\(\int (c+d x)^3 \sec (a+b x) \sin (3 a+3 b x) \, dx\) [383]

Optimal result

Integrand size = 23, antiderivative size = 242 \[ \int (c+d x)^3 \sec (a+b x) \sin (3 a+3 b x) \, dx=\frac {3 d^3 x}{2 b^3}-\frac {(c+d x)^3}{b}-\frac {i (c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{2 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{b} \]

[Out]

3/2*d^3*x/b^3-(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,-ex
p(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^
4-3/2*d^3*cos(b*x+a)*sin(b*x+a)/b^4+3*d*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)/b^2-3*d^2*(d*x+c)*sin(b*x+a)^2/b^3+2*(
d*x+c)^3*sin(b*x+a)^2/b

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {4516, 4489, 3392, 32, 2715, 8, 4492, 3800, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \sec (a+b x) \sin (3 a+3 b x) \, dx=\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 d^3 \sin (a+b x) \cos (a+b x)}{2 b^4}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{b^2}+\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{b}+\frac {3 d^3 x}{2 b^3}-\frac {(c+d x)^3}{b}-\frac {i (c+d x)^4}{4 d} \]

[In]

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

[Out]

(3*d^3*x)/(2*b^3) - (c + d*x)^3/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])/(2*b^4)
+ (3*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/b^2 - (3*d^2*(c + d*x)*Sin[a + b*x]^2)/b^3 + (2*(c + d*x)^3*Sin[
a + b*x]^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 4516

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 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 \left (3 (c+d x)^3 \cos (a+b x) \sin (a+b x)-(c+d x)^3 \sin ^2(a+b x) \tan (a+b x)\right ) \, dx \\ & = 3 \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx-\int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx \\ & = \frac {3 (c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {(9 d) \int (c+d x)^2 \sin ^2(a+b x) \, dx}{2 b}+\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 {9 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {9 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{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 {(9 d) \int (c+d x)^2 \, dx}{4 b}+\frac {\left (9 d^3\right ) \int \sin ^2(a+b x) \, dx}{4 b^3} \\ & = -\frac {3 (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 {9 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)}{b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{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 {\left (9 d^3\right ) \int 1 \, dx}{8 b^3} \\ & = \frac {9 d^3 x}{8 b^3}-\frac {(c+d x)^3}{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)}{2 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{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}{2 b^3}-\frac {(c+d x)^3}{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)}{2 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{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}{2 b^3}-\frac {(c+d x)^3}{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)}{2 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{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}{2 b^3}-\frac {(c+d x)^3}{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)}{2 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}+\frac {2 (c+d x)^3 \sin ^2(a+b x)}{b} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1730\) vs. \(2(242)=484\).

Time = 6.36 (sec) , antiderivative size = 1730, normalized size of antiderivative = 7.15 \[ \int (c+d x)^3 \sec (a+b x) \sin (3 a+3 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)}{16 b^4}-\frac {i \sin (2 a+2 b x)}{16 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)-8 i b^4 c^3 x \cos (a+2 b x)-12 i b^4 c^2 d x^2 \cos (a+2 b x)-8 i b^4 c d^2 x^3 \cos (a+2 b x)-2 i b^4 d^3 x^4 \cos (a+2 b x)+8 i b^4 c^3 x \cos (3 a+2 b x)+12 i b^4 c^2 d x^2 \cos (3 a+2 b x)+8 i b^4 c d^2 x^3 \cos (3 a+2 b x)+2 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)+8 b^4 c^3 x \sin (a+2 b x)+12 b^4 c^2 d x^2 \sin (a+2 b x)+8 b^4 c d^2 x^3 \sin (a+2 b x)+2 b^4 d^3 x^4 \sin (a+2 b x)-8 b^4 c^3 x \sin (3 a+2 b x)-12 b^4 c^2 d x^2 \sin (3 a+2 b x)-8 b^4 c d^2 x^3 \sin (3 a+2 b x)-2 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*Sec[a + b*x]*Sin[3*a + 3*b*x],x]

[Out]

((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*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*Po
lyLog[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*S
ec[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*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*S
qrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + Sec[a]*(Cos[2*a + 2*b*x]/(16*b^4) - ((I/16)*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] + (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] - (8*I)*b^4*c^3*x*Cos[a + 2*b*x] - (12*I)*b^4*c^2*d*x^2*Cos[a + 2*b*x] - (8*I)*b^4*c*d^2*x^3*Cos[a +
2*b*x] - (2*I)*b^4*d^3*x^4*Cos[a + 2*b*x] + (8*I)*b^4*c^3*x*Cos[3*a + 2*b*x] + (12*I)*b^4*c^2*d*x^2*Cos[3*a +
2*b*x] + (8*I)*b^4*c*d^2*x^3*Cos[3*a + 2*b*x] + (2*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] + 8*b^4*c^3*x*Sin[a + 2*b*x]
+ 12*b^4*c^2*d*x^2*Sin[a + 2*b*x] + 8*b^4*c*d^2*x^3*Sin[a + 2*b*x] + 2*b^4*d^3*x^4*Sin[a + 2*b*x] - 8*b^4*c^3*
x*Sin[3*a + 2*b*x] - 12*b^4*c^2*d*x^2*Sin[3*a + 2*b*x] - 8*b^4*c*d^2*x^3*Sin[3*a + 2*b*x] - 2*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. 647 vs. \(2 (219 ) = 438\).

Time = 2.88 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.68

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 {\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 )}}{8 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 )}}{8 b^{4}}-\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 {i d^{3} x^{4}}{4}\)	\(648\)

[In]

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

[Out]

-3/2*I/b^2*d^3*polylog(2,-exp(2*I*(b*x+a)))*x^2-1/8*(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))-3/2*I/b^2*c^2*
d*polylog(2,-exp(2*I*(b*x+a)))-1/8*(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))+3/2/b^3*d^3*polylog(3,-exp(2*I
*(b*x+a)))*x+3/2/b^3*c*d^2*polylog(3,-exp(2*I*(b*x+a)))+1/b*d^3*ln(exp(2*I*(b*x+a))+1)*x^3+1/b*c^3*ln(exp(2*I*
(b*x+a))+1)-6*I/b*d*c^2*x*a+6*I/b^2*c*d^2*a^2*x-I*d^2*c*x^3-3/2*I*d*c^2*x^2-6/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)))-2/b*c^3*ln(exp(I*(b*x+a)))-1/4*I*d^3*x^4+3/b*c*d^2*ln(exp(2*I*(b*x+a))+1)*x^
2+3/b*c^2*d*ln(exp(2*I*(b*x+a))+1)*x-3*I/b^2*d*c^2*a^2+4*I/b^3*c*d^2*a^3-2*I/b^3*d^3*a^3*x+3/4*I*d^3*polylog(4
,-exp(2*I*(b*x+a)))/b^4+2/b^4*d^3*a^3*ln(exp(I*(b*x+a)))-3/2*I/b^4*d^3*a^4+I*c^3*x+1/4*I/d*c^4-3*I/b^2*d^2*c*p
olylog(2,-exp(2*I*(b*x+a)))*x

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. 1126 vs. \(2 (215) = 430\).

Time = 0.34 (sec) , antiderivative size = 1126, normalized size of antiderivative = 4.65 \[ \int (c+d x)^3 \sec (a+b x) \sin (3 a+3 b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 - 6*I*d^3*polylog(4, I*cos(b*x + a) + sin(b*x + a)) + 6*I*d^3*polylog(4,
I*cos(b*x + a) - sin(b*x + a)) + 6*I*d^3*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) - 6*I*d^3*polylog(4, -I*co
s(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 - 3*(-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)
) - 3*(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)) - 3*(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)) - 3*(-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)) + (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) + (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) + (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) + (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) + (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) + (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) + (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) + (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) + 6*(b*d^3*x + b*c*d^2)*polylog(3, I*c
os(b*x + a) + sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2)*polylog(3, I*cos(b*x + a) - sin(b*x + a)) + 6*(b*d^3*x + b
*c*d^2)*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2)*polylog(3, -I*cos(b*x + a) - sin(b*
x + a)))/b^4

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^3 \sec (a+b x) \sin (3 a+3 b x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

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. 444 vs. \(2 (215) = 430\).

Time = 0.34 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.83 \[ \int (c+d x)^3 \sec (a+b x) \sin (3 a+3 b x) \, dx=-\frac {c^{3} {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )\right )}}{2 \, b} + \frac {-3 i \, b^{4} d^{3} x^{4} - 12 i \, b^{4} c d^{2} x^{3} - 18 i \, b^{4} c^{2} d x^{2} + 12 i \, d^{3} {\rm Li}_{4}(-e^{\left (2 i \, b x + 2 i \, a\right )}) - 4 \, {\left (-4 i \, b^{3} d^{3} x^{3} - 9 i \, b^{3} c d^{2} x^{2} - 9 i \, b^{3} c^{2} d x\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 6 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} - 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (4 i \, b^{2} d^{3} x^{2} + 6 i \, b^{2} c d^{2} x + 3 i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 2 \, {\left (4 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 6 \, {\left (4 \, b d^{3} x + 3 \, b c d^{2}\right )} {\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )}) + 9 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{12 \, b^{4}} \]

[In]

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

[Out]

-1/2*c^3*(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 + 1/12*(-3*I*b^4*d^3*x^4 - 12*I*b^4*c*d^2*x^3 - 18*I*b^4*c^2*d*x^2 + 12*I*d^3*p
olylog(4, -e^(2*I*b*x + 2*I*a)) - 4*(-4*I*b^3*d^3*x^3 - 9*I*b^3*c*d^2*x^2 - 9*I*b^3*c^2*d*x)*arctan2(sin(2*b*x
+ 2*a), cos(2*b*x + 2*a) + 1) - 6*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 - 3*b*c*d^2 + 3*(2*b^3*c^2*d - b*d^3)*x)*c
os(2*b*x + 2*a) - 6*(4*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d)*dilog(-e^(2*I*b*x + 2*I*a)) + 2*(4*b^3
*d^3*x^3 + 9*b^3*c*d^2*x^2 + 9*b^3*c^2*d*x)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) +
1) + 6*(4*b*d^3*x + 3*b*c*d^2)*polylog(3, -e^(2*I*b*x + 2*I*a)) + 9*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^
2*d - d^3)*sin(2*b*x + 2*a))/b^4

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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