3.241 \(\int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx\)
Optimal. Leaf size=325 \[ -\frac {3 i d^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d^3 \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 i d^3 \text {Li}_4\left (e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b} \]
[Out]
-3/2*I*d*(d*x+c)^2/b^2-1/2*(d*x+c)^3/b-2*(d*x+c)^3*arctanh(exp(2*I*(b*x+a)))/b-3/2*d*(d*x+c)^2*cot(b*x+a)/b^2-
1/2*(d*x+c)^3*cot(b*x+a)^2/b+3*d^2*(d*x+c)*ln(1-exp(2*I*(b*x+a)))/b^3+3/2*I*d*(d*x+c)^2*polylog(2,-exp(2*I*(b*
x+a)))/b^2-3/2*I*d^3*polylog(2,exp(2*I*(b*x+a)))/b^4-3/2*I*d*(d*x+c)^2*polylog(2,exp(2*I*(b*x+a)))/b^2-3/2*d^2
*(d*x+c)*polylog(3,-exp(2*I*(b*x+a)))/b^3+3/2*d^2*(d*x+c)*polylog(3,exp(2*I*(b*x+a)))/b^3-3/4*I*d^3*polylog(4,
-exp(2*I*(b*x+a)))/b^4+3/4*I*d^3*polylog(4,exp(2*I*(b*x+a)))/b^4
________________________________________________________________________________________
Rubi [A] time = 0.82, antiderivative size = 325, normalized size of antiderivative = 1.00,
number of steps used = 22, number of rules used = 18, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used
= {2620, 14, 4420, 6741, 12, 6742, 3720, 3717, 2190, 2279, 2391, 32, 2551, 4183, 2531, 6609, 2282,
6589} \[ -\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d^3 \text {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 i d^3 \text {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x],x]
[Out]
(((-3*I)/2)*d*(c + d*x)^2)/b^2 - (c + d*x)^3/(2*b) - (2*(c + d*x)^3*ArcTanh[E^((2*I)*(a + b*x))])/b - (3*d*(c
+ d*x)^2*Cot[a + b*x])/(2*b^2) - ((c + d*x)^3*Cot[a + b*x]^2)/(2*b) + (3*d^2*(c + d*x)*Log[1 - E^((2*I)*(a + b
*x))])/b^3 + (((3*I)/2)*d*(c + d*x)^2*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 - (((3*I)/2)*d^3*PolyLog[2, E^((2*
I)*(a + b*x))])/b^4 - (((3*I)/2)*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(a + b*x))])/b^2 - (3*d^2*(c + d*x)*PolyLog
[3, -E^((2*I)*(a + b*x))])/(2*b^3) + (3*d^2*(c + d*x)*PolyLog[3, E^((2*I)*(a + b*x))])/(2*b^3) - (((3*I)/4)*d^
3*PolyLog[4, -E^((2*I)*(a + b*x))])/b^4 + (((3*I)/4)*d^3*PolyLog[4, E^((2*I)*(a + b*x))])/b^4
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 2279
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2391
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
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 3717
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && 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)^3 \csc ^3(a+b x) \sec (a+b x) \, dx &=-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}-(3 d) \int (c+d x)^2 \left (-\frac {\cot ^2(a+b x)}{2 b}+\frac {\log (\tan (a+b x))}{b}\right ) \, dx\\ &=-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}-(3 d) \int \frac {(c+d x)^2 \left (-\cot ^2(a+b x)+2 \log (\tan (a+b x))\right )}{2 b} \, dx\\ &=-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}-\frac {(3 d) \int (c+d x)^2 \left (-\cot ^2(a+b x)+2 \log (\tan (a+b x))\right ) \, dx}{2 b}\\ &=-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}-\frac {(3 d) \int \left (-(c+d x)^2 \cot ^2(a+b x)+2 (c+d x)^2 \log (\tan (a+b x))\right ) \, dx}{2 b}\\ &=-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}+\frac {(3 d) \int (c+d x)^2 \cot ^2(a+b x) \, dx}{2 b}-\frac {(3 d) \int (c+d x)^2 \log (\tan (a+b x)) \, dx}{b}\\ &=-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {\int 2 b (c+d x)^3 \csc (2 a+2 b x) \, dx}{b}-\frac {(3 d) \int (c+d x)^2 \, dx}{2 b}+\frac {\left (3 d^2\right ) \int (c+d x) \cot (a+b x) \, dx}{b^2}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+2 \int (c+d x)^3 \csc (2 a+2 b x) \, dx-\frac {\left (6 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)}{1-e^{2 i (a+b x)}} \, dx}{b^2}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {(3 d) \int (c+d x)^2 \log \left (1-e^{i (2 a+2 b x)}\right ) \, dx}{b}+\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{i (2 a+2 b x)}\right ) \, dx}{b}-\frac {\left (3 d^3\right ) \int \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}-\frac {\left (3 i d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{i (2 a+2 b x)}\right ) \, dx}{b^2}+\frac {\left (3 i d^2\right ) \int (c+d x) \text {Li}_2\left (e^{i (2 a+2 b x)}\right ) \, dx}{b^2}+\frac {\left (3 i d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}+\frac {\left (3 d^3\right ) \int \text {Li}_3\left (-e^{i (2 a+2 b x)}\right ) \, dx}{2 b^3}-\frac {\left (3 d^3\right ) \int \text {Li}_3\left (e^{i (2 a+2 b x)}\right ) \, dx}{2 b^3}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}-\frac {\left (3 i d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{4 b^4}+\frac {\left (3 i d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{4 b^4}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d^3 \text {Li}_4\left (-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 i d^3 \text {Li}_4\left (e^{2 i (a+b x)}\right )}{4 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 6.97, size = 1477, normalized size = 4.54 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x],x]
[Out]
-1/2*((c + d*x)^3*Csc[a + b*x]^2)/b - (c*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)))/(2*b^3)
- (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*Log[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*PolyLog[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*PolyLog[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)))/(4*b^4) + (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Csc[
a]*Sec[a])/4 - ((I/4)*c*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))*PolyLog[3, -E^((-2*I)*(a + b*x)
)])*Sec[a])/(b^3*E^(I*a)) - (I/8)*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] - (c^3*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)) + (c^3*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)) + (3*c*d^2*Csc[a]*(-(b*x*Cos[a])
+ Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) - (3*c^2*d*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])/(2*b^2*Sqrt[Csc[a]^2*(Co
s[a]^2 + Sin[a]^2)]) + (3*Csc[a]*Csc[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2
) - (3*c^2*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]))/(2*b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*d^3*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]))/(2*b^4*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])
________________________________________________________________________________________
fricas [C] time = 0.77, size = 3459, normalized size = 10.64 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a),x, algorithm="fricas")
[Out]
1/2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3 + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cos
(b*x + a)*sin(b*x + a) + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 3*I*d^3 + (-3*I*b^2*d^3*x^2 - 6*
I*b^2*c*d^2*x - 3*I*b^2*c^2*d - 3*I*d^3)*cos(b*x + a)^2)*dilog(cos(b*x + a) + I*sin(b*x + a)) + (-3*I*b^2*d^3*
x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d - 3*I*d^3 + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 3*I*d^3
)*cos(b*x + a)^2)*dilog(cos(b*x + a) - I*sin(b*x + a)) + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d +
(-3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d)*cos(b*x + a)^2)*dilog(I*cos(b*x + a) + sin(b*x + a)) + (-
3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d)*cos(b*
x + a)^2)*dilog(I*cos(b*x + a) - sin(b*x + a)) + (-3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d + (3*I*b^
2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d)*cos(b*x + a)^2)*dilog(-I*cos(b*x + a) + sin(b*x + a)) + (3*I*b^2*
d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + (-3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d)*cos(b*x + a)^
2)*dilog(-I*cos(b*x + a) - sin(b*x + a)) + (-3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d - 3*I*d^3 + (3*
I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 3*I*d^3)*cos(b*x + a)^2)*dilog(-cos(b*x + a) + I*sin(b*x + a
)) + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 3*I*d^3 + (-3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*
b^2*c^2*d - 3*I*d^3)*cos(b*x + a)^2)*dilog(-cos(b*x + a) - I*sin(b*x + a)) - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 +
b^3*c^3 + 3*b*c*d^2 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 3*b*c*d^2 + 3*(b^3*c^2*d + b*d^3)*x)*cos(b*x
+ a)^2 + 3*(b^3*c^2*d + b*d^3)*x)*log(cos(b*x + a) + I*sin(b*x + a) + 1) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*
c*d^2 - a^3*d^3 - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cos(b*x + a)^2)*log(cos(b*x + a) + I*sin
(b*x + a) + I) - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 3*b*c*d^2 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c
^3 + 3*b*c*d^2 + 3*(b^3*c^2*d + b*d^3)*x)*cos(b*x + a)^2 + 3*(b^3*c^2*d + b*d^3)*x)*log(cos(b*x + a) - I*sin(b
*x + a) + 1) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 -
a^3*d^3)*cos(b*x + a)^2)*log(cos(b*x + a) - I*sin(b*x + a) + I) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*
d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d
- 3*a^2*b*c*d^2 + a^3*d^3)*cos(b*x + a)^2)*log(I*cos(b*x + a) + sin(b*x + a) + 1) + (b^3*d^3*x^3 + 3*b^3*c*d^
2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d
*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*cos(b*x + a)^2)*log(I*cos(b*x + a) - sin(b*x + a) + 1) + (b^3*d^
3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2
*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*cos(b*x + a)^2)*log(-I*cos(b*x + a) + sin(b*x
+ a) + 1) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - (b^3*d^
3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*cos(b*x + a)^2)*log(-I*cos(
b*x + a) - sin(b*x + a) + 1) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*(a^2 + 1)*b*c*d^2 - (a^3 + 3*a)*d^3 - (b^3*c^3 - 3
*a*b^2*c^2*d + 3*(a^2 + 1)*b*c*d^2 - (a^3 + 3*a)*d^3)*cos(b*x + a)^2)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x +
a) + 1/2) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*(a^2 + 1)*b*c*d^2 - (a^3 + 3*a)*d^3 - (b^3*c^3 - 3*a*b^2*c^2*d + 3*(a
^2 + 1)*b*c*d^2 - (a^3 + 3*a)*d^3)*cos(b*x + a)^2)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) - (b^3*d^
3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^3 + 3*a)*d^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3
*a*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^3 + 3*a)*d^3 + 3*(b^3*c^2*d + b*d^3)*x)*cos(b*x + a)^2 + 3*(b^3*c^2*d + b*d^
3)*x)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - (b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cos(b*x + a)^2)*log(-cos(b*x + a) + I*sin(b*x + a) + I) - (b^3*d^3*
x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^3 + 3*a)*d^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a
*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^3 + 3*a)*d^3 + 3*(b^3*c^2*d + b*d^3)*x)*cos(b*x + a)^2 + 3*(b^3*c^2*d + b*d^3)
*x)*log(-cos(b*x + a) - I*sin(b*x + a) + 1) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - (b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cos(b*x + a)^2)*log(-cos(b*x + a) - I*sin(b*x + a) + I) + (6*I*d^3*co
s(b*x + a)^2 - 6*I*d^3)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) + (-6*I*d^3*cos(b*x + a)^2 + 6*I*d^3)*polylo
g(4, cos(b*x + a) - I*sin(b*x + a)) + (6*I*d^3*cos(b*x + a)^2 - 6*I*d^3)*polylog(4, I*cos(b*x + a) + sin(b*x +
a)) + (-6*I*d^3*cos(b*x + a)^2 + 6*I*d^3)*polylog(4, I*cos(b*x + a) - sin(b*x + a)) + (-6*I*d^3*cos(b*x + a)^
2 + 6*I*d^3)*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) + (6*I*d^3*cos(b*x + a)^2 - 6*I*d^3)*polylog(4, -I*cos
(b*x + a) - sin(b*x + a)) + (-6*I*d^3*cos(b*x + a)^2 + 6*I*d^3)*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) + (
6*I*d^3*cos(b*x + a)^2 - 6*I*d^3)*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) - 6*(b*d^3*x + b*c*d^2 - (b*d^3*x
+ b*c*d^2)*cos(b*x + a)^2)*polylog(3, cos(b*x + a) + I*sin(b*x + a)) - 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*
d^2)*cos(b*x + a)^2)*polylog(3, cos(b*x + a) - I*sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*co
s(b*x + a)^2)*polylog(3, I*cos(b*x + a) + sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cos(b*x +
a)^2)*polylog(3, I*cos(b*x + a) - sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cos(b*x + a)^2)*
polylog(3, -I*cos(b*x + a) + sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cos(b*x + a)^2)*polylo
g(3, -I*cos(b*x + a) - sin(b*x + a)) - 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cos(b*x + a)^2)*polylog(3, -
cos(b*x + a) + I*sin(b*x + a)) - 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cos(b*x + a)^2)*polylog(3, -cos(b*
x + a) - I*sin(b*x + a)))/(b^4*cos(b*x + a)^2 - b^4)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a),x, algorithm="giac")
[Out]
integrate((d*x + c)^3*csc(b*x + a)^3*sec(b*x + a), x)
________________________________________________________________________________________
maple [B] time = 0.19, size = 1223, normalized size = 3.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a),x)
[Out]
-6*I/b^2*c*d^2*polylog(2,exp(I*(b*x+a)))*x-6*I/b^2*c*d^2*polylog(2,-exp(I*(b*x+a)))*x+6*I*d^3*polylog(4,exp(I*
(b*x+a)))/b^4-1/b*c^3*ln(1+exp(2*I*(b*x+a)))-1/b^4*d^3*a^3*ln(exp(I*(b*x+a))-1)+6/b^3*c*d^2*polylog(3,-exp(I*(
b*x+a)))+6/b^3*c*d^2*polylog(3,exp(I*(b*x+a)))+6/b^3*d^3*polylog(3,exp(I*(b*x+a)))*x+6/b^3*d^3*polylog(3,-exp(
I*(b*x+a)))*x+6*I/b^4*d^3*polylog(4,-exp(I*(b*x+a)))-3*I*d^3*polylog(2,exp(I*(b*x+a)))/b^4-3/2/b^3*c*d^2*polyl
og(3,-exp(2*I*(b*x+a)))-3/2/b^3*d^3*polylog(3,-exp(2*I*(b*x+a)))*x-3/4*I*d^3*polylog(4,-exp(2*I*(b*x+a)))/b^4+
1/b*c^3*ln(exp(I*(b*x+a))-1)+1/b*c^3*ln(exp(I*(b*x+a))+1)+3*I/b^2*c*d^2*polylog(2,-exp(2*I*(b*x+a)))*x+(2*b*d^
3*x^3*exp(2*I*(b*x+a))-3*I*d^3*x^2*exp(2*I*(b*x+a))+6*b*c*d^2*x^2*exp(2*I*(b*x+a))-6*I*c*d^2*x*exp(2*I*(b*x+a)
)+6*b*c^2*d*x*exp(2*I*(b*x+a))-3*I*c^2*d*exp(2*I*(b*x+a))+3*I*d^3*x^2+2*b*c^3*exp(2*I*(b*x+a))+6*I*c*d^2*x+3*I
*c^2*d)/b^2/(exp(2*I*(b*x+a))-1)^2+3/b^3*d^2*c*ln(exp(I*(b*x+a))-1)+3/b^3*d^2*c*ln(exp(I*(b*x+a))+1)-6/b^3*d^2
*c*ln(exp(I*(b*x+a)))+3/b^3*d^3*ln(exp(I*(b*x+a))+1)*x+3/b^3*d^3*ln(1-exp(I*(b*x+a)))*x+3/b^4*d^3*ln(1-exp(I*(
b*x+a)))*a-3/b^4*d^3*a*ln(exp(I*(b*x+a))-1)+6/b^4*d^3*a*ln(exp(I*(b*x+a)))-3*I/b^2*d^3*x^2-3*I/b^4*d^3*a^2-3*I
/b^4*d^3*polylog(2,-exp(I*(b*x+a)))+3/b^3*c*d^2*a^2*ln(exp(I*(b*x+a))-1)-3*I/b^2*c^2*d*polylog(2,exp(I*(b*x+a)
))-3*I/b^2*c^2*d*polylog(2,-exp(I*(b*x+a)))-3*I/b^2*d^3*polylog(2,exp(I*(b*x+a)))*x^2-3*I/b^2*d^3*polylog(2,-e
xp(I*(b*x+a)))*x^2+3/b*c^2*d*ln(exp(I*(b*x+a))+1)*x+3/b*c^2*d*ln(1-exp(I*(b*x+a)))*x+3/b^2*c^2*d*ln(1-exp(I*(b
*x+a)))*a-3/b^3*c*d^2*a^2*ln(1-exp(I*(b*x+a)))+3/b*c*d^2*ln(1-exp(I*(b*x+a)))*x^2+3/b*c*d^2*ln(exp(I*(b*x+a))+
1)*x^2-3/b^2*c^2*d*a*ln(exp(I*(b*x+a))-1)+1/b*d^3*ln(1-exp(I*(b*x+a)))*x^3+1/b^4*d^3*ln(1-exp(I*(b*x+a)))*a^3+
1/b*d^3*ln(exp(I*(b*x+a))+1)*x^3-1/b*d^3*ln(1+exp(2*I*(b*x+a)))*x^3+3/2*I/b^2*d^3*polylog(2,-exp(2*I*(b*x+a)))
*x^2-6*I/b^3*d^3*a*x+3/2*I/b^2*c^2*d*polylog(2,-exp(2*I*(b*x+a)))-3/b*c^2*d*ln(1+exp(2*I*(b*x+a)))*x-3/b*c*d^2
*ln(1+exp(2*I*(b*x+a)))*x^2
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maxima [B] time = 2.14, size = 5140, normalized size = 15.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a),x, algorithm="maxima")
[Out]
-1/2*(c^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2)) - 3*a*c^2*d*(1/sin(b*x + a)^2 + l
og(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b + 3*a^2*c*d^2*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) - lo
g(sin(b*x + a)^2))/b^2 - a^3*d^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b^3 - 2*(1
8*b^2*c^2*d - 36*a*b*c*d^2 + 18*a^2*d^3 - (8*(b*x + a)^3*d^3 + 18*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 18*(b^2*c^2*
d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a) + 2*(4*(b*x + a)^3*d^3 + 9*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(b^2*c^2*d -
2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 4*(4*(b*x + a)^3*d^3 + 9*(b*c*d^2 - a*d^3)*(b*x + a)^2 +
9*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (8*I*(b*x + a)^3*d^3 + (18*I*b*c*d^2 - 18
*I*a*d^3)*(b*x + a)^2 + (18*I*b^2*c^2*d - 36*I*a*b*c*d^2 + 18*I*a^2*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-16*I*
(b*x + a)^3*d^3 + (-36*I*b*c*d^2 + 36*I*a*d^3)*(b*x + a)^2 + (-36*I*b^2*c^2*d + 72*I*a*b*c*d^2 - 36*I*a^2*d^3)
*(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)^3*d^3 + 18*b*c*d^
2 - 18*a*d^3 + 18*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 18*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(b*x + a) + 6*(
(b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 +
1)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 12*((b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a
)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-6*I*(b*x + a)^3*d^3 - 18*I*b
*c*d^2 + 18*I*a*d^3 + (-18*I*b*c*d^2 + 18*I*a*d^3)*(b*x + a)^2 + (-18*I*b^2*c^2*d + 36*I*a*b*c*d^2 + (-18*I*a^
2 - 18*I)*d^3)*(b*x + a))*sin(4*b*x + 4*a) - (12*I*(b*x + a)^3*d^3 + 36*I*b*c*d^2 - 36*I*a*d^3 + (36*I*b*c*d^2
- 36*I*a*d^3)*(b*x + a)^2 + (36*I*b^2*c^2*d - 72*I*a*b*c*d^2 + (36*I*a^2 + 36*I)*d^3)*(b*x + a))*sin(2*b*x +
2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + (18*b*c*d^2 - 18*a*d^3 + 18*(b*c*d^2 - a*d^3)*cos(4*b*x + 4*a)
- 36*(b*c*d^2 - a*d^3)*cos(2*b*x + 2*a) - (-18*I*b*c*d^2 + 18*I*a*d^3)*sin(4*b*x + 4*a) - (36*I*b*c*d^2 - 36*
I*a*d^3)*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) - 1) - (6*(b*x + a)^3*d^3 + 18*(b*c*d^2 - a*d^3)
*(b*x + a)^2 + 18*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(b*x + a) + 6*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^
3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 12*((b*x + a)^3*d^3
+ 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(b*x + a))*cos(2*b*x + 2*a) +
(6*I*(b*x + a)^3*d^3 + (18*I*b*c*d^2 - 18*I*a*d^3)*(b*x + a)^2 + (18*I*b^2*c^2*d - 36*I*a*b*c*d^2 + (18*I*a^2
+ 18*I)*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-12*I*(b*x + a)^3*d^3 + (-36*I*b*c*d^2 + 36*I*a*d^3)*(b*x + a)^2
+ (-36*I*b^2*c^2*d + 72*I*a*b*c*d^2 + (-36*I*a^2 - 36*I)*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a
), -cos(b*x + a) + 1) - 18*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(4*b*x + 4*a) - (12*I*(b*x + a
)^3*d^3 + 18*b^2*c^2*d - 36*a*b*c*d^2 + 18*a^2*d^3 + (36*I*b*c*d^2 - 18*(2*I*a + 1)*d^3)*(b*x + a)^2 + (36*I*b
^2*c^2*d - 36*(2*I*a + 1)*b*c*d^2 + (36*I*a^2 + 36*a)*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (9*b^2*c^2*d - 18*a*b
*c*d^2 + 12*(b*x + a)^2*d^3 + 9*a^2*d^3 + 18*(b*c*d^2 - a*d^3)*(b*x + a) + 3*(3*b^2*c^2*d - 6*a*b*c*d^2 + 4*(b
*x + a)^2*d^3 + 3*a^2*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 6*(3*b^2*c^2*d - 6*a*b*c*d^2 + 4
*(b*x + a)^2*d^3 + 3*a^2*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-9*I*b^2*c^2*d + 18*I*a*b*c*
d^2 - 12*I*(b*x + a)^2*d^3 - 9*I*a^2*d^3 + (-18*I*b*c*d^2 + 18*I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) - (18*I*b^
2*c^2*d - 36*I*a*b*c*d^2 + 24*I*(b*x + a)^2*d^3 + 18*I*a^2*d^3 + (36*I*b*c*d^2 - 36*I*a*d^3)*(b*x + a))*sin(2*
b*x + 2*a))*dilog(-e^(2*I*b*x + 2*I*a)) - (18*b^2*c^2*d - 36*a*b*c*d^2 + 18*(b*x + a)^2*d^3 + 18*(a^2 + 1)*d^3
+ 36*(b*c*d^2 - a*d^3)*(b*x + a) + 18*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 1)*d^3 + 2*(b*c*d^2
- a*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 36*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 1)*d^3 + 2*(b*c
*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (18*I*b^2*c^2*d - 36*I*a*b*c*d^2 + 18*I*(b*x + a)^2*d^3 + (18*I*a^
2 + 18*I)*d^3 + (36*I*b*c*d^2 - 36*I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-36*I*b^2*c^2*d + 72*I*a*b*c*d^2 -
36*I*(b*x + a)^2*d^3 + (-36*I*a^2 - 36*I)*d^3 + (-72*I*b*c*d^2 + 72*I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*dilo
g(-e^(I*b*x + I*a)) - (18*b^2*c^2*d - 36*a*b*c*d^2 + 18*(b*x + a)^2*d^3 + 18*(a^2 + 1)*d^3 + 36*(b*c*d^2 - a*d
^3)*(b*x + a) + 18*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 1)*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))
*cos(4*b*x + 4*a) - 36*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 1)*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x +
a))*cos(2*b*x + 2*a) + (18*I*b^2*c^2*d - 36*I*a*b*c*d^2 + 18*I*(b*x + a)^2*d^3 + (18*I*a^2 + 18*I)*d^3 + (36*
I*b*c*d^2 - 36*I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-36*I*b^2*c^2*d + 72*I*a*b*c*d^2 - 36*I*(b*x + a)^2*d^3
+ (-36*I*a^2 - 36*I)*d^3 + (-72*I*b*c*d^2 + 72*I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) -
(-4*I*(b*x + a)^3*d^3 + (-9*I*b*c*d^2 + 9*I*a*d^3)*(b*x + a)^2 + (-9*I*b^2*c^2*d + 18*I*a*b*c*d^2 - 9*I*a^2*d
^3)*(b*x + a) + (-4*I*(b*x + a)^3*d^3 + (-9*I*b*c*d^2 + 9*I*a*d^3)*(b*x + a)^2 + (-9*I*b^2*c^2*d + 18*I*a*b*c*
d^2 - 9*I*a^2*d^3)*(b*x + a))*cos(4*b*x + 4*a) + (8*I*(b*x + a)^3*d^3 + (18*I*b*c*d^2 - 18*I*a*d^3)*(b*x + a)^
2 + (18*I*b^2*c^2*d - 36*I*a*b*c*d^2 + 18*I*a^2*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (4*(b*x + a)^3*d^3 + 9*(b*c
*d^2 - a*d^3)*(b*x + a)^2 + 9*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*sin(4*b*x + 4*a) - 2*(4*(b*x + a)
^3*d^3 + 9*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(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)^3*d^3 + 9*I*b*c*d^2 - 9
*I*a*d^3 + (9*I*b*c*d^2 - 9*I*a*d^3)*(b*x + a)^2 + (9*I*b^2*c^2*d - 18*I*a*b*c*d^2 + (9*I*a^2 + 9*I)*d^3)*(b*x
+ a) + (3*I*(b*x + a)^3*d^3 + 9*I*b*c*d^2 - 9*I*a*d^3 + (9*I*b*c*d^2 - 9*I*a*d^3)*(b*x + a)^2 + (9*I*b^2*c^2*
d - 18*I*a*b*c*d^2 + (9*I*a^2 + 9*I)*d^3)*(b*x + a))*cos(4*b*x + 4*a) + (-6*I*(b*x + a)^3*d^3 - 18*I*b*c*d^2 +
18*I*a*d^3 + (-18*I*b*c*d^2 + 18*I*a*d^3)*(b*x + a)^2 + (-18*I*b^2*c^2*d + 36*I*a*b*c*d^2 + (-18*I*a^2 - 18*I
)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - 3*((b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^
2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(b*x + a))*sin(4*b*x + 4*a) + 6*((b*x + a)^3*d^3 + 3*b*c*d^2 -
3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(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)^3*d^3 + 9*I*b*c*d^2 - 9*I*a
*d^3 + (9*I*b*c*d^2 - 9*I*a*d^3)*(b*x + a)^2 + (9*I*b^2*c^2*d - 18*I*a*b*c*d^2 + (9*I*a^2 + 9*I)*d^3)*(b*x + a
) + (3*I*(b*x + a)^3*d^3 + 9*I*b*c*d^2 - 9*I*a*d^3 + (9*I*b*c*d^2 - 9*I*a*d^3)*(b*x + a)^2 + (9*I*b^2*c^2*d -
18*I*a*b*c*d^2 + (9*I*a^2 + 9*I)*d^3)*(b*x + a))*cos(4*b*x + 4*a) + (-6*I*(b*x + a)^3*d^3 - 18*I*b*c*d^2 + 18*
I*a*d^3 + (-18*I*b*c*d^2 + 18*I*a*d^3)*(b*x + a)^2 + (-18*I*b^2*c^2*d + 36*I*a*b*c*d^2 + (-18*I*a^2 - 18*I)*d^
3)*(b*x + a))*cos(2*b*x + 2*a) - 3*((b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 +
3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(b*x + a))*sin(4*b*x + 4*a) + 6*((b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a
*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 1)*d^3)*(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) - (6*d^3*cos(4*b*x + 4*a) - 12*d^3*cos(2*b*x + 2
*a) + 6*I*d^3*sin(4*b*x + 4*a) - 12*I*d^3*sin(2*b*x + 2*a) + 6*d^3)*polylog(4, -e^(2*I*b*x + 2*I*a)) + (36*d^3
*cos(4*b*x + 4*a) - 72*d^3*cos(2*b*x + 2*a) + 36*I*d^3*sin(4*b*x + 4*a) - 72*I*d^3*sin(2*b*x + 2*a) + 36*d^3)*
polylog(4, -e^(I*b*x + I*a)) + (36*d^3*cos(4*b*x + 4*a) - 72*d^3*cos(2*b*x + 2*a) + 36*I*d^3*sin(4*b*x + 4*a)
- 72*I*d^3*sin(2*b*x + 2*a) + 36*d^3)*polylog(4, e^(I*b*x + I*a)) - (-9*I*b*c*d^2 - 12*I*(b*x + a)*d^3 + 9*I*a
*d^3 + (-9*I*b*c*d^2 - 12*I*(b*x + a)*d^3 + 9*I*a*d^3)*cos(4*b*x + 4*a) + (18*I*b*c*d^2 + 24*I*(b*x + a)*d^3 -
18*I*a*d^3)*cos(2*b*x + 2*a) + 3*(3*b*c*d^2 + 4*(b*x + a)*d^3 - 3*a*d^3)*sin(4*b*x + 4*a) - 6*(3*b*c*d^2 + 4*
(b*x + a)*d^3 - 3*a*d^3)*sin(2*b*x + 2*a))*polylog(3, -e^(2*I*b*x + 2*I*a)) - (36*I*b*c*d^2 + 36*I*(b*x + a)*d
^3 - 36*I*a*d^3 + (36*I*b*c*d^2 + 36*I*(b*x + a)*d^3 - 36*I*a*d^3)*cos(4*b*x + 4*a) + (-72*I*b*c*d^2 - 72*I*(b
*x + a)*d^3 + 72*I*a*d^3)*cos(2*b*x + 2*a) - 36*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(4*b*x + 4*a) + 72*(b*c*d
^2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(3, -e^(I*b*x + I*a)) - (36*I*b*c*d^2 + 36*I*(b*x + a)*d^
3 - 36*I*a*d^3 + (36*I*b*c*d^2 + 36*I*(b*x + a)*d^3 - 36*I*a*d^3)*cos(4*b*x + 4*a) + (-72*I*b*c*d^2 - 72*I*(b*
x + a)*d^3 + 72*I*a*d^3)*cos(2*b*x + 2*a) - 36*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(4*b*x + 4*a) + 72*(b*c*d^
2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) - (18*I*(b*x + a)^2*d^3 + (36*I*b*c*d
^2 - 36*I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (12*(b*x + a)^3*d^3 - 18*I*b^2*c^2*d + 36*I*a*b*c*d^2 - 18*I*a^
2*d^3 + (36*b*c*d^2 - (36*a - 18*I)*d^3)*(b*x + a)^2 + (36*b^2*c^2*d - (72*a - 36*I)*b*c*d^2 + 36*(a^2 - I*a)*
d^3)*(b*x + a))*sin(2*b*x + 2*a))/(-6*I*b^3*cos(4*b*x + 4*a) + 12*I*b^3*cos(2*b*x + 2*a) + 6*b^3*sin(4*b*x + 4
*a) - 12*b^3*sin(2*b*x + 2*a) - 6*I*b^3))/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)^3/(cos(a + b*x)*sin(a + b*x)^3),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)**3*csc(b*x+a)**3*sec(b*x+a),x)
[Out]
Timed out
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