b*x + a)^2*d^4 + 3*(a^2 + 1)*d^
4 + 8*(b*c*d^3 - a*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 12*(3*b^2*c^2*d^2 - 6*a*b*c*d^3 + 6*(b*x + a)^2*d^4 + 3*
(a^2 + 1)*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*polylog(3, -e^(2*I*b*x + 2*I*a)) + (72*I*b^2*
c^2*d^2 - 144*I*a*b*c*d^3 + 72*I*(b*x + a)^2*d^4 + 72*I*a^2*d^4 + (144*I*b*c*d^3 - 144*I*a*d^4)*(b*x + a) + (7
2*I*b^2*c^2*d^2 - 144*I*a*b*c*d^3 + 72*I*(b*x + a)^2*d^4 + 72*I*a^2*d^4 + (144*I*b*c*d^3 - 144*I*a*d^4)*(b*x +
a))*cos(4*b*x + 4*a) + (144*I*b^2*c^2*d^2 - 288*I*a*b*c*d^3 + 144*I*(b*x + a)^2*d^4 + 144*I*a^2*d^4 + (288*I*
b*c*d^3 - 288*I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - 72*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4
+ 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 144*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^
4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*polylog(3, -e^(I*b*x + I*a)) + (72*I*b^2*c^2*d^2 - 144*I*
a*b*c*d^3 + 72*I*(b*x + a)^2*d^4 + 72*I*a^2*d^4 + (144*I*b*c*d^3 - 144*I*a*d^4)*(b*x + a) + (72*I*b^2*c^2*d^2
- 144*I*a*b*c*d^3 + 72*I*(b*x + a)^2*d^4 + 72*I*a^2*d^4 + (144*I*b*c*d^3 - 144*I*a*d^4)*(b*x + a))*cos(4*b*x +
4*a) + (144*I*b^2*c^2*d^2 - 288*I*a*b*c*d^3 + 144*I*(b*x + a)^2*d^4 + 144*I*a^2*d^4 + (288*I*b*c*d^3 - 288*I*
a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - 72*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 -
a*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 144*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 -
a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) + (-24*I*(b*x + a)^3*d^4 + (-72*I*b*c*d^3 + 7
2*I*a*d^4)*(b*x + a)^2 + (-72*I*b^2*c^2*d^2 + 144*I*a*b*c*d^3 - 72*I*a^2*d^4)*(b*x + a))*sin(4*b*x + 4*a) - (1
2*(b*x + a)^4*d^4 - 24*I*b^3*c^3*d + 72*I*a*b^2*c^2*d^2 - 72*I*a^2*b*c*d^3 + 24*I*a^3*d^4 + (48*b*c*d^3 - (48*
a - 24*I)*d^4)*(b*x + a)^3 + (72*b^2*c^2*d^2 - (144*a - 72*I)*b*c*d^3 + 72*(a^2 - I*a)*d^4)*(b*x + a)^2 + (48*
b^3*c^3*d - (144*a - 72*I)*b^2*c^2*d^2 + 144*(a^2 - I*a)*b*c*d^3 - 24*(2*a^3 - 3*I*a^2)*d^4)*(b*x + a))*sin(2*
b*x + 2*a))/(-6*I*b^4*cos(4*b*x + 4*a) - 12*I*b^4*cos(2*b*x + 2*a) + 6*b^4*sin(4*b*x + 4*a) + 12*b^4*sin(2*b*x
+ 2*a) - 6*I*b^4))/b
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^4/(cos(a + b*x)^3*sin(a + b*x)),x)
[Out]
\text{Hanged}
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**4*csc(b*x+a)*sec(b*x+a)**3,x)
[Out]
Timed out
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