3.282 \(\int \frac{(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=431 \[ \frac{3 i f (e+f x) \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 i f (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 f^2 \text{PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac{3 f^2 \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}+\frac{f (e+f x) \tan (c+d x)}{3 a d^2}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}+\frac{f (e+f x) \tan (c+d x) \sec ^2(c+d x)}{6 a d^2}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}+\frac{5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}+\frac{f^2 \log (\cos (c+d x))}{3 a d^3}+\frac{f^2 \tan (c+d x) \sec (c+d x)}{12 a d^3}-\frac{3 i (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{4 a d}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{(e+f x)^2 \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 (e+f x)^2 \tan (c+d x) \sec (c+d x)}{8 a d} \]

[Out]

(((-3*I)/4)*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(a*d) + (5*f^2*ArcTanh[Sin[c + d*x]])/(6*a*d^3) + (f^2*Log[Co
s[c + d*x]])/(3*a*d^3) + (((3*I)/4)*f*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^2) - (((3*I)/4)*f*(e +
f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2) - (3*f^2*PolyLog[3, (-I)*E^(I*(c + d*x))])/(4*a*d^3) + (3*f^2*Poly
Log[3, I*E^(I*(c + d*x))])/(4*a*d^3) - (3*f*(e + f*x)*Sec[c + d*x])/(4*a*d^2) - (f^2*Sec[c + d*x]^2)/(12*a*d^3
) - (f*(e + f*x)*Sec[c + d*x]^3)/(6*a*d^2) - ((e + f*x)^2*Sec[c + d*x]^4)/(4*a*d) + (f*(e + f*x)*Tan[c + d*x])
/(3*a*d^2) + (f^2*Sec[c + d*x]*Tan[c + d*x])/(12*a*d^3) + (3*(e + f*x)^2*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) +
(f*(e + f*x)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d^2) + ((e + f*x)^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.397982, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4531, 4186, 3768, 3770, 4181, 2531, 2282, 6589, 4409, 4185, 4184, 3475} \[ \frac{3 i f (e+f x) \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 i f (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 f^2 \text{PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac{3 f^2 \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}+\frac{f (e+f x) \tan (c+d x)}{3 a d^2}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}+\frac{f (e+f x) \tan (c+d x) \sec ^2(c+d x)}{6 a d^2}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}+\frac{5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}+\frac{f^2 \log (\cos (c+d x))}{3 a d^3}+\frac{f^2 \tan (c+d x) \sec (c+d x)}{12 a d^3}-\frac{3 i (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{4 a d}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{(e+f x)^2 \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 (e+f x)^2 \tan (c+d x) \sec (c+d x)}{8 a d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^2*Sec[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

(((-3*I)/4)*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(a*d) + (5*f^2*ArcTanh[Sin[c + d*x]])/(6*a*d^3) + (f^2*Log[Co
s[c + d*x]])/(3*a*d^3) + (((3*I)/4)*f*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^2) - (((3*I)/4)*f*(e +
f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2) - (3*f^2*PolyLog[3, (-I)*E^(I*(c + d*x))])/(4*a*d^3) + (3*f^2*Poly
Log[3, I*E^(I*(c + d*x))])/(4*a*d^3) - (3*f*(e + f*x)*Sec[c + d*x])/(4*a*d^2) - (f^2*Sec[c + d*x]^2)/(12*a*d^3
) - (f*(e + f*x)*Sec[c + d*x]^3)/(6*a*d^2) - ((e + f*x)^2*Sec[c + d*x]^4)/(4*a*d) + (f*(e + f*x)*Tan[c + d*x])
/(3*a*d^2) + (f^2*Sec[c + d*x]*Tan[c + d*x])/(12*a*d^3) + (3*(e + f*x)^2*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) +
(f*(e + f*x)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d^2) + ((e + f*x)^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)

Rule 4531

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sec[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Sec[c + d*x]^(n + 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Sec[c + d*x]^(n + 1)*
Tan[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]

Rule 4186

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

Rule 3768

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && 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 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]

Rule 4409

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
((c + d*x)^m*Sec[a + b*x]^n)/(b*n), x] - Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int (e+f x)^2 \sec ^5(c+d x) \, dx}{a}-\frac{\int (e+f x)^2 \sec ^4(c+d x) \tan (c+d x) \, dx}{a}\\ &=-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac{3 \int (e+f x)^2 \sec ^3(c+d x) \, dx}{4 a}+\frac{f \int (e+f x) \sec ^4(c+d x) \, dx}{2 a d}+\frac{f^2 \int \sec ^3(c+d x) \, dx}{6 a d^2}\\ &=-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{f^2 \sec (c+d x) \tan (c+d x)}{12 a d^3}+\frac{3 (e+f x)^2 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac{f (e+f x) \sec ^2(c+d x) \tan (c+d x)}{6 a d^2}+\frac{(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac{3 \int (e+f x)^2 \sec (c+d x) \, dx}{8 a}+\frac{f \int (e+f x) \sec ^2(c+d x) \, dx}{3 a d}+\frac{f^2 \int \sec (c+d x) \, dx}{12 a d^2}+\frac{\left (3 f^2\right ) \int \sec (c+d x) \, dx}{4 a d^2}\\ &=-\frac{3 i (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{4 a d}+\frac{5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{f (e+f x) \tan (c+d x)}{3 a d^2}+\frac{f^2 \sec (c+d x) \tan (c+d x)}{12 a d^3}+\frac{3 (e+f x)^2 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac{f (e+f x) \sec ^2(c+d x) \tan (c+d x)}{6 a d^2}+\frac{(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac{(3 f) \int (e+f x) \log \left (1-i e^{i (c+d x)}\right ) \, dx}{4 a d}+\frac{(3 f) \int (e+f x) \log \left (1+i e^{i (c+d x)}\right ) \, dx}{4 a d}-\frac{f^2 \int \tan (c+d x) \, dx}{3 a d^2}\\ &=-\frac{3 i (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{4 a d}+\frac{5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}+\frac{f^2 \log (\cos (c+d x))}{3 a d^3}+\frac{3 i f (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 i f (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{f (e+f x) \tan (c+d x)}{3 a d^2}+\frac{f^2 \sec (c+d x) \tan (c+d x)}{12 a d^3}+\frac{3 (e+f x)^2 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac{f (e+f x) \sec ^2(c+d x) \tan (c+d x)}{6 a d^2}+\frac{(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac{\left (3 i f^2\right ) \int \text{Li}_2\left (-i e^{i (c+d x)}\right ) \, dx}{4 a d^2}+\frac{\left (3 i f^2\right ) \int \text{Li}_2\left (i e^{i (c+d x)}\right ) \, dx}{4 a d^2}\\ &=-\frac{3 i (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{4 a d}+\frac{5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}+\frac{f^2 \log (\cos (c+d x))}{3 a d^3}+\frac{3 i f (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 i f (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{f (e+f x) \tan (c+d x)}{3 a d^2}+\frac{f^2 \sec (c+d x) \tan (c+d x)}{12 a d^3}+\frac{3 (e+f x)^2 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac{f (e+f x) \sec ^2(c+d x) \tan (c+d x)}{6 a d^2}+\frac{(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^3}+\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^3}\\ &=-\frac{3 i (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{4 a d}+\frac{5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}+\frac{f^2 \log (\cos (c+d x))}{3 a d^3}+\frac{3 i f (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 i f (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{4 a d^2}-\frac{3 f^2 \text{Li}_3\left (-i e^{i (c+d x)}\right )}{4 a d^3}+\frac{3 f^2 \text{Li}_3\left (i e^{i (c+d x)}\right )}{4 a d^3}-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{f (e+f x) \tan (c+d x)}{3 a d^2}+\frac{f^2 \sec (c+d x) \tan (c+d x)}{12 a d^3}+\frac{3 (e+f x)^2 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac{f (e+f x) \sec ^2(c+d x) \tan (c+d x)}{6 a d^2}+\frac{(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d}\\ \end{align*}

Mathematica [B]  time = 8.92612, size = 1468, normalized size = 3.41 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^2*Sec[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

-((Cos[c] + I*Sin[c])*(3*d^2*e*f*x^2*Cos[c] + 6*e*f*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*(-1
+ Sin[c])) + (3*d^2*e^2 + 4*f^2)*x*(Cos[c] - I*Sin[c]) + d^2*f^2*x^3*(Cos[c] - I*Sin[c]) + 6*d*e*f*x*Log[1 - I
*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c]))*(Cos[c] - I*Sin[c]) + 3*d*f^2*x^2*Log[1 - I*Cos[c + d
*x] - Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c]))*(Cos[c] - I*Sin[c]) + (6*f^2*(I*d*x*PolyLog[2, I*Cos[c + d*x] +
Sin[c + d*x]] + PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]])*(Cos[c] + I*(-1 + Sin[c]))*(Cos[c] - I*Sin[c]))/d
- (3*I)*d^2*e*f*x^2*Sin[c] + ((3*d^2*e^2 + 4*f^2)*(d*x + I*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])])*(Cos[c]
- I*Sin[c])*(-1 - I*Cos[c] + Sin[c]))/d))/(8*a*d^2*(Cos[c] + I*(-1 + Sin[c]))) - ((Cos[c] + I*Sin[c])*(9*d^2*
e*f*x^2*Cos[c] + 3*d^2*f^2*x^3*Cos[c] + (9*d^2*e^2 + 28*f^2)*x*(Cos[c] - I*Sin[c]) - (9*I)*d^2*e*f*x^2*Sin[c]
- (3*I)*d^2*f^2*x^3*Sin[c] + (18*f^2*(d*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos
[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d + 18*e*f*PolyLog[2, (-I)*Cos[c + d*x
] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])) - 18*d*e*f*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Si
n[c])*(Cos[c] + I*(1 + Sin[c])) - 9*d*f^2*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[
c] + I*(1 + Sin[c])) + ((9*d^2*e^2 + 28*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(I*Cos[c] + Si
n[c])*(Cos[c] + I*(1 + Sin[c])))/d))/(24*a*d^2*(Cos[c] + I*(1 + Sin[c]))) + ((3*e^2*x*Cos[c])/(4*a) + (((3*I)/
4)*e^2*x*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((3*e*f*x^2*Cos[c])/(4*a) + (((3*I)/4)*e*f*x^2*Sin[c])/a)/(1
+ Cos[2*c] + I*Sin[2*c]) + ((f^2*x^3*Cos[c])/(4*a) + ((I/4)*f^2*x^3*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) +
(e^2 + 2*e*f*x + f^2*x^2)/(8*a*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (-(e*f*Sin[(d*x)/2]) - f^2*x*S
in[(d*x)/2])/(2*a*d^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (-e^2 - 2*e*f*x - f^2
*x^2)/(8*a*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (e*f*Sin[(d*x)/2] + f^2*x*Sin[(d*x)/2])/(6*a*d^2*(
Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (-3*d^2*e^2*Cos[c/2] - d*e*f*Cos[c/2] - f^
2*Cos[c/2] - 6*d^2*e*f*x*Cos[c/2] - d*f^2*x*Cos[c/2] - 3*d^2*f^2*x^2*Cos[c/2] - 3*d^2*e^2*Sin[c/2] + d*e*f*Sin
[c/2] - f^2*Sin[c/2] - 6*d^2*e*f*x*Sin[c/2] + d*f^2*x*Sin[c/2] - 3*d^2*f^2*x^2*Sin[c/2])/(12*a*d^3*(Cos[c/2] +
Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (7*(e*f*Sin[(d*x)/2] + f^2*x*Sin[(d*x)/2]))/(6*a*d^2
*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

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Maple [B]  time = 0.361, size = 1119, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*sec(d*x+c)^3/(a+a*sin(d*x+c)),x)

[Out]

3/4*I/d^2/a*polylog(2,-I*exp(I*(d*x+c)))*f^2*x-1/2/d^3/a*f^2*ln(exp(I*(d*x+c))-I)-3/4*f^2*polylog(3,-I*exp(I*(
d*x+c)))/a/d^3+3/4/a/d*f*e*ln(1-I*exp(I*(d*x+c)))*x+3/4/a/d^2*f*e*ln(1-I*exp(I*(d*x+c)))*c-3/4/a/d^2*f*e*c*ln(
exp(I*(d*x+c))+I)+3/8/a/d*f^2*ln(1-I*exp(I*(d*x+c)))*x^2-3/8/a/d^3*f^2*ln(1-I*exp(I*(d*x+c)))*c^2+3/4/d^2/a*e*
f*c*ln(exp(I*(d*x+c))-I)+3/8/a/d*ln(exp(I*(d*x+c))+I)*e^2+3/8/d^3/a*ln(1+I*exp(I*(d*x+c)))*c^2*f^2-3/8/d/a*ln(
1+I*exp(I*(d*x+c)))*f^2*x^2-3/4/d/a*ln(1+I*exp(I*(d*x+c)))*e*f*x-3/4/d^2/a*ln(1+I*exp(I*(d*x+c)))*c*e*f+3/8/a/
d^3*f^2*c^2*ln(exp(I*(d*x+c))+I)+3/4*f^2*polylog(3,I*exp(I*(d*x+c)))/a/d^3-3/8/d^3/a*f^2*c^2*ln(exp(I*(d*x+c))
-I)-2/3/d^3/a*f^2*ln(exp(I*(d*x+c)))+7/6/d^3/a*f^2*ln(exp(I*(d*x+c))+I)-3/8/d/a*e^2*ln(exp(I*(d*x+c))-I)-1/12*
I*(-18*I*d^2*f^2*x^2*exp(2*I*(d*x+c))-16*I*d*f^2*x*exp(3*I*(d*x+c))-16*I*d*e*f*exp(3*I*(d*x+c))+6*d^2*f^2*x^2*
exp(3*I*(d*x+c))+2*f^2*exp(I*(d*x+c))+18*I*d^2*e^2*exp(4*I*(d*x+c))+18*d^2*e*f*x*exp(I*(d*x+c))+2*I*d*e*f*exp(
I*(d*x+c))+2*I*d*f^2*x*exp(I*(d*x+c))+9*d^2*e^2*exp(I*(d*x+c))+6*d^2*e^2*exp(3*I*(d*x+c))+9*d^2*e^2*exp(5*I*(d
*x+c))+44*d*f^2*x*exp(2*I*(d*x+c))+36*d*e*f*exp(4*I*(d*x+c))+36*I*d^2*e*f*x*exp(4*I*(d*x+c))-36*I*d^2*e*f*x*ex
p(2*I*(d*x+c))+8*d*f^2*x+2*f^2*exp(5*I*(d*x+c))+18*I*d^2*f^2*x^2*exp(4*I*(d*x+c))+18*d^2*e*f*x*exp(5*I*(d*x+c)
)+12*d^2*e*f*x*exp(3*I*(d*x+c))-18*I*d*f^2*x*exp(5*I*(d*x+c))-18*I*d*e*f*exp(5*I*(d*x+c))+9*d^2*f^2*x^2*exp(I*
(d*x+c))+8*d*e*f+4*f^2*exp(3*I*(d*x+c))+44*d*e*f*exp(2*I*(d*x+c))+9*d^2*f^2*x^2*exp(5*I*(d*x+c))+36*d*f^2*x*ex
p(4*I*(d*x+c))-18*I*d^2*e^2*exp(2*I*(d*x+c)))/(exp(I*(d*x+c))+I)^4/d^3/(exp(I*(d*x+c))-I)^2/a-3/4*I/d^2/a*e*f*
polylog(2,I*exp(I*(d*x+c)))-3/4*I/d^2/a*polylog(2,I*exp(I*(d*x+c)))*f^2*x+3/4*I/d^2/a*e*f*polylog(2,-I*exp(I*(
d*x+c)))

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Maxima [B]  time = 40.6915, size = 7104, normalized size = 16.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sec(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/16*(2*c*e*f*(2*(3*sin(d*x + c)^2 + 3*sin(d*x + c) - 2)/(a*d*sin(d*x + c)^3 + a*d*sin(d*x + c)^2 - a*d*sin(d*
x + c) - a*d) - 3*log(sin(d*x + c) + 1)/(a*d) + 3*log(sin(d*x + c) - 1)/(a*d)) - e^2*(2*(3*sin(d*x + c)^2 + 3*
sin(d*x + c) - 2)/(a*sin(d*x + c)^3 + a*sin(d*x + c)^2 - a*sin(d*x + c) - a) - 3*log(sin(d*x + c) + 1)/a + 3*l
og(sin(d*x + c) - 1)/a) - 16*(32*(d*x + c)*f^2*cos(6*d*x + 6*c) + 32*I*(d*x + c)*f^2*sin(6*d*x + 6*c) + 32*d*e
*f - 32*c*f^2 - (2*(9*c^2 + 28)*f^2*cos(6*d*x + 6*c) + (36*I*c^2 + 112*I)*f^2*cos(5*d*x + 5*c) + 2*(9*c^2 + 28
)*f^2*cos(4*d*x + 4*c) + (72*I*c^2 + 224*I)*f^2*cos(3*d*x + 3*c) - 2*(9*c^2 + 28)*f^2*cos(2*d*x + 2*c) + (36*I
*c^2 + 112*I)*f^2*cos(d*x + c) + (18*I*c^2 + 56*I)*f^2*sin(6*d*x + 6*c) - 4*(9*c^2 + 28)*f^2*sin(5*d*x + 5*c)
+ (18*I*c^2 + 56*I)*f^2*sin(4*d*x + 4*c) - 8*(9*c^2 + 28)*f^2*sin(3*d*x + 3*c) + (-18*I*c^2 - 56*I)*f^2*sin(2*
d*x + 2*c) - 4*(9*c^2 + 28)*f^2*sin(d*x + c) - 2*(9*c^2 + 28)*f^2)*arctan2(sin(d*x + c) + 1, cos(d*x + c)) + (
6*(3*c^2 + 4)*f^2*cos(6*d*x + 6*c) - (-36*I*c^2 - 48*I)*f^2*cos(5*d*x + 5*c) + 6*(3*c^2 + 4)*f^2*cos(4*d*x + 4
*c) - (-72*I*c^2 - 96*I)*f^2*cos(3*d*x + 3*c) - 6*(3*c^2 + 4)*f^2*cos(2*d*x + 2*c) - (-36*I*c^2 - 48*I)*f^2*co
s(d*x + c) - (-18*I*c^2 - 24*I)*f^2*sin(6*d*x + 6*c) - 12*(3*c^2 + 4)*f^2*sin(5*d*x + 5*c) - (-18*I*c^2 - 24*I
)*f^2*sin(4*d*x + 4*c) - 24*(3*c^2 + 4)*f^2*sin(3*d*x + 3*c) - (18*I*c^2 + 24*I)*f^2*sin(2*d*x + 2*c) - 12*(3*
c^2 + 4)*f^2*sin(d*x + c) - 6*(3*c^2 + 4)*f^2)*arctan2(sin(d*x + c) - 1, cos(d*x + c)) - (18*(d*x + c)^2*f^2 +
36*(d*e*f - c*f^2)*(d*x + c) - 18*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*cos(6*d*x + 6*c) + (-36*I*(
d*x + c)^2*f^2 + (-72*I*d*e*f + 72*I*c*f^2)*(d*x + c))*cos(5*d*x + 5*c) - 18*((d*x + c)^2*f^2 + 2*(d*e*f - c*f
^2)*(d*x + c))*cos(4*d*x + 4*c) + (-72*I*(d*x + c)^2*f^2 + (-144*I*d*e*f + 144*I*c*f^2)*(d*x + c))*cos(3*d*x +
3*c) + 18*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*cos(2*d*x + 2*c) + (-36*I*(d*x + c)^2*f^2 + (-72*I*
d*e*f + 72*I*c*f^2)*(d*x + c))*cos(d*x + c) + (-18*I*(d*x + c)^2*f^2 + (-36*I*d*e*f + 36*I*c*f^2)*(d*x + c))*s
in(6*d*x + 6*c) + 36*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*sin(5*d*x + 5*c) + (-18*I*(d*x + c)^2*f^2
+ (-36*I*d*e*f + 36*I*c*f^2)*(d*x + c))*sin(4*d*x + 4*c) + 72*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))
*sin(3*d*x + 3*c) + (18*I*(d*x + c)^2*f^2 + (36*I*d*e*f - 36*I*c*f^2)*(d*x + c))*sin(2*d*x + 2*c) + 36*((d*x +
c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) - (18*(d*x + c)
^2*f^2 + 36*(d*e*f - c*f^2)*(d*x + c) - 18*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*cos(6*d*x + 6*c) +
(-36*I*(d*x + c)^2*f^2 + (-72*I*d*e*f + 72*I*c*f^2)*(d*x + c))*cos(5*d*x + 5*c) - 18*((d*x + c)^2*f^2 + 2*(d*e
*f - c*f^2)*(d*x + c))*cos(4*d*x + 4*c) + (-72*I*(d*x + c)^2*f^2 + (-144*I*d*e*f + 144*I*c*f^2)*(d*x + c))*cos
(3*d*x + 3*c) + 18*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*cos(2*d*x + 2*c) + (-36*I*(d*x + c)^2*f^2 +
(-72*I*d*e*f + 72*I*c*f^2)*(d*x + c))*cos(d*x + c) + (-18*I*(d*x + c)^2*f^2 + (-36*I*d*e*f + 36*I*c*f^2)*(d*x
+ c))*sin(6*d*x + 6*c) + 36*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*sin(5*d*x + 5*c) + (-18*I*(d*x +
c)^2*f^2 + (-36*I*d*e*f + 36*I*c*f^2)*(d*x + c))*sin(4*d*x + 4*c) + 72*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d
*x + c))*sin(3*d*x + 3*c) + (18*I*(d*x + c)^2*f^2 + (36*I*d*e*f - 36*I*c*f^2)*(d*x + c))*sin(2*d*x + 2*c) + 36
*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), -sin(d*x + c) + 1) + (36*
(d*x + c)^2*f^2 - 72*I*d*e*f + 4*(9*c^2 + 18*I*c + 2)*f^2 + (72*d*e*f - (72*c + 8*I)*f^2)*(d*x + c))*cos(5*d*x
+ 5*c) - (-72*I*(d*x + c)^2*f^2 - 144*d*e*f + (-72*I*c^2 + 144*c)*f^2 - 16*(9*I*d*e*f + (-9*I*c + 11)*f^2)*(d
*x + c))*cos(4*d*x + 4*c) + (24*(d*x + c)^2*f^2 - 64*I*d*e*f + 8*(3*c^2 + 8*I*c + 2)*f^2 + (48*d*e*f - (48*c -
64*I)*f^2)*(d*x + c))*cos(3*d*x + 3*c) - (72*I*(d*x + c)^2*f^2 - 176*d*e*f + (72*I*c^2 + 176*c)*f^2 - 144*(-I
*d*e*f + (I*c + 1)*f^2)*(d*x + c))*cos(2*d*x + 2*c) + (36*(d*x + c)^2*f^2 + 8*I*d*e*f + 4*(9*c^2 - 2*I*c + 2)*
f^2 + (72*d*e*f - (72*c - 72*I)*f^2)*(d*x + c))*cos(d*x + c) - (36*d*e*f + 36*(d*x + c)*f^2 - 36*c*f^2 - 36*(d
*e*f + (d*x + c)*f^2 - c*f^2)*cos(6*d*x + 6*c) + (-72*I*d*e*f - 72*I*(d*x + c)*f^2 + 72*I*c*f^2)*cos(5*d*x + 5
*c) - 36*(d*e*f + (d*x + c)*f^2 - c*f^2)*cos(4*d*x + 4*c) + (-144*I*d*e*f - 144*I*(d*x + c)*f^2 + 144*I*c*f^2)
*cos(3*d*x + 3*c) + 36*(d*e*f + (d*x + c)*f^2 - c*f^2)*cos(2*d*x + 2*c) + (-72*I*d*e*f - 72*I*(d*x + c)*f^2 +
72*I*c*f^2)*cos(d*x + c) + (-36*I*d*e*f - 36*I*(d*x + c)*f^2 + 36*I*c*f^2)*sin(6*d*x + 6*c) + 72*(d*e*f + (d*x
+ c)*f^2 - c*f^2)*sin(5*d*x + 5*c) + (-36*I*d*e*f - 36*I*(d*x + c)*f^2 + 36*I*c*f^2)*sin(4*d*x + 4*c) + 144*(
d*e*f + (d*x + c)*f^2 - c*f^2)*sin(3*d*x + 3*c) + (36*I*d*e*f + 36*I*(d*x + c)*f^2 - 36*I*c*f^2)*sin(2*d*x + 2
*c) + 72*(d*e*f + (d*x + c)*f^2 - c*f^2)*sin(d*x + c))*dilog(I*e^(I*d*x + I*c)) + (36*d*e*f + 36*(d*x + c)*f^2
- 36*c*f^2 - 36*(d*e*f + (d*x + c)*f^2 - c*f^2)*cos(6*d*x + 6*c) - (72*I*d*e*f + 72*I*(d*x + c)*f^2 - 72*I*c*
f^2)*cos(5*d*x + 5*c) - 36*(d*e*f + (d*x + c)*f^2 - c*f^2)*cos(4*d*x + 4*c) - (144*I*d*e*f + 144*I*(d*x + c)*f
^2 - 144*I*c*f^2)*cos(3*d*x + 3*c) + 36*(d*e*f + (d*x + c)*f^2 - c*f^2)*cos(2*d*x + 2*c) - (72*I*d*e*f + 72*I*
(d*x + c)*f^2 - 72*I*c*f^2)*cos(d*x + c) - (36*I*d*e*f + 36*I*(d*x + c)*f^2 - 36*I*c*f^2)*sin(6*d*x + 6*c) + 7
2*(d*e*f + (d*x + c)*f^2 - c*f^2)*sin(5*d*x + 5*c) - (36*I*d*e*f + 36*I*(d*x + c)*f^2 - 36*I*c*f^2)*sin(4*d*x
+ 4*c) + 144*(d*e*f + (d*x + c)*f^2 - c*f^2)*sin(3*d*x + 3*c) - (-36*I*d*e*f - 36*I*(d*x + c)*f^2 + 36*I*c*f^2
)*sin(2*d*x + 2*c) + 72*(d*e*f + (d*x + c)*f^2 - c*f^2)*sin(d*x + c))*dilog(-I*e^(I*d*x + I*c)) - (9*I*(d*x +
c)^2*f^2 + (9*I*c^2 + 28*I)*f^2 + (18*I*d*e*f - 18*I*c*f^2)*(d*x + c) + (-9*I*(d*x + c)^2*f^2 + (-9*I*c^2 - 28
*I)*f^2 + (-18*I*d*e*f + 18*I*c*f^2)*(d*x + c))*cos(6*d*x + 6*c) + 2*(9*(d*x + c)^2*f^2 + (9*c^2 + 28)*f^2 + 1
8*(d*e*f - c*f^2)*(d*x + c))*cos(5*d*x + 5*c) + (-9*I*(d*x + c)^2*f^2 + (-9*I*c^2 - 28*I)*f^2 + (-18*I*d*e*f +
18*I*c*f^2)*(d*x + c))*cos(4*d*x + 4*c) + 4*(9*(d*x + c)^2*f^2 + (9*c^2 + 28)*f^2 + 18*(d*e*f - c*f^2)*(d*x +
c))*cos(3*d*x + 3*c) + (9*I*(d*x + c)^2*f^2 + (9*I*c^2 + 28*I)*f^2 + (18*I*d*e*f - 18*I*c*f^2)*(d*x + c))*cos
(2*d*x + 2*c) + 2*(9*(d*x + c)^2*f^2 + (9*c^2 + 28)*f^2 + 18*(d*e*f - c*f^2)*(d*x + c))*cos(d*x + c) + (9*(d*x
+ c)^2*f^2 + (9*c^2 + 28)*f^2 + 18*(d*e*f - c*f^2)*(d*x + c))*sin(6*d*x + 6*c) + (18*I*(d*x + c)^2*f^2 + (18*
I*c^2 + 56*I)*f^2 + (36*I*d*e*f - 36*I*c*f^2)*(d*x + c))*sin(5*d*x + 5*c) + (9*(d*x + c)^2*f^2 + (9*c^2 + 28)*
f^2 + 18*(d*e*f - c*f^2)*(d*x + c))*sin(4*d*x + 4*c) + (36*I*(d*x + c)^2*f^2 + (36*I*c^2 + 112*I)*f^2 + (72*I*
d*e*f - 72*I*c*f^2)*(d*x + c))*sin(3*d*x + 3*c) - (9*(d*x + c)^2*f^2 + (9*c^2 + 28)*f^2 + 18*(d*e*f - c*f^2)*(
d*x + c))*sin(2*d*x + 2*c) + (18*I*(d*x + c)^2*f^2 + (18*I*c^2 + 56*I)*f^2 + (36*I*d*e*f - 36*I*c*f^2)*(d*x +
c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) - (-9*I*(d*x + c)^2*f^2 + (-9*I*c^
2 - 12*I)*f^2 + (-18*I*d*e*f + 18*I*c*f^2)*(d*x + c) + (9*I*(d*x + c)^2*f^2 + (9*I*c^2 + 12*I)*f^2 + (18*I*d*e
*f - 18*I*c*f^2)*(d*x + c))*cos(6*d*x + 6*c) - 6*(3*(d*x + c)^2*f^2 + (3*c^2 + 4)*f^2 + 6*(d*e*f - c*f^2)*(d*x
+ c))*cos(5*d*x + 5*c) + (9*I*(d*x + c)^2*f^2 + (9*I*c^2 + 12*I)*f^2 + (18*I*d*e*f - 18*I*c*f^2)*(d*x + c))*c
os(4*d*x + 4*c) - 12*(3*(d*x + c)^2*f^2 + (3*c^2 + 4)*f^2 + 6*(d*e*f - c*f^2)*(d*x + c))*cos(3*d*x + 3*c) + (-
9*I*(d*x + c)^2*f^2 + (-9*I*c^2 - 12*I)*f^2 + (-18*I*d*e*f + 18*I*c*f^2)*(d*x + c))*cos(2*d*x + 2*c) - 6*(3*(d
*x + c)^2*f^2 + (3*c^2 + 4)*f^2 + 6*(d*e*f - c*f^2)*(d*x + c))*cos(d*x + c) - 3*(3*(d*x + c)^2*f^2 + (3*c^2 +
4)*f^2 + 6*(d*e*f - c*f^2)*(d*x + c))*sin(6*d*x + 6*c) + (-18*I*(d*x + c)^2*f^2 + (-18*I*c^2 - 24*I)*f^2 + (-3
6*I*d*e*f + 36*I*c*f^2)*(d*x + c))*sin(5*d*x + 5*c) - 3*(3*(d*x + c)^2*f^2 + (3*c^2 + 4)*f^2 + 6*(d*e*f - c*f^
2)*(d*x + c))*sin(4*d*x + 4*c) + (-36*I*(d*x + c)^2*f^2 + (-36*I*c^2 - 48*I)*f^2 + (-72*I*d*e*f + 72*I*c*f^2)*
(d*x + c))*sin(3*d*x + 3*c) + 3*(3*(d*x + c)^2*f^2 + (3*c^2 + 4)*f^2 + 6*(d*e*f - c*f^2)*(d*x + c))*sin(2*d*x
+ 2*c) + (-18*I*(d*x + c)^2*f^2 + (-18*I*c^2 - 24*I)*f^2 + (-36*I*d*e*f + 36*I*c*f^2)*(d*x + c))*sin(d*x + c))
*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) - (-36*I*f^2*cos(6*d*x + 6*c) + 72*f^2*cos(5*d*x +
5*c) - 36*I*f^2*cos(4*d*x + 4*c) + 144*f^2*cos(3*d*x + 3*c) + 36*I*f^2*cos(2*d*x + 2*c) + 72*f^2*cos(d*x + c)
+ 36*f^2*sin(6*d*x + 6*c) + 72*I*f^2*sin(5*d*x + 5*c) + 36*f^2*sin(4*d*x + 4*c) + 144*I*f^2*sin(3*d*x + 3*c) -
36*f^2*sin(2*d*x + 2*c) + 72*I*f^2*sin(d*x + c) + 36*I*f^2)*polylog(3, I*e^(I*d*x + I*c)) - (36*I*f^2*cos(6*d
*x + 6*c) - 72*f^2*cos(5*d*x + 5*c) + 36*I*f^2*cos(4*d*x + 4*c) - 144*f^2*cos(3*d*x + 3*c) - 36*I*f^2*cos(2*d*
x + 2*c) - 72*f^2*cos(d*x + c) - 36*f^2*sin(6*d*x + 6*c) - 72*I*f^2*sin(5*d*x + 5*c) - 36*f^2*sin(4*d*x + 4*c)
- 144*I*f^2*sin(3*d*x + 3*c) + 36*f^2*sin(2*d*x + 2*c) - 72*I*f^2*sin(d*x + c) - 36*I*f^2)*polylog(3, -I*e^(I
*d*x + I*c)) - (-36*I*(d*x + c)^2*f^2 - 72*d*e*f + (-36*I*c^2 + 72*c - 8*I)*f^2 - 8*(9*I*d*e*f + (-9*I*c + 1)*
f^2)*(d*x + c))*sin(5*d*x + 5*c) - (72*(d*x + c)^2*f^2 - 144*I*d*e*f + 72*(c^2 + 2*I*c)*f^2 + (144*d*e*f - (14
4*c + 176*I)*f^2)*(d*x + c))*sin(4*d*x + 4*c) - (-24*I*(d*x + c)^2*f^2 - 64*d*e*f + (-24*I*c^2 + 64*c - 16*I)*
f^2 - 16*(3*I*d*e*f + (-3*I*c - 4)*f^2)*(d*x + c))*sin(3*d*x + 3*c) + (72*(d*x + c)^2*f^2 + 176*I*d*e*f + 8*(9
*c^2 - 22*I*c)*f^2 + (144*d*e*f - (144*c - 144*I)*f^2)*(d*x + c))*sin(2*d*x + 2*c) - (-36*I*(d*x + c)^2*f^2 +
8*d*e*f + (-36*I*c^2 - 8*c - 8*I)*f^2 - 72*(I*d*e*f + (-I*c - 1)*f^2)*(d*x + c))*sin(d*x + c))/(-48*I*a*d^2*co
s(6*d*x + 6*c) + 96*a*d^2*cos(5*d*x + 5*c) - 48*I*a*d^2*cos(4*d*x + 4*c) + 192*a*d^2*cos(3*d*x + 3*c) + 48*I*a
*d^2*cos(2*d*x + 2*c) + 96*a*d^2*cos(d*x + c) + 48*a*d^2*sin(6*d*x + 6*c) + 96*I*a*d^2*sin(5*d*x + 5*c) + 48*a
*d^2*sin(4*d*x + 4*c) + 192*I*a*d^2*sin(3*d*x + 3*c) - 48*a*d^2*sin(2*d*x + 2*c) + 96*I*a*d^2*sin(d*x + c) + 4
8*I*a*d^2))/d

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Fricas [C]  time = 3.18991, size = 3775, normalized size = 8.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sec(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/48*(6*d^2*f^2*x^2 + 12*d^2*e*f*x + 6*d^2*e^2 - 16*(d*f^2*x + d*e*f)*cos(d*x + c)^3 - 2*(9*d^2*f^2*x^2 + 18*d
^2*e*f*x + 9*d^2*e^2 + 2*f^2)*cos(d*x + c)^2 - 28*(d*f^2*x + d*e*f)*cos(d*x + c) + ((-18*I*d*f^2*x - 18*I*d*e*
f)*cos(d*x + c)^2*sin(d*x + c) + (-18*I*d*f^2*x - 18*I*d*e*f)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) + sin(d*x +
c)) + ((-18*I*d*f^2*x - 18*I*d*e*f)*cos(d*x + c)^2*sin(d*x + c) + (-18*I*d*f^2*x - 18*I*d*e*f)*cos(d*x + c)^2
)*dilog(I*cos(d*x + c) - sin(d*x + c)) + ((18*I*d*f^2*x + 18*I*d*e*f)*cos(d*x + c)^2*sin(d*x + c) + (18*I*d*f^
2*x + 18*I*d*e*f)*cos(d*x + c)^2)*dilog(-I*cos(d*x + c) + sin(d*x + c)) + ((18*I*d*f^2*x + 18*I*d*e*f)*cos(d*x
+ c)^2*sin(d*x + c) + (18*I*d*f^2*x + 18*I*d*e*f)*cos(d*x + c)^2)*dilog(-I*cos(d*x + c) - sin(d*x + c)) + ((9
*d^2*e^2 - 18*c*d*e*f + (9*c^2 + 28)*f^2)*cos(d*x + c)^2*sin(d*x + c) + (9*d^2*e^2 - 18*c*d*e*f + (9*c^2 + 28)
*f^2)*cos(d*x + c)^2)*log(cos(d*x + c) + I*sin(d*x + c) + I) - 3*((3*d^2*e^2 - 6*c*d*e*f + (3*c^2 + 4)*f^2)*co
s(d*x + c)^2*sin(d*x + c) + (3*d^2*e^2 - 6*c*d*e*f + (3*c^2 + 4)*f^2)*cos(d*x + c)^2)*log(cos(d*x + c) - I*sin
(d*x + c) + I) + 9*((d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*cos(d*x + c)^2*sin(d*x + c) + (d^2*f^2*x
^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*cos(d*x + c)^2)*log(I*cos(d*x + c) + sin(d*x + c) + 1) - 9*((d^2*f^2*x
^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*cos(d*x + c)^2*sin(d*x + c) + (d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f -
c^2*f^2)*cos(d*x + c)^2)*log(I*cos(d*x + c) - sin(d*x + c) + 1) + 9*((d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f -
c^2*f^2)*cos(d*x + c)^2*sin(d*x + c) + (d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*cos(d*x + c)^2)*log(
-I*cos(d*x + c) + sin(d*x + c) + 1) - 9*((d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*cos(d*x + c)^2*sin(
d*x + c) + (d^2*f^2*x^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*cos(d*x + c)^2)*log(-I*cos(d*x + c) - sin(d*x + c
) + 1) + ((9*d^2*e^2 - 18*c*d*e*f + (9*c^2 + 28)*f^2)*cos(d*x + c)^2*sin(d*x + c) + (9*d^2*e^2 - 18*c*d*e*f +
(9*c^2 + 28)*f^2)*cos(d*x + c)^2)*log(-cos(d*x + c) + I*sin(d*x + c) + I) - 3*((3*d^2*e^2 - 6*c*d*e*f + (3*c^2
+ 4)*f^2)*cos(d*x + c)^2*sin(d*x + c) + (3*d^2*e^2 - 6*c*d*e*f + (3*c^2 + 4)*f^2)*cos(d*x + c)^2)*log(-cos(d*
x + c) - I*sin(d*x + c) + I) - 18*(f^2*cos(d*x + c)^2*sin(d*x + c) + f^2*cos(d*x + c)^2)*polylog(3, I*cos(d*x
+ c) + sin(d*x + c)) + 18*(f^2*cos(d*x + c)^2*sin(d*x + c) + f^2*cos(d*x + c)^2)*polylog(3, I*cos(d*x + c) - s
in(d*x + c)) - 18*(f^2*cos(d*x + c)^2*sin(d*x + c) + f^2*cos(d*x + c)^2)*polylog(3, -I*cos(d*x + c) + sin(d*x
+ c)) + 18*(f^2*cos(d*x + c)^2*sin(d*x + c) + f^2*cos(d*x + c)^2)*polylog(3, -I*cos(d*x + c) - sin(d*x + c)) +
2*(9*d^2*f^2*x^2 + 18*d^2*e*f*x + 9*d^2*e^2 - 10*(d*f^2*x + d*e*f)*cos(d*x + c))*sin(d*x + c))/(a*d^3*cos(d*x
+ c)^2*sin(d*x + c) + a*d^3*cos(d*x + c)^2)

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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((f*x+e)**2*sec(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} \sec \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*sec(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*sec(d*x + c)^3/(a*sin(d*x + c) + a), x)