3.382 \(\int (c+d x)^4 \sec (a+b x) \sin (3 a+3 b x) \, dx\)

Optimal. Leaf size=299 \[ \frac{3 d^2 (c+d x)^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 i d^3 (c+d x) \text{PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}-\frac{2 i d (c+d x)^3 \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{3 d^4 \text{PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}-\frac{6 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}+\frac{6 c d^3 x}{b^3}+\frac{3 d^4 x^2}{b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d} \]

[Out]

(6*c*d^3*x)/b^3 + (3*d^4*x^2)/b^3 - (c + d*x)^4/b - ((I/5)*(c + d*x)^5)/d + ((c + d*x)^4*Log[1 + E^((2*I)*(a +
b*x))])/b - ((2*I)*d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 + (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*d^4*PolyLog[5, -E^((2*I
)*(a + b*x))])/(2*b^5) - (6*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/b^4 + (4*d*(c + d*x)^3*Cos[a + b*x]*Sin[a
+ b*x])/b^2 + (3*d^4*Sin[a + b*x]^2)/b^5 - (6*d^2*(c + d*x)^2*Sin[a + b*x]^2)/b^3 + (2*(c + d*x)^4*Sin[a + b*
x]^2)/b

________________________________________________________________________________________

Rubi [A]  time = 0.503125, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {4431, 4404, 3311, 32, 3310, 4407, 3719, 2190, 2531, 6609, 2282, 6589} \[ \frac{3 d^2 (c+d x)^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 i d^3 (c+d x) \text{PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}-\frac{2 i d (c+d x)^3 \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{3 d^4 \text{PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}-\frac{6 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}+\frac{6 c d^3 x}{b^3}+\frac{3 d^4 x^2}{b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Sec[a + b*x]*Sin[3*a + 3*b*x],x]

[Out]

(6*c*d^3*x)/b^3 + (3*d^4*x^2)/b^3 - (c + d*x)^4/b - ((I/5)*(c + d*x)^5)/d + ((c + d*x)^4*Log[1 + E^((2*I)*(a +
b*x))])/b - ((2*I)*d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 + (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*d^4*PolyLog[5, -E^((2*I
)*(a + b*x))])/(2*b^5) - (6*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/b^4 + (4*d*(c + d*x)^3*Cos[a + b*x]*Sin[a
+ b*x])/b^2 + (3*d^4*Sin[a + b*x]^2)/b^5 - (6*d^2*(c + d*x)^2*Sin[a + b*x]^2)/b^3 + (2*(c + d*x)^4*Sin[a + b*
x]^2)/b

Rule 4431

Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && M
emberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d,
1]

Rule 4404

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c +
d*x)^m*Sin[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n +
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

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 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 4407

Int[((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Int[
(c + d*x)^m*Sin[a + b*x]^n*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sin[a + b*x]^(n - 2)*Tan[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

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 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 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 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 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 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]

Rubi steps

\begin{align*} \int (c+d x)^4 \sec (a+b x) \sin (3 a+3 b x) \, dx &=\int \left (3 (c+d x)^4 \cos (a+b x) \sin (a+b x)-(c+d x)^4 \sin ^2(a+b x) \tan (a+b x)\right ) \, dx\\ &=3 \int (c+d x)^4 \cos (a+b x) \sin (a+b x) \, dx-\int (c+d x)^4 \sin ^2(a+b x) \tan (a+b x) \, dx\\ &=\frac{3 (c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac{(6 d) \int (c+d x)^3 \sin ^2(a+b x) \, dx}{b}+\int (c+d x)^4 \cos (a+b x) \sin (a+b x) \, dx-\int (c+d x)^4 \tan (a+b x) \, dx\\ &=-\frac{i (c+d x)^5}{5 d}+\frac{3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac{9 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}+2 i \int \frac{e^{2 i (a+b x)} (c+d x)^4}{1+e^{2 i (a+b x)}} \, dx-\frac{(2 d) \int (c+d x)^3 \sin ^2(a+b x) \, dx}{b}-\frac{(3 d) \int (c+d x)^3 \, dx}{b}+\frac{\left (9 d^3\right ) \int (c+d x) \sin ^2(a+b x) \, dx}{b^3}\\ &=-\frac{3 (c+d x)^4}{4 b}-\frac{i (c+d x)^5}{5 d}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac{9 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}+\frac{9 d^4 \sin ^2(a+b x)}{4 b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}-\frac{d \int (c+d x)^3 \, dx}{b}-\frac{(4 d) \int (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}+\frac{\left (3 d^3\right ) \int (c+d x) \sin ^2(a+b x) \, dx}{b^3}+\frac{\left (9 d^3\right ) \int (c+d x) \, dx}{2 b^3}\\ &=\frac{9 c d^3 x}{2 b^3}+\frac{9 d^4 x^2}{4 b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac{2 i d (c+d x)^3 \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{6 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}+\frac{\left (6 i d^2\right ) \int (c+d x)^2 \text{Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^2}+\frac{\left (3 d^3\right ) \int (c+d x) \, dx}{2 b^3}\\ &=\frac{6 c d^3 x}{b^3}+\frac{3 d^4 x^2}{b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\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{6 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}-\frac{\left (6 d^3\right ) \int (c+d x) \text{Li}_3\left (-e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=\frac{6 c d^3 x}{b^3}+\frac{3 d^4 x^2}{b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\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 i d^3 (c+d x) \text{Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}-\frac{6 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}-\frac{\left (3 i d^4\right ) \int \text{Li}_4\left (-e^{2 i (a+b x)}\right ) \, dx}{b^4}\\ &=\frac{6 c d^3 x}{b^3}+\frac{3 d^4 x^2}{b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\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 i d^3 (c+d x) \text{Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}-\frac{6 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}-\frac{\left (3 d^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^5}\\ &=\frac{6 c d^3 x}{b^3}+\frac{3 d^4 x^2}{b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d}+\frac{(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\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 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{6 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}\\ \end{align*}

Mathematica [B]  time = 6.74213, size = 2482, normalized size = 8.3 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x)^4*Sec[a + b*x]*Sin[3*a + 3*b*x],x]

[Out]

((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))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(
b^3*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*Po
lyLog[5, -E^((-2*I)*(a + b*x))]))/b^5)*Sec[a])/(20*E^(I*a)) + (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)) + (2*c^3*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])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + Sec[a
]*(Cos[2*a + 2*b*x]/(40*b^5) - ((I/40)*Sin[2*a + 2*b*x])/b^5)*(-20*b^4*c^4*Cos[a] + (40*I)*b^3*c^3*d*Cos[a] +
60*b^2*c^2*d^2*Cos[a] - (60*I)*b*c*d^3*Cos[a] - 30*d^4*Cos[a] - 80*b^4*c^3*d*x*Cos[a] + (120*I)*b^3*c^2*d^2*x*
Cos[a] + 120*b^2*c*d^3*x*Cos[a] - (60*I)*b*d^4*x*Cos[a] - 120*b^4*c^2*d^2*x^2*Cos[a] + (120*I)*b^3*c*d^3*x^2*C
os[a] + 60*b^2*d^4*x^2*Cos[a] - 80*b^4*c*d^3*x^3*Cos[a] + (40*I)*b^3*d^4*x^3*Cos[a] - 20*b^4*d^4*x^4*Cos[a] -
(20*I)*b^5*c^4*x*Cos[a + 2*b*x] - (40*I)*b^5*c^3*d*x^2*Cos[a + 2*b*x] - (40*I)*b^5*c^2*d^2*x^3*Cos[a + 2*b*x]
- (20*I)*b^5*c*d^3*x^4*Cos[a + 2*b*x] - (4*I)*b^5*d^4*x^5*Cos[a + 2*b*x] + (20*I)*b^5*c^4*x*Cos[3*a + 2*b*x] +
(40*I)*b^5*c^3*d*x^2*Cos[3*a + 2*b*x] + (40*I)*b^5*c^2*d^2*x^3*Cos[3*a + 2*b*x] + (20*I)*b^5*c*d^3*x^4*Cos[3*
a + 2*b*x] + (4*I)*b^5*d^4*x^5*Cos[3*a + 2*b*x] - 10*b^4*c^4*Cos[3*a + 4*b*x] - (20*I)*b^3*c^3*d*Cos[3*a + 4*b
*x] + 30*b^2*c^2*d^2*Cos[3*a + 4*b*x] + (30*I)*b*c*d^3*Cos[3*a + 4*b*x] - 15*d^4*Cos[3*a + 4*b*x] - 40*b^4*c^3
*d*x*Cos[3*a + 4*b*x] - (60*I)*b^3*c^2*d^2*x*Cos[3*a + 4*b*x] + 60*b^2*c*d^3*x*Cos[3*a + 4*b*x] + (30*I)*b*d^4
*x*Cos[3*a + 4*b*x] - 60*b^4*c^2*d^2*x^2*Cos[3*a + 4*b*x] - (60*I)*b^3*c*d^3*x^2*Cos[3*a + 4*b*x] + 30*b^2*d^4
*x^2*Cos[3*a + 4*b*x] - 40*b^4*c*d^3*x^3*Cos[3*a + 4*b*x] - (20*I)*b^3*d^4*x^3*Cos[3*a + 4*b*x] - 10*b^4*d^4*x
^4*Cos[3*a + 4*b*x] - 10*b^4*c^4*Cos[5*a + 4*b*x] - (20*I)*b^3*c^3*d*Cos[5*a + 4*b*x] + 30*b^2*c^2*d^2*Cos[5*a
+ 4*b*x] + (30*I)*b*c*d^3*Cos[5*a + 4*b*x] - 15*d^4*Cos[5*a + 4*b*x] - 40*b^4*c^3*d*x*Cos[5*a + 4*b*x] - (60*
I)*b^3*c^2*d^2*x*Cos[5*a + 4*b*x] + 60*b^2*c*d^3*x*Cos[5*a + 4*b*x] + (30*I)*b*d^4*x*Cos[5*a + 4*b*x] - 60*b^4
*c^2*d^2*x^2*Cos[5*a + 4*b*x] - (60*I)*b^3*c*d^3*x^2*Cos[5*a + 4*b*x] + 30*b^2*d^4*x^2*Cos[5*a + 4*b*x] - 40*b
^4*c*d^3*x^3*Cos[5*a + 4*b*x] - (20*I)*b^3*d^4*x^3*Cos[5*a + 4*b*x] - 10*b^4*d^4*x^4*Cos[5*a + 4*b*x] + 20*b^5
*c^4*x*Sin[a + 2*b*x] + 40*b^5*c^3*d*x^2*Sin[a + 2*b*x] + 40*b^5*c^2*d^2*x^3*Sin[a + 2*b*x] + 20*b^5*c*d^3*x^4
*Sin[a + 2*b*x] + 4*b^5*d^4*x^5*Sin[a + 2*b*x] - 20*b^5*c^4*x*Sin[3*a + 2*b*x] - 40*b^5*c^3*d*x^2*Sin[3*a + 2*
b*x] - 40*b^5*c^2*d^2*x^3*Sin[3*a + 2*b*x] - 20*b^5*c*d^3*x^4*Sin[3*a + 2*b*x] - 4*b^5*d^4*x^5*Sin[3*a + 2*b*x
] - (10*I)*b^4*c^4*Sin[3*a + 4*b*x] + 20*b^3*c^3*d*Sin[3*a + 4*b*x] + (30*I)*b^2*c^2*d^2*Sin[3*a + 4*b*x] - 30
*b*c*d^3*Sin[3*a + 4*b*x] - (15*I)*d^4*Sin[3*a + 4*b*x] - (40*I)*b^4*c^3*d*x*Sin[3*a + 4*b*x] + 60*b^3*c^2*d^2
*x*Sin[3*a + 4*b*x] + (60*I)*b^2*c*d^3*x*Sin[3*a + 4*b*x] - 30*b*d^4*x*Sin[3*a + 4*b*x] - (60*I)*b^4*c^2*d^2*x
^2*Sin[3*a + 4*b*x] + 60*b^3*c*d^3*x^2*Sin[3*a + 4*b*x] + (30*I)*b^2*d^4*x^2*Sin[3*a + 4*b*x] - (40*I)*b^4*c*d
^3*x^3*Sin[3*a + 4*b*x] + 20*b^3*d^4*x^3*Sin[3*a + 4*b*x] - (10*I)*b^4*d^4*x^4*Sin[3*a + 4*b*x] - (10*I)*b^4*c
^4*Sin[5*a + 4*b*x] + 20*b^3*c^3*d*Sin[5*a + 4*b*x] + (30*I)*b^2*c^2*d^2*Sin[5*a + 4*b*x] - 30*b*c*d^3*Sin[5*a
+ 4*b*x] - (15*I)*d^4*Sin[5*a + 4*b*x] - (40*I)*b^4*c^3*d*x*Sin[5*a + 4*b*x] + 60*b^3*c^2*d^2*x*Sin[5*a + 4*b
*x] + (60*I)*b^2*c*d^3*x*Sin[5*a + 4*b*x] - 30*b*d^4*x*Sin[5*a + 4*b*x] - (60*I)*b^4*c^2*d^2*x^2*Sin[5*a + 4*b
*x] + 60*b^3*c*d^3*x^2*Sin[5*a + 4*b*x] + (30*I)*b^2*d^4*x^2*Sin[5*a + 4*b*x] - (40*I)*b^4*c*d^3*x^3*Sin[5*a +
4*b*x] + 20*b^3*d^4*x^3*Sin[5*a + 4*b*x] - (10*I)*b^4*d^4*x^4*Sin[5*a + 4*b*x])

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Maple [B]  time = 0.255, size = 956, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*sec(b*x+a)*sin(3*b*x+3*a),x)

[Out]

-3/2*d^4*polylog(5,-exp(2*I*(b*x+a)))/b^5+6/b^3*c*d^3*polylog(3,-exp(2*I*(b*x+a)))*x+1/b*d^4*ln(exp(2*I*(b*x+a
))+1)*x^4+I*c^4*x-1/4*(2*d^4*x^4*b^4+4*I*b^3*d^4*x^3+8*b^4*c*d^3*x^3+12*I*b^3*c*d^3*x^2+12*b^4*c^2*d^2*x^2+12*
I*b^3*c^2*d^2*x+8*b^4*c^3*d*x+4*I*b^3*c^3*d+2*b^4*c^4-6*b^2*d^4*x^2-6*I*b*d^4*x-12*b^2*c*d^3*x-6*I*b*c*d^3-6*c
^2*d^2*b^2+3*d^4)/b^5*exp(2*I*(b*x+a))+6/b*c^2*d^2*ln(exp(2*I*(b*x+a))+1)*x^2+4/b*c^3*d*ln(exp(2*I*(b*x+a))+1)
*x+8/5*I/b^5*d^4*a^5-I*c*d^3*x^4-2*I*c^2*d^2*x^3-2*I*c^3*d*x^2+4/b*c*d^3*ln(exp(2*I*(b*x+a))+1)*x^3-6*I/b^2*c^
2*d^2*polylog(2,-exp(2*I*(b*x+a)))*x-6*I/b^2*c*d^3*polylog(2,-exp(2*I*(b*x+a)))*x^2+3*I/b^4*c*d^3*polylog(4,-e
xp(2*I*(b*x+a)))-2*I/b^2*c^3*d*polylog(2,-exp(2*I*(b*x+a)))-2*I/b^2*d^4*polylog(2,-exp(2*I*(b*x+a)))*x^3+3/b^3
*d^4*polylog(3,-exp(2*I*(b*x+a)))*x^2+3/b^3*c^2*d^2*polylog(3,-exp(2*I*(b*x+a)))-2/b^5*d^4*a^4*ln(exp(I*(b*x+a
)))+1/b*c^4*ln(exp(2*I*(b*x+a))+1)-1/5*I*d^4*x^5+8/b^2*c^3*d*a*ln(exp(I*(b*x+a)))-12/b^3*c^2*d^2*a^2*ln(exp(I*
(b*x+a)))+8/b^4*c*d^3*a^3*ln(exp(I*(b*x+a)))+2*I/b^4*d^4*a^4*x+8*I/b^3*c^2*d^2*a^3-4*I/b^2*c^3*d*a^2-6*I/b^4*c
*d^3*a^4-8*I/b*c^3*d*a*x+12*I/b^2*c^2*d^2*a^2*x-2/b*c^4*ln(exp(I*(b*x+a)))-1/4*(2*d^4*x^4*b^4-4*I*b^3*d^4*x^3+
8*b^4*c*d^3*x^3-12*I*b^3*c*d^3*x^2+12*b^4*c^2*d^2*x^2-12*I*b^3*c^2*d^2*x+8*b^4*c^3*d*x-4*I*b^3*c^3*d+2*b^4*c^4
-6*b^2*d^4*x^2+6*I*b*d^4*x-12*b^2*c*d^3*x+6*I*b*c*d^3-6*c^2*d^2*b^2+3*d^4)/b^5*exp(-2*I*(b*x+a))-8*I/b^3*c*d^3
*a^3*x+3*I/b^4*d^4*polylog(4,-exp(2*I*(b*x+a)))*x

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Maxima [B]  time = 1.6788, size = 819, normalized size = 2.74 \begin{align*} -\frac{c^{4}{\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )\right )}}{2 \, b} + \frac{-6 i \, b^{5} d^{4} x^{5} - 30 i \, b^{5} c d^{3} x^{4} - 60 i \, b^{5} c^{2} d^{2} x^{3} - 60 i \, b^{5} c^{3} d x^{2} - 90 \, d^{4}{\rm Li}_{5}(-e^{\left (2 i \, b x + 2 i \, a\right )}) +{\left (60 i \, b^{4} d^{4} x^{4} + 160 i \, b^{4} c d^{3} x^{3} + 180 i \, b^{4} c^{2} d^{2} x^{2} + 120 i \, b^{4} c^{3} d x\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 15 \,{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 6 \,{\left (2 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 4 \,{\left (2 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (2 \, b x + 2 \, a\right ) +{\left (-120 i \, b^{3} d^{4} x^{3} - 240 i \, b^{3} c d^{3} x^{2} - 180 i \, b^{3} c^{2} d^{2} x - 60 i \, b^{3} c^{3} d\right )}{\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 10 \,{\left (3 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 9 \, b^{4} c^{2} d^{2} x^{2} + 6 \, b^{4} c^{3} d x\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) +{\left (180 i \, b d^{4} x + 120 i \, b c d^{3}\right )}{\rm Li}_{4}(-e^{\left (2 i \, b x + 2 i \, a\right )}) + 30 \,{\left (6 \, b^{2} d^{4} x^{2} + 8 \, b^{2} c d^{3} x + 3 \, b^{2} c^{2} d^{2}\right )}{\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )}) + 30 \,{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \,{\left (2 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \sin \left (2 \, b x + 2 \, a\right )}{30 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*sec(b*x+a)*sin(3*b*x+3*a),x, algorithm="maxima")

[Out]

-1/2*c^4*(2*cos(2*b*x + 2*a) - log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a) + cos(2*a)^2 + sin(2*b*x)^2 - 2*sin(2*
b*x)*sin(2*a) + sin(2*a)^2))/b + 1/30*(-6*I*b^5*d^4*x^5 - 30*I*b^5*c*d^3*x^4 - 60*I*b^5*c^2*d^2*x^3 - 60*I*b^5
*c^3*d*x^2 - 90*d^4*polylog(5, -e^(2*I*b*x + 2*I*a)) + (60*I*b^4*d^4*x^4 + 160*I*b^4*c*d^3*x^3 + 180*I*b^4*c^2
*d^2*x^2 + 120*I*b^4*c^3*d*x)*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - 15*(2*b^4*d^4*x^4 + 8*b^4*c*d^
3*x^3 - 6*b^2*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(2*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(2*b*x +
2*a) + (-120*I*b^3*d^4*x^3 - 240*I*b^3*c*d^3*x^2 - 180*I*b^3*c^2*d^2*x - 60*I*b^3*c^3*d)*dilog(-e^(2*I*b*x +
2*I*a)) + 10*(3*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 9*b^4*c^2*d^2*x^2 + 6*b^4*c^3*d*x)*log(cos(2*b*x + 2*a)^2 + si
n(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + (180*I*b*d^4*x + 120*I*b*c*d^3)*polylog(4, -e^(2*I*b*x + 2*I*a))
+ 30*(6*b^2*d^4*x^2 + 8*b^2*c*d^3*x + 3*b^2*c^2*d^2)*polylog(3, -e^(2*I*b*x + 2*I*a)) + 30*(2*b^3*d^4*x^3 + 6*
b^3*c*d^3*x^2 + 2*b^3*c^3*d - 3*b*c*d^3 + 3*(2*b^3*c^2*d^2 - b*d^4)*x)*sin(2*b*x + 2*a))/b^5

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Fricas [C]  time = 1.00211, size = 3866, normalized size = 12.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*sec(b*x+a)*sin(3*b*x+3*a),x, algorithm="fricas")

[Out]

1/2*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 - 24*d^4*polylog(5, I*cos(b*x + a) + sin(b*x + a)) - 24*d^4*polylog(5, I*
cos(b*x + a) - sin(b*x + a)) - 24*d^4*polylog(5, -I*cos(b*x + a) + sin(b*x + a)) - 24*d^4*polylog(5, -I*cos(b*
x + a) - sin(b*x + a)) + 6*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 - 2*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 2*b^4*c^4 - 6*
b^2*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(2*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)^2 + 4*(2
*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d - 3*b*c*d^3 + 3*(2*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)*sin(b*x +
a) + 4*(2*b^4*c^3*d - 3*b^2*c*d^3)*x + (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)*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)*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)*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)*dilog(-I*cos(b*x + a) - sin(b*x + a)) + (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)*log(cos(b*x + a) + I*sin(b*x + a) + I) + (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)*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)*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 + 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)*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 + 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)*log(-I*c
os(b*x + a) + sin(b*x + a) + 1) + (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)*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)*log(-cos(b*x + a) + I*sin(b*x + a) + I) + (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)*log(-cos(b*x + a) - I*sin(b*x + a) + I) + (-24*I
*b*d^4*x - 24*I*b*c*d^3)*polylog(4, I*cos(b*x + a) + sin(b*x + a)) + (24*I*b*d^4*x + 24*I*b*c*d^3)*polylog(4,
I*cos(b*x + a) - sin(b*x + a)) + (24*I*b*d^4*x + 24*I*b*c*d^3)*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) + (-
24*I*b*d^4*x - 24*I*b*c*d^3)*polylog(4, -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)*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)*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)*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)*polylog(3, -I*cos(b*x + a) - sin(b*x + a)))/b
^5

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4*sec(b*x+a)*sin(3*b*x+3*a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{4} \sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*sec(b*x+a)*sin(3*b*x+3*a),x, algorithm="giac")

[Out]

integrate((d*x + c)^4*sec(b*x + a)*sin(3*b*x + 3*a), x)