\(\int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx\) [222]

Optimal result

Integrand size = 22, antiderivative size = 251 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=-\frac {3 d^3 x}{8 b^3}+\frac {(c+d x)^3}{4 b}+\frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b} \]

[Out]

-3/8*d^3*x/b^3+1/4*(d*x+c)^3/b+1/4*I*(d*x+c)^4/d-(d*x+c)^3*ln(1+exp(2*I*(b*x+a)))/b+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^3-3/4*I*d^3*polylog(4,-exp(2*I*(b*x+a)
))/b^4+3/8*d^3*cos(b*x+a)*sin(b*x+a)/b^4-3/4*d*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)/b^2+3/4*d^2*(d*x+c)*sin(b*x+a)^
2/b^3-1/2*(d*x+c)^3*sin(b*x+a)^2/b

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {4492, 4489, 3392, 32, 2715, 8, 3800, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=-\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^3 \sin (a+b x) \cos (a+b x)}{8 b^4}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{4 b^2}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d^3 x}{8 b^3}+\frac {(c+d x)^3}{4 b}+\frac {i (c+d x)^4}{4 d} \]

[In]

Int[(c + d*x)^3*Sin[a + b*x]^2*Tan[a + b*x],x]

[Out]

(-3*d^3*x)/(8*b^3) + (c + d*x)^3/(4*b) + ((I/4)*(c + d*x)^4)/d - ((c + d*x)^3*Log[1 + E^((2*I)*(a + b*x))])/b
+ (((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*I)/4)*d^3*PolyLog[4, -E^((2*I)*(a + b*x))])/b^4 + (3*d^3*Cos[a + b*x]*Sin[a + b*x])/(8*
b^4) - (3*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(4*b^2) + (3*d^2*(c + d*x)*Sin[a + b*x]^2)/(4*b^3) - ((c +
d*x)^3*Sin[a + b*x]^2)/(2*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 2715

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

Rule 3392

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 3800

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 4489

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 4492

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 6724

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 6744

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]

Rubi steps \begin{align*} \text {integral}& = -\int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx+\int (c+d x)^3 \tan (a+b x) \, dx \\ & = \frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-2 i \int \frac {e^{2 i (a+b x)} (c+d x)^3}{1+e^{2 i (a+b x)}} \, dx+\frac {(3 d) \int (c+d x)^2 \sin ^2(a+b x) \, dx}{2 b} \\ & = \frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {(3 d) \int (c+d x)^2 \, dx}{4 b}+\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}-\frac {\left (3 d^3\right ) \int \sin ^2(a+b x) \, dx}{4 b^3} \\ & = \frac {(c+d x)^3}{4 b}+\frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {\left (3 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (3 d^3\right ) \int 1 \, dx}{8 b^3} \\ & = -\frac {3 d^3 x}{8 b^3}+\frac {(c+d x)^3}{4 b}+\frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {\left (3 d^3\right ) \int \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right ) \, dx}{2 b^3} \\ & = -\frac {3 d^3 x}{8 b^3}+\frac {(c+d x)^3}{4 b}+\frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{4 b^4} \\ & = -\frac {3 d^3 x}{8 b^3}+\frac {(c+d x)^3}{4 b}+\frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1731\) vs. \(2(251)=502\).

Time = 6.53 (sec) , antiderivative size = 1731, normalized size of antiderivative = 6.90 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=-\frac {i c d^2 e^{-i a} \left (2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )+6 b \left (1+e^{2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{4 b^3}-\frac {i d^3 e^{i a} \left (2 b^4 e^{-2 i a} x^4-4 i b^3 \left (1+e^{-2 i a}\right ) x^3 \log \left (1+e^{-2 i (a+b x)}\right )+6 b^2 \left (1+e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-6 i b \left (1+e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )-3 \left (1+e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{8 b^4}-\frac {c^3 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a))}{b \left (\cos ^2(a)+\sin ^2(a)\right )}-\frac {3 c^2 d \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{2 b^2 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}+\sec (a) \left (\frac {\cos (2 a+2 b x)}{64 b^4}-\frac {i \sin (2 a+2 b x)}{64 b^4}\right ) \left (8 b^3 c^3 \cos (a)-12 i b^2 c^2 d \cos (a)-12 b c d^2 \cos (a)+6 i d^3 \cos (a)+24 b^3 c^2 d x \cos (a)-24 i b^2 c d^2 x \cos (a)-12 b d^3 x \cos (a)+24 b^3 c d^2 x^2 \cos (a)-12 i b^2 d^3 x^2 \cos (a)+8 b^3 d^3 x^3 \cos (a)+32 i b^4 c^3 x \cos (a+2 b x)+48 i b^4 c^2 d x^2 \cos (a+2 b x)+32 i b^4 c d^2 x^3 \cos (a+2 b x)+8 i b^4 d^3 x^4 \cos (a+2 b x)-32 i b^4 c^3 x \cos (3 a+2 b x)-48 i b^4 c^2 d x^2 \cos (3 a+2 b x)-32 i b^4 c d^2 x^3 \cos (3 a+2 b x)-8 i b^4 d^3 x^4 \cos (3 a+2 b x)+4 b^3 c^3 \cos (3 a+4 b x)+6 i b^2 c^2 d \cos (3 a+4 b x)-6 b c d^2 \cos (3 a+4 b x)-3 i d^3 \cos (3 a+4 b x)+12 b^3 c^2 d x \cos (3 a+4 b x)+12 i b^2 c d^2 x \cos (3 a+4 b x)-6 b d^3 x \cos (3 a+4 b x)+12 b^3 c d^2 x^2 \cos (3 a+4 b x)+6 i b^2 d^3 x^2 \cos (3 a+4 b x)+4 b^3 d^3 x^3 \cos (3 a+4 b x)+4 b^3 c^3 \cos (5 a+4 b x)+6 i b^2 c^2 d \cos (5 a+4 b x)-6 b c d^2 \cos (5 a+4 b x)-3 i d^3 \cos (5 a+4 b x)+12 b^3 c^2 d x \cos (5 a+4 b x)+12 i b^2 c d^2 x \cos (5 a+4 b x)-6 b d^3 x \cos (5 a+4 b x)+12 b^3 c d^2 x^2 \cos (5 a+4 b x)+6 i b^2 d^3 x^2 \cos (5 a+4 b x)+4 b^3 d^3 x^3 \cos (5 a+4 b x)-32 b^4 c^3 x \sin (a+2 b x)-48 b^4 c^2 d x^2 \sin (a+2 b x)-32 b^4 c d^2 x^3 \sin (a+2 b x)-8 b^4 d^3 x^4 \sin (a+2 b x)+32 b^4 c^3 x \sin (3 a+2 b x)+48 b^4 c^2 d x^2 \sin (3 a+2 b x)+32 b^4 c d^2 x^3 \sin (3 a+2 b x)+8 b^4 d^3 x^4 \sin (3 a+2 b x)+4 i b^3 c^3 \sin (3 a+4 b x)-6 b^2 c^2 d \sin (3 a+4 b x)-6 i b c d^2 \sin (3 a+4 b x)+3 d^3 \sin (3 a+4 b x)+12 i b^3 c^2 d x \sin (3 a+4 b x)-12 b^2 c d^2 x \sin (3 a+4 b x)-6 i b d^3 x \sin (3 a+4 b x)+12 i b^3 c d^2 x^2 \sin (3 a+4 b x)-6 b^2 d^3 x^2 \sin (3 a+4 b x)+4 i b^3 d^3 x^3 \sin (3 a+4 b x)+4 i b^3 c^3 \sin (5 a+4 b x)-6 b^2 c^2 d \sin (5 a+4 b x)-6 i b c d^2 \sin (5 a+4 b x)+3 d^3 \sin (5 a+4 b x)+12 i b^3 c^2 d x \sin (5 a+4 b x)-12 b^2 c d^2 x \sin (5 a+4 b x)-6 i b d^3 x \sin (5 a+4 b x)+12 i b^3 c d^2 x^2 \sin (5 a+4 b x)-6 b^2 d^3 x^2 \sin (5 a+4 b x)+4 i b^3 d^3 x^3 \sin (5 a+4 b x)\right ) \]

[In]

Integrate[(c + d*x)^3*Sin[a + b*x]^2*Tan[a + b*x],x]

[Out]

((-1/4*I)*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*b^4*x^4)/E^((2*I)*a) - (4*I)*b^3*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*
I)*(a + b*x))] + 6*b^2*(1 + E^((-2*I)*a))*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (6*I)*b*(1 + E^((-2*I)*a))*x
*PolyLog[3, -E^((-2*I)*(a + b*x))] - 3*(1 + E^((-2*I)*a))*PolyLog[4, -E^((-2*I)*(a + b*x))])*Sec[a])/b^4 - (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)) - (3*c^2*d*Cs
c[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[S
in[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*(Cos[a]^2 + Sin[a]^2)]) + Sec[a]*(Cos[2*a + 2*b*x]/(64*b^4) - ((I/64)*Sin[2*a + 2*b*x])/b^4)*(
8*b^3*c^3*Cos[a] - (12*I)*b^2*c^2*d*Cos[a] - 12*b*c*d^2*Cos[a] + (6*I)*d^3*Cos[a] + 24*b^3*c^2*d*x*Cos[a] - (2
4*I)*b^2*c*d^2*x*Cos[a] - 12*b*d^3*x*Cos[a] + 24*b^3*c*d^2*x^2*Cos[a] - (12*I)*b^2*d^3*x^2*Cos[a] + 8*b^3*d^3*
x^3*Cos[a] + (32*I)*b^4*c^3*x*Cos[a + 2*b*x] + (48*I)*b^4*c^2*d*x^2*Cos[a + 2*b*x] + (32*I)*b^4*c*d^2*x^3*Cos[
a + 2*b*x] + (8*I)*b^4*d^3*x^4*Cos[a + 2*b*x] - (32*I)*b^4*c^3*x*Cos[3*a + 2*b*x] - (48*I)*b^4*c^2*d*x^2*Cos[3
*a + 2*b*x] - (32*I)*b^4*c*d^2*x^3*Cos[3*a + 2*b*x] - (8*I)*b^4*d^3*x^4*Cos[3*a + 2*b*x] + 4*b^3*c^3*Cos[3*a +
4*b*x] + (6*I)*b^2*c^2*d*Cos[3*a + 4*b*x] - 6*b*c*d^2*Cos[3*a + 4*b*x] - (3*I)*d^3*Cos[3*a + 4*b*x] + 12*b^3*
c^2*d*x*Cos[3*a + 4*b*x] + (12*I)*b^2*c*d^2*x*Cos[3*a + 4*b*x] - 6*b*d^3*x*Cos[3*a + 4*b*x] + 12*b^3*c*d^2*x^2
*Cos[3*a + 4*b*x] + (6*I)*b^2*d^3*x^2*Cos[3*a + 4*b*x] + 4*b^3*d^3*x^3*Cos[3*a + 4*b*x] + 4*b^3*c^3*Cos[5*a +
4*b*x] + (6*I)*b^2*c^2*d*Cos[5*a + 4*b*x] - 6*b*c*d^2*Cos[5*a + 4*b*x] - (3*I)*d^3*Cos[5*a + 4*b*x] + 12*b^3*c
^2*d*x*Cos[5*a + 4*b*x] + (12*I)*b^2*c*d^2*x*Cos[5*a + 4*b*x] - 6*b*d^3*x*Cos[5*a + 4*b*x] + 12*b^3*c*d^2*x^2*
Cos[5*a + 4*b*x] + (6*I)*b^2*d^3*x^2*Cos[5*a + 4*b*x] + 4*b^3*d^3*x^3*Cos[5*a + 4*b*x] - 32*b^4*c^3*x*Sin[a +
2*b*x] - 48*b^4*c^2*d*x^2*Sin[a + 2*b*x] - 32*b^4*c*d^2*x^3*Sin[a + 2*b*x] - 8*b^4*d^3*x^4*Sin[a + 2*b*x] + 32
*b^4*c^3*x*Sin[3*a + 2*b*x] + 48*b^4*c^2*d*x^2*Sin[3*a + 2*b*x] + 32*b^4*c*d^2*x^3*Sin[3*a + 2*b*x] + 8*b^4*d^
3*x^4*Sin[3*a + 2*b*x] + (4*I)*b^3*c^3*Sin[3*a + 4*b*x] - 6*b^2*c^2*d*Sin[3*a + 4*b*x] - (6*I)*b*c*d^2*Sin[3*a
+ 4*b*x] + 3*d^3*Sin[3*a + 4*b*x] + (12*I)*b^3*c^2*d*x*Sin[3*a + 4*b*x] - 12*b^2*c*d^2*x*Sin[3*a + 4*b*x] - (
6*I)*b*d^3*x*Sin[3*a + 4*b*x] + (12*I)*b^3*c*d^2*x^2*Sin[3*a + 4*b*x] - 6*b^2*d^3*x^2*Sin[3*a + 4*b*x] + (4*I)
*b^3*d^3*x^3*Sin[3*a + 4*b*x] + (4*I)*b^3*c^3*Sin[5*a + 4*b*x] - 6*b^2*c^2*d*Sin[5*a + 4*b*x] - (6*I)*b*c*d^2*
Sin[5*a + 4*b*x] + 3*d^3*Sin[5*a + 4*b*x] + (12*I)*b^3*c^2*d*x*Sin[5*a + 4*b*x] - 12*b^2*c*d^2*x*Sin[5*a + 4*b
*x] - (6*I)*b*d^3*x*Sin[5*a + 4*b*x] + (12*I)*b^3*c*d^2*x^2*Sin[5*a + 4*b*x] - 6*b^2*d^3*x^2*Sin[5*a + 4*b*x]
+ (4*I)*b^3*d^3*x^3*Sin[5*a + 4*b*x])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (220 ) = 440\).

Time = 3.11 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.59

method	result	size
risch	\(\frac {6 i d \,c^{2} x a}{b}+\frac {3 i c \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {6 i c \,d^{2} a^{2} x}{b^{2}}-i c^{3} x -\frac {i c^{4}}{4 d}-\frac {c^{3} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b}+\frac {2 c^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {6 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {6 c^{2} d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b}-\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{2}}{b}+\frac {3 i d \,c^{2} a^{2}}{b^{2}}+\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x^{2}}{2 b^{2}}-\frac {4 i c \,d^{2} a^{3}}{b^{3}}+\frac {3 i d \,c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{2}}+\frac {2 i d^{3} a^{3} x}{b^{3}}+i d^{2} c \,x^{3}+\frac {3 i d \,c^{2} x^{2}}{2}-\frac {3 c \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{3}}-\frac {d^{3} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{3}}{b}-\frac {3 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{2 b^{3}}-\frac {3 i d^{3} \operatorname {polylog}\left (4, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{4 b^{4}}-\frac {2 d^{3} a^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {3 i d^{3} a^{4}}{2 b^{4}}+\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}-6 i b^{2} d^{3} x^{2}+12 b^{3} c^{2} d x -12 i b^{2} c \,d^{2} x +4 b^{3} c^{3}-6 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b +3 i d^{3}\right ) {\mathrm e}^{-2 i \left (x b +a \right )}}{32 b^{4}}+\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+6 i b^{2} d^{3} x^{2}+12 b^{3} c^{2} d x +12 i b^{2} c \,d^{2} x +4 b^{3} c^{3}+6 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b -3 i d^{3}\right ) {\mathrm e}^{2 i \left (x b +a \right )}}{32 b^{4}}+\frac {i d^{3} x^{4}}{4}\)	\(650\)

[In]

int((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-3/4*I*d^3*polylog(4,-exp(2*I*(b*x+a)))/b^4+3/2*I*d*c^2*x^2-1/b*c^3*ln(exp(2*I*(b*x+a))+1)-3/b*d*c^2*ln(exp(2*
I*(b*x+a))+1)*x-3/b*c*d^2*ln(exp(2*I*(b*x+a))+1)*x^2+2*I/b^3*d^3*a^3*x+3/2*I/b^2*d^3*polylog(2,-exp(2*I*(b*x+a
)))*x^2+3/2*I/b^2*d*c^2*polylog(2,-exp(2*I*(b*x+a)))+3*I/b^2*d*c^2*a^2-4*I/b^3*c*d^2*a^3+1/4*I*d^3*x^4+6/b^3*c
*d^2*a^2*ln(exp(I*(b*x+a)))-6/b^2*c^2*d*a*ln(exp(I*(b*x+a)))-1/b*d^3*ln(exp(2*I*(b*x+a))+1)*x^3-3/2/b^3*d^3*po
lylog(3,-exp(2*I*(b*x+a)))*x-3/2/b^3*c*d^2*polylog(3,-exp(2*I*(b*x+a)))+3/2*I/b^4*d^3*a^4-I*c^3*x-1/4*I/d*c^4+
2/b*c^3*ln(exp(I*(b*x+a)))-2/b^4*d^3*a^3*ln(exp(I*(b*x+a)))-6*I/b^2*c*d^2*a^2*x+6*I/b*d*c^2*x*a+3*I/b^2*c*d^2*
polylog(2,-exp(2*I*(b*x+a)))*x+1/32*(4*d^3*x^3*b^3-6*I*b^2*d^3*x^2+12*b^3*c*d^2*x^2-12*I*b^2*c*d^2*x+12*b^3*c^
2*d*x-6*I*b^2*c^2*d+4*b^3*c^3-6*b*d^3*x+3*I*d^3-6*c*d^2*b)/b^4*exp(-2*I*(b*x+a))+1/32*(4*d^3*x^3*b^3+6*I*b^2*d
^3*x^2+12*b^3*c*d^2*x^2+12*I*b^2*c*d^2*x+12*b^3*c^2*d*x+6*I*b^2*c^2*d+4*b^3*c^3-6*b*d^3*x-3*I*d^3-6*c*d^2*b)/b
^4*exp(2*I*(b*x+a))+I*d^2*c*x^3

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (216) = 432\).

Time = 0.32 (sec) , antiderivative size = 1134, normalized size of antiderivative = 4.52 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 - 24*I*d^3*polylog(4, I*cos(b*x + a) + sin(b*x + a)) + 24*I*d^3*polylog(
4, I*cos(b*x + a) - sin(b*x + a)) + 24*I*d^3*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) - 24*I*d^3*polylog(4,
-I*cos(b*x + a) - sin(b*x + a)) - 2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*c^3 - 3*b*c*d^2 + 3*(2*b^3*c^2*d
- b*d^3)*x)*cos(b*x + a)^2 + 3*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - d^3)*cos(b*x + a)*sin(b*x + a) +
3*(2*b^3*c^2*d - b*d^3)*x + 12*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dilog(I*cos(b*x + a) + sin(b*x
+ a)) + 12*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(I*cos(b*x + a) - sin(b*x + a)) + 12*(-I*b^2
*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(-I*cos(b*x + a) + sin(b*x + a)) + 12*(I*b^2*d^3*x^2 + 2*I*b^2*
c*d^2*x + I*b^2*c^2*d)*dilog(-I*cos(b*x + a) - sin(b*x + a)) + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^
3*d^3)*log(cos(b*x + a) + I*sin(b*x + a) + I) + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(cos(
b*x + a) - I*sin(b*x + a) + I) + 4*(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)*log(I*cos(b*x + a) + sin(b*x + a) + 1) + 4*(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)*log(I*cos(b*x + a) - sin(b*x + a) + 1) + 4*(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)*log(-I*cos(b*x + a) + sin(b*x + a) + 1) + 4*(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)*log(-I*cos(b*x + a) - sin
(b*x + a) + 1) + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-cos(b*x + a) + I*sin(b*x + a) + I)
+ 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-cos(b*x + a) - I*sin(b*x + a) + I) + 24*(b*d^3*x
+ b*c*d^2)*polylog(3, I*cos(b*x + a) + sin(b*x + a)) + 24*(b*d^3*x + b*c*d^2)*polylog(3, I*cos(b*x + a) - sin
(b*x + a)) + 24*(b*d^3*x + b*c*d^2)*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) + 24*(b*d^3*x + b*c*d^2)*polylo
g(3, -I*cos(b*x + a) - sin(b*x + a)))/b^4

Sympy [F]

\[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{3} \sin ^{3}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (216) = 432\).

Time = 0.44 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.76 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=-\frac {24 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} c^{3} - \frac {72 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a c^{2} d}{b} + \frac {72 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a^{2} c d^{2}}{b^{2}} - \frac {24 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a^{3} d^{3}}{b^{3}} + \frac {-12 i \, {\left (b x + a\right )}^{4} d^{3} - 48 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}^{3} + 48 i \, d^{3} {\rm Li}_{4}(-e^{\left (2 i \, b x + 2 i \, a\right )}) - 72 \, {\left (i \, b^{2} c^{2} d - 2 i \, a b c d^{2} + i \, a^{2} d^{3}\right )} {\left (b x + a\right )}^{2} - 16 \, {\left (-4 i \, {\left (b x + a\right )}^{3} d^{3} + 9 \, {\left (-i \, b c d^{2} + i \, a d^{3}\right )} {\left (b x + a\right )}^{2} + 9 \, {\left (-i \, b^{2} c^{2} d + 2 i \, a b c d^{2} - i \, a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 6 \, {\left (2 \, {\left (b x + a\right )}^{3} d^{3} - 3 \, b c d^{2} + 3 \, a d^{3} + 6 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (2 \, b^{2} c^{2} d - 4 \, a b c d^{2} + {\left (2 \, a^{2} - 1\right )} d^{3}\right )} {\left (b x + a\right )}\right )} \cos \left (2 \, b x + 2 \, a\right ) - 24 \, {\left (3 i \, b^{2} c^{2} d - 6 i \, a b c d^{2} + 4 i \, {\left (b x + a\right )}^{2} d^{3} + 3 i \, a^{2} d^{3} + 6 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 8 \, {\left (4 \, {\left (b x + a\right )}^{3} d^{3} + 9 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 9 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (b x + a\right )}\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 ) + 24 \, {\left (3 \, b c d^{2} + 4 \, {\left (b x + a\right )} d^{3} - 3 \, a d^{3}\right )} {\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )}) + 9 \, {\left (2 \, b^{2} c^{2} d - 4 \, a b c d^{2} + 2 \, {\left (b x + a\right )}^{2} d^{3} + {\left (2 \, a^{2} - 1\right )} d^{3} + 4 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}\right )} \sin \left (2 \, b x + 2 \, a\right )}{b^{3}}}{48 \, b} \]

[In]

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

[Out]

-1/48*(24*(sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1))*c^3 - 72*(sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1))*a*c^2
*d/b + 72*(sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1))*a^2*c*d^2/b^2 - 24*(sin(b*x + a)^2 + log(sin(b*x + a)^2 -
1))*a^3*d^3/b^3 + (-12*I*(b*x + a)^4*d^3 - 48*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^3 + 48*I*d^3*polylog(4, -e^(2*I
*b*x + 2*I*a)) - 72*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*a^2*d^3)*(b*x + a)^2 - 16*(-4*I*(b*x + a)^3*d^3 + 9*(-I*b
*c*d^2 + I*a*d^3)*(b*x + a)^2 + 9*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*a^2*d^3)*(b*x + a))*arctan2(sin(2*b*x + 2*
a), cos(2*b*x + 2*a) + 1) - 6*(2*(b*x + a)^3*d^3 - 3*b*c*d^2 + 3*a*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(
2*b^2*c^2*d - 4*a*b*c*d^2 + (2*a^2 - 1)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - 24*(3*I*b^2*c^2*d - 6*I*a*b*c*d^2 +
4*I*(b*x + a)^2*d^3 + 3*I*a^2*d^3 + 6*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*dilog(-e^(2*I*b*x + 2*I*a)) + 8*(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))*log(cos(2*b*
x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + 24*(3*b*c*d^2 + 4*(b*x + a)*d^3 - 3*a*d^3)*polylog
(3, -e^(2*I*b*x + 2*I*a)) + 9*(2*b^2*c^2*d - 4*a*b*c*d^2 + 2*(b*x + a)^2*d^3 + (2*a^2 - 1)*d^3 + 4*(b*c*d^2 -
a*d^3)*(b*x + a))*sin(2*b*x + 2*a))/b^3)/b

Giac [F]

\[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^3}{\cos \left (a+b\,x\right )} \,d x \]

[In]

int((sin(a + b*x)^3*(c + d*x)^3)/cos(a + b*x),x)

[Out]

int((sin(a + b*x)^3*(c + d*x)^3)/cos(a + b*x), x)