\(\int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [275]

Optimal result

Integrand size = 28, antiderivative size = 475 \[ \int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 i (e+f x)^3}{3 a d}-\frac {i f (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{a d^2}+\frac {f^3 \text {arctanh}(\sin (c+d x))}{a d^4}+\frac {2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}+\frac {f^3 \log (\cos (c+d x))}{a d^4}+\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{a d^3}-\frac {f^3 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{a d^4}-\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}+\frac {f (e+f x)^2 \sec (c+d x) \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d} \]

[Out]

-2/3*I*(f*x+e)^3/a/d-I*f^2*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^3+f^3*arctanh(sin(d*x+c))/a/d^4+2*f*(f*x+e)
^2*ln(1+exp(2*I*(d*x+c)))/a/d^2+f^3*ln(cos(d*x+c))/a/d^4-I*f*(f*x+e)^2*arctan(exp(I*(d*x+c)))/a/d^2-2*I*f^2*(f
*x+e)*polylog(2,-exp(2*I*(d*x+c)))/a/d^3+I*f^2*(f*x+e)*polylog(2,-I*exp(I*(d*x+c)))/a/d^3-f^3*polylog(3,-I*exp
(I*(d*x+c)))/a/d^4+f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+f^3*polylog(3,-exp(2*I*(d*x+c)))/a/d^4-f^2*(f*x+e)*se
c(d*x+c)/a/d^3-1/2*f*(f*x+e)^2*sec(d*x+c)^2/a/d^2-1/3*(f*x+e)^3*sec(d*x+c)^3/a/d+f^2*(f*x+e)*tan(d*x+c)/a/d^3+
2/3*(f*x+e)^3*tan(d*x+c)/a/d+1/2*f*(f*x+e)^2*sec(d*x+c)*tan(d*x+c)/a/d^2+1/3*(f*x+e)^3*sec(d*x+c)^2*tan(d*x+c)
/a/d

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4627, 4271, 4269, 3556, 3800, 2221, 2611, 2320, 6724, 4494, 3855, 4266} \[ \int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {i f (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{a d^2}+\frac {f^3 \text {arctanh}(\sin (c+d x))}{a d^4}-\frac {f^3 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{a d^4}+\frac {f^3 \log (\cos (c+d x))}{a d^4}+\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{a d^3}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}-\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}+\frac {2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}+\frac {f (e+f x)^2 \tan (c+d x) \sec (c+d x)}{2 a d^2}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {(e+f x)^3 \tan (c+d x) \sec ^2(c+d x)}{3 a d}-\frac {2 i (e+f x)^3}{3 a d} \]

[In]

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

[Out]

(((-2*I)/3)*(e + f*x)^3)/(a*d) - (I*f*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(a*d^2) + (f^3*ArcTanh[Sin[c + d*x]
])/(a*d^4) + (2*f*(e + f*x)^2*Log[1 + E^((2*I)*(c + d*x))])/(a*d^2) + (f^3*Log[Cos[c + d*x]])/(a*d^4) + (I*f^2
*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^3) - (I*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3)
- ((2*I)*f^2*(e + f*x)*PolyLog[2, -E^((2*I)*(c + d*x))])/(a*d^3) - (f^3*PolyLog[3, (-I)*E^(I*(c + d*x))])/(a*
d^4) + (f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (f^3*PolyLog[3, -E^((2*I)*(c + d*x))])/(a*d^4) - (f^2*(e
+ f*x)*Sec[c + d*x])/(a*d^3) - (f*(e + f*x)^2*Sec[c + d*x]^2)/(2*a*d^2) - ((e + f*x)^3*Sec[c + d*x]^3)/(3*a*d)
+ (f^2*(e + f*x)*Tan[c + d*x])/(a*d^3) + (2*(e + f*x)^3*Tan[c + d*x])/(3*a*d) + (f*(e + f*x)^2*Sec[c + d*x]*T
an[c + d*x])/(2*a*d^2) + ((e + f*x)^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)

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 3556

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

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 3855

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

Rule 4266

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 4269

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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4494

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 4627

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \sec ^4(c+d x) \, dx}{a}-\frac {\int (e+f x)^3 \sec ^3(c+d x) \tan (c+d x) \, dx}{a} \\ & = -\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {2 \int (e+f x)^3 \sec ^2(c+d x) \, dx}{3 a}+\frac {f \int (e+f x)^2 \sec ^3(c+d x) \, dx}{a d}+\frac {f^2 \int (e+f x) \sec ^2(c+d x) \, dx}{a d^2} \\ & = -\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}+\frac {f (e+f x)^2 \sec (c+d x) \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {f \int (e+f x)^2 \sec (c+d x) \, dx}{2 a d}-\frac {(2 f) \int (e+f x)^2 \tan (c+d x) \, dx}{a d}+\frac {f^3 \int \sec (c+d x) \, dx}{a d^3}-\frac {f^3 \int \tan (c+d x) \, dx}{a d^3} \\ & = -\frac {2 i (e+f x)^3}{3 a d}-\frac {i f (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{a d^2}+\frac {f^3 \text {arctanh}(\sin (c+d x))}{a d^4}+\frac {f^3 \log (\cos (c+d x))}{a d^4}-\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}+\frac {f (e+f x)^2 \sec (c+d x) \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {(4 i f) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{1+e^{2 i (c+d x)}} \, dx}{a d}-\frac {f^2 \int (e+f x) \log \left (1-i e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {f^2 \int (e+f x) \log \left (1+i e^{i (c+d x)}\right ) \, dx}{a d^2} \\ & = -\frac {2 i (e+f x)^3}{3 a d}-\frac {i f (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{a d^2}+\frac {f^3 \text {arctanh}(\sin (c+d x))}{a d^4}+\frac {2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}+\frac {f^3 \log (\cos (c+d x))}{a d^4}+\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}+\frac {f (e+f x)^2 \sec (c+d x) \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d}-\frac {\left (4 f^2\right ) \int (e+f x) \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (i f^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (i f^3\right ) \int \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) \, dx}{a d^3} \\ & = -\frac {2 i (e+f x)^3}{3 a d}-\frac {i f (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{a d^2}+\frac {f^3 \text {arctanh}(\sin (c+d x))}{a d^4}+\frac {2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}+\frac {f^3 \log (\cos (c+d x))}{a d^4}+\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{a d^3}-\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}+\frac {f (e+f x)^2 \sec (c+d x) \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d}-\frac {f^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (2 i f^3\right ) \int \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right ) \, dx}{a d^3} \\ & = -\frac {2 i (e+f x)^3}{3 a d}-\frac {i f (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{a d^2}+\frac {f^3 \text {arctanh}(\sin (c+d x))}{a d^4}+\frac {2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}+\frac {f^3 \log (\cos (c+d x))}{a d^4}+\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{a d^3}-\frac {f^3 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}+\frac {f (e+f x)^2 \sec (c+d x) \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {f^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{a d^4} \\ & = -\frac {2 i (e+f x)^3}{3 a d}-\frac {i f (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{a d^2}+\frac {f^3 \text {arctanh}(\sin (c+d x))}{a d^4}+\frac {2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}+\frac {f^3 \log (\cos (c+d x))}{a d^4}+\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac {i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{a d^3}-\frac {f^3 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {f^3 \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{a d^4}-\frac {f^2 (e+f x) \sec (c+d x)}{a d^3}-\frac {f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac {f^2 (e+f x) \tan (c+d x)}{a d^3}+\frac {2 (e+f x)^3 \tan (c+d x)}{3 a d}+\frac {f (e+f x)^2 \sec (c+d x) \tan (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \sec ^2(c+d x) \tan (c+d x)}{3 a d} \\ \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. \(1173\) vs. \(2(475)=950\).

Time = 8.38 (sec) , antiderivative size = 1173, normalized size of antiderivative = 2.47 \[ \int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {d^3 (e+f x)^3}{-i+e^{i c}}+3 d^2 f (e+f x)^2 \log \left (1-i e^{-i (c+d x)}\right )+6 f^2 \left (i d (e+f x) \operatorname {PolyLog}\left (2,i e^{-i (c+d x)}\right )+f \operatorname {PolyLog}\left (3,i e^{-i (c+d x)}\right )\right )}{2 a d^4}-\frac {f (\cos (c)+i \sin (c)) \left (5 d^2 e^2 x \cos (c)+4 f^2 x \cos (c)+5 d^2 e f x^2 \cos (c)+\frac {5}{3} d^2 f^2 x^3 (\cos (c)-i \sin (c))-5 i d^2 e^2 x \sin (c)-4 i f^2 x \sin (c)-5 i d^2 e f x^2 \sin (c)+10 e f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))+10 f^2 x \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))-10 d e f x \log (1+i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))-5 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)))-\frac {\left (5 d^2 e^2+4 f^2\right ) \log (\cos (c+d x)+i (1+\sin (c+d x))) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))}{d}-\frac {10 f^2 \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))}{d}+\left (5 d^2 e^2+4 f^2\right ) x (i \cos (c)+\sin (c)) (\cos (c)+i (1+\sin (c)))\right )}{2 a d^3 (\cos (c)+i (1+\sin (c)))}+\frac {e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )}{2 a d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )}{3 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {-d e^3 \cos \left (\frac {c}{2}\right )-3 e^2 f \cos \left (\frac {c}{2}\right )-3 d e^2 f x \cos \left (\frac {c}{2}\right )-6 e f^2 x \cos \left (\frac {c}{2}\right )-3 d e f^2 x^2 \cos \left (\frac {c}{2}\right )-3 f^3 x^2 \cos \left (\frac {c}{2}\right )-d f^3 x^3 \cos \left (\frac {c}{2}\right )+d e^3 \sin \left (\frac {c}{2}\right )-3 e^2 f \sin \left (\frac {c}{2}\right )+3 d e^2 f x \sin \left (\frac {c}{2}\right )-6 e f^2 x \sin \left (\frac {c}{2}\right )+3 d e f^2 x^2 \sin \left (\frac {c}{2}\right )-3 f^3 x^2 \sin \left (\frac {c}{2}\right )+d f^3 x^3 \sin \left (\frac {c}{2}\right )}{6 a d^2 \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {5 d^2 e^3 \sin \left (\frac {d x}{2}\right )+12 e f^2 \sin \left (\frac {d x}{2}\right )+15 d^2 e^2 f x \sin \left (\frac {d x}{2}\right )+12 f^3 x \sin \left (\frac {d x}{2}\right )+15 d^2 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+5 d^2 f^3 x^3 \sin \left (\frac {d x}{2}\right )}{6 a d^3 \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]

[In]

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

[Out]

((d^3*(e + f*x)^3)/(-I + E^(I*c)) + 3*d^2*f*(e + f*x)^2*Log[1 - I/E^(I*(c + d*x))] + 6*f^2*(I*d*(e + f*x)*Poly
Log[2, I/E^(I*(c + d*x))] + f*PolyLog[3, I/E^(I*(c + d*x))]))/(2*a*d^4) - (f*(Cos[c] + I*Sin[c])*(5*d^2*e^2*x*
Cos[c] + 4*f^2*x*Cos[c] + 5*d^2*e*f*x^2*Cos[c] + (5*d^2*f^2*x^3*(Cos[c] - I*Sin[c]))/3 - (5*I)*d^2*e^2*x*Sin[c
] - (4*I)*f^2*x*Sin[c] - (5*I)*d^2*e*f*x^2*Sin[c] + 10*e*f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c
] - I*(1 + Sin[c])) + 10*f^2*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])) - 10*d*e
*f*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - 5*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])) - ((5*d^2*e^2 + 4*f^2)*Log[Cos[
c + d*x] + I*(1 + Sin[c + d*x])]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d - (10*f^2*PolyLog[3, (-I)*Co
s[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d + (5*d^2*e^2 + 4*f^2)*x*(I*Cos[c]
+ Sin[c])*(Cos[c] + I*(1 + Sin[c]))))/(2*a*d^3*(Cos[c] + I*(1 + Sin[c]))) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[
(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2])/(2*a*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] -
Sin[c/2 + (d*x)/2])) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d
*x)/2])/(3*a*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (-(d*e^3*Cos[c/2]) - 3*e^2
*f*Cos[c/2] - 3*d*e^2*f*x*Cos[c/2] - 6*e*f^2*x*Cos[c/2] - 3*d*e*f^2*x^2*Cos[c/2] - 3*f^3*x^2*Cos[c/2] - d*f^3*
x^3*Cos[c/2] + d*e^3*Sin[c/2] - 3*e^2*f*Sin[c/2] + 3*d*e^2*f*x*Sin[c/2] - 6*e*f^2*x*Sin[c/2] + 3*d*e*f^2*x^2*S
in[c/2] - 3*f^3*x^2*Sin[c/2] + d*f^3*x^3*Sin[c/2])/(6*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/
2 + (d*x)/2])^2) + (5*d^2*e^3*Sin[(d*x)/2] + 12*e*f^2*Sin[(d*x)/2] + 15*d^2*e^2*f*x*Sin[(d*x)/2] + 12*f^3*x*Si
n[(d*x)/2] + 15*d^2*e*f^2*x^2*Sin[(d*x)/2] + 5*d^2*f^3*x^3*Sin[(d*x)/2])/(6*a*d^3*(Cos[c/2] + Sin[c/2])*(Cos[c
/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (438 ) = 876\).

Time = 0.86 (sec) , antiderivative size = 1135, normalized size of antiderivative = 2.39

method	result	size
risch	\(\text {Expression too large to display}\)	\(1135\)

[In]

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

[Out]

5/a/d^3*f^2*e*ln(1-I*exp(I*(d*x+c)))*c+5/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x+8/a/d^3*e*f^2*c*ln(exp(I*(d*x+c)
))-4/a/d^3*c*f^2*e*ln(1+exp(2*I*(d*x+c)))+3/2/a/d^2*f^3*ln(1+I*exp(I*(d*x+c)))*x^2-4/3*I/a/d*x^3*f^3-2*I/a/d^4
*f^3*arctan(exp(I*(d*x+c)))+8/3*I/a/d^4*c^3*f^3+3/a/d^2*e*f^2*ln(1+I*exp(I*(d*x+c)))*x+3/a/d^3*e*f^2*ln(1+I*ex
p(I*(d*x+c)))*c-I/a/d^2*e^2*f*arctan(exp(I*(d*x+c)))-5*I/a/d^3*e*f^2*polylog(2,I*exp(I*(d*x+c)))-3*I/a/d^3*e*f
^2*polylog(2,-I*exp(I*(d*x+c)))-4*I/a/d^3*e*f^2*c^2-I/a/d^4*f^3*c^2*arctan(exp(I*(d*x+c)))+4*I/a/d^3*c^2*f^3*x
-5*I/a/d^3*f^3*polylog(2,I*exp(I*(d*x+c)))*x-3*I/a/d^3*f^3*polylog(2,-I*exp(I*(d*x+c)))*x-4*I/a/d*e*f^2*x^2-3/
2/a/d^4*c^2*f^3*ln(1+I*exp(I*(d*x+c)))+2*I/a/d^3*e*f^2*c*arctan(exp(I*(d*x+c)))-8*I/a/d^2*e*f^2*c*x+1/a/d^4*f^
3*ln(1+exp(2*I*(d*x+c)))-2/a/d^4*f^3*ln(exp(I*(d*x+c)))-1/3*(6*I*f^3*x+6*I*e*f^2*exp(2*I*(d*x+c))+12*I*d^2*e*f
^2*x^2+6*I*d*e*f^2*x*exp(I*(d*x+c))+6*I*d*e*f^2*x*exp(3*I*(d*x+c))+6*I*e*f^2+24*d^2*e*f^2*x^2*exp(I*(d*x+c))+2
4*d^2*e^2*f*x*exp(I*(d*x+c))+3*I*d*f^3*x^2*exp(I*(d*x+c))+3*I*d*f^3*x^2*exp(3*I*(d*x+c))+8*d^2*e^3*exp(I*(d*x+
c))+4*I*d^2*e^3+8*d^2*f^3*x^3*exp(I*(d*x+c))+6*I*f^3*x*exp(2*I*(d*x+c))+3*I*d*e^2*f*exp(3*I*(d*x+c))+4*I*d^2*f
^3*x^3+6*f^3*x*exp(3*I*(d*x+c))+6*e*f^2*exp(3*I*(d*x+c))+6*f^3*x*exp(I*(d*x+c))+6*e*f^2*exp(I*(d*x+c))+12*I*d^
2*e^2*f*x+3*I*d*e^2*f*exp(I*(d*x+c)))/(-I+exp(I*(d*x+c)))/(exp(I*(d*x+c))+I)^3/d^3/a+3*f^3*polylog(3,-I*exp(I*
(d*x+c)))/a/d^4-4/a/d^4*f^3*c^2*ln(exp(I*(d*x+c)))+2/a/d^2*e^2*f*ln(1+exp(2*I*(d*x+c)))+5*f^3*polylog(3,I*exp(
I*(d*x+c)))/a/d^4-5/2/a/d^4*c^2*f^3*ln(1-I*exp(I*(d*x+c)))+2/a/d^4*c^2*f^3*ln(1+exp(2*I*(d*x+c)))+5/2/a/d^2*f^
3*ln(1-I*exp(I*(d*x+c)))*x^2-4/a/d^2*e^2*f*ln(exp(I*(d*x+c)))

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. 1531 vs. \(2 (426) = 852\).

Time = 0.36 (sec) , antiderivative size = 1531, normalized size of antiderivative = 3.22 \[ \int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/12*(4*d^3*f^3*x^3 + 12*d^3*e*f^2*x^2 + 12*d^3*e^2*f*x + 4*d^3*e^3 - 4*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 + 2*d
^3*e^3 + 3*d*e*f^2 + 3*(2*d^3*e^2*f + d*f^3)*x)*cos(d*x + c)^2 - 6*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*c
os(d*x + c) - 18*((-I*d*f^3*x - I*d*e*f^2)*cos(d*x + c)*sin(d*x + c) + (-I*d*f^3*x - I*d*e*f^2)*cos(d*x + c))*
dilog(I*cos(d*x + c) + sin(d*x + c)) - 30*((I*d*f^3*x + I*d*e*f^2)*cos(d*x + c)*sin(d*x + c) + (I*d*f^3*x + I*
d*e*f^2)*cos(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) - 18*((I*d*f^3*x + I*d*e*f^2)*cos(d*x + c)*sin(d*x
+ c) + (I*d*f^3*x + I*d*e*f^2)*cos(d*x + c))*dilog(-I*cos(d*x + c) + sin(d*x + c)) - 30*((-I*d*f^3*x - I*d*e*
f^2)*cos(d*x + c)*sin(d*x + c) + (-I*d*f^3*x - I*d*e*f^2)*cos(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c))
+ 3*((5*d^2*e^2*f - 10*c*d*e*f^2 + (5*c^2 + 4)*f^3)*cos(d*x + c)*sin(d*x + c) + (5*d^2*e^2*f - 10*c*d*e*f^2 +
(5*c^2 + 4)*f^3)*cos(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) + 9*((d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)
*cos(d*x + c)*sin(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c))*log(cos(d*x + c) - I*sin(d*x +
c) + I) + 15*((d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*f^3*x^2 +
2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c))*log(I*cos(d*x + c) + sin(d*x + c) + 1) + 9*((d^2*f^3*x^2
+ 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f
^2 - c^2*f^3)*cos(d*x + c))*log(I*cos(d*x + c) - sin(d*x + c) + 1) + 15*((d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*
e*f^2 - c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c
))*log(-I*cos(d*x + c) + sin(d*x + c) + 1) + 9*((d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x
+ c)*sin(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c))*log(-I*cos(d*x + c) -
sin(d*x + c) + 1) + 3*((5*d^2*e^2*f - 10*c*d*e*f^2 + (5*c^2 + 4)*f^3)*cos(d*x + c)*sin(d*x + c) + (5*d^2*e^2*f
- 10*c*d*e*f^2 + (5*c^2 + 4)*f^3)*cos(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + I) + 9*((d^2*e^2*f - 2*c
*d*e*f^2 + c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c))*log(-cos(d*x
+ c) - I*sin(d*x + c) + I) + 18*(f^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(d*x + c))*polylog(3, I*cos(d*x + c)
+ sin(d*x + c)) + 30*(f^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(d*x + c))*polylog(3, I*cos(d*x + c) - sin(d*x +
c)) + 18*(f^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(d*x + c))*polylog(3, -I*cos(d*x + c) + sin(d*x + c)) + 30*(f
^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(d*x + c))*polylog(3, -I*cos(d*x + c) - sin(d*x + c)) + 8*(d^3*f^3*x^3 +
3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*sin(d*x + c))/(a*d^4*cos(d*x + c)*sin(d*x + c) + a*d^4*cos(d*x + c
))

Sympy [F]

\[ \int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

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

[Out]

(Integral(e**3*sec(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(f**3*x**3*sec(c + d*x)**2/(sin(c + d*x) + 1),
x) + Integral(3*e*f**2*x**2*sec(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*sec(c + d*x)**2/(sin
(c + d*x) + 1), x))/a

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. 5130 vs. \(2 (426) = 852\).

Time = 1.12 (sec) , antiderivative size = 5130, normalized size of antiderivative = 10.80 \[ \int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/12*(24*c^2*e*f^2*(sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 - 1)/(a*d^2 + 2*a*d^2*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*d^2*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 - a*d^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + 6*(4*(8*(d*x + c)*cos(d*x + c) - sin(3*d*x + 3*c) -
sin(d*x + c))*cos(4*d*x + 4*c) + 16*(2*d*x + 4*(d*x + c)*sin(d*x + c) + 2*c + cos(d*x + c))*cos(3*d*x + 3*c)
+ 8*cos(3*d*x + 3*c)^2 + 8*cos(d*x + c)^2 + 5*(2*(2*sin(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) -
cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*c)^2 - 8*cos(3*d*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x
+ 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin
(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 - 4*sin(d*x + c) - 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c)
+ 1) + 3*(2*(2*sin(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*
c)^2 - 8*cos(3*d*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*
c) - sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 -
4*sin(d*x + c) - 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 4*(4*d*x + 8*(d*x + c)*sin(d*x
+ c) + 4*c + cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - 4*(16*(d*x + c)*cos(d*x + c) - 4*sin(d*x + c
) - 1)*sin(3*d*x + 3*c) + 8*sin(3*d*x + 3*c)^2 + 8*sin(d*x + c)^2 + 4*sin(d*x + c))*c*e*f^2/(a*d^2*cos(4*d*x +
4*c)^2 + 4*a*d^2*cos(3*d*x + 3*c)^2 + 8*a*d^2*cos(3*d*x + 3*c)*cos(d*x + c) + 4*a*d^2*cos(d*x + c)^2 + a*d^2*
sin(4*d*x + 4*c)^2 + 4*a*d^2*sin(3*d*x + 3*c)^2 + 4*a*d^2*sin(d*x + c)^2 + 4*a*d^2*sin(d*x + c) + a*d^2 - 2*(2
*a*d^2*sin(3*d*x + 3*c) + 2*a*d^2*sin(d*x + c) + a*d^2)*cos(4*d*x + 4*c) + 4*(a*d^2*cos(3*d*x + 3*c) + a*d^2*c
os(d*x + c))*sin(4*d*x + 4*c) + 4*(2*a*d^2*sin(d*x + c) + a*d^2)*sin(3*d*x + 3*c)) - 24*c*e^2*f*(sin(d*x + c)/
(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)/(a*d +
2*a*d*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*d*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - a*d*sin(d*x + c)^4/(cos(d
*x + c) + 1)^4) - 3*(4*(8*(d*x + c)*cos(d*x + c) - sin(3*d*x + 3*c) - sin(d*x + c))*cos(4*d*x + 4*c) + 16*(2*d
*x + 4*(d*x + c)*sin(d*x + c) + 2*c + cos(d*x + c))*cos(3*d*x + 3*c) + 8*cos(3*d*x + 3*c)^2 + 8*cos(d*x + c)^2
+ 5*(2*(2*sin(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*c)^2
- 8*cos(3*d*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) -
sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 - 4*si
n(d*x + c) - 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 3*(2*(2*sin(3*d*x + 3*c) + 2*sin(d
*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*c)^2 - 8*cos(3*d*x + 3*c)*cos(d*x + c) -
4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x +
c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 - 4*sin(d*x + c) - 1)*log(cos(d*x + c)^2 +
sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 4*(4*d*x + 8*(d*x + c)*sin(d*x + c) + 4*c + cos(3*d*x + 3*c) + cos(d*x
+ c))*sin(4*d*x + 4*c) - 4*(16*(d*x + c)*cos(d*x + c) - 4*sin(d*x + c) - 1)*sin(3*d*x + 3*c) + 8*sin(3*d*x + 3
*c)^2 + 8*sin(d*x + c)^2 + 4*sin(d*x + c))*e^2*f/(a*d*cos(4*d*x + 4*c)^2 + 4*a*d*cos(3*d*x + 3*c)^2 + 8*a*d*co
s(3*d*x + 3*c)*cos(d*x + c) + 4*a*d*cos(d*x + c)^2 + a*d*sin(4*d*x + 4*c)^2 + 4*a*d*sin(3*d*x + 3*c)^2 + 4*a*d
*sin(d*x + c)^2 + 4*a*d*sin(d*x + c) + a*d - 2*(2*a*d*sin(3*d*x + 3*c) + 2*a*d*sin(d*x + c) + a*d)*cos(4*d*x +
4*c) + 4*(a*d*cos(3*d*x + 3*c) + a*d*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(2*a*d*sin(d*x + c) + a*d)*sin(3*d*x
+ 3*c)) + 8*e^3*(sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 1)/(a + 2*a*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - a
*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 12*(24*d*e*f^2 - 8*(2*c^3 + 3*c)*f^3 - 6*((5*c^2 + 4)*f^3*cos(4*d*x +
4*c) - 2*(-5*I*c^2 - 4*I)*f^3*cos(3*d*x + 3*c) - 2*(-5*I*c^2 - 4*I)*f^3*cos(d*x + c) - (-5*I*c^2 - 4*I)*f^3*si
n(4*d*x + 4*c) - 2*(5*c^2 + 4)*f^3*sin(3*d*x + 3*c) - 2*(5*c^2 + 4)*f^3*sin(d*x + c) - (5*c^2 + 4)*f^3)*arctan
2(sin(d*x + c) + 1, cos(d*x + c)) - 18*(c^2*f^3*cos(4*d*x + 4*c) + 2*I*c^2*f^3*cos(3*d*x + 3*c) + 2*I*c^2*f^3*
cos(d*x + c) + I*c^2*f^3*sin(4*d*x + 4*c) - 2*c^2*f^3*sin(3*d*x + 3*c) - 2*c^2*f^3*sin(d*x + c) - c^2*f^3)*arc
tan2(sin(d*x + c) - 1, cos(d*x + c)) - 30*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c) - ((d*x + c)^2*f^3
+ 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(4*d*x + 4*c) - 2*(I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*
cos(3*d*x + 3*c) - 2*(I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*cos(d*x + c) - (I*(d*x + c)^2*f^3
+ 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 2*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*s
in(3*d*x + 3*c) + 2*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(
d*x + c) + 1) + 18*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c) - ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(
d*x + c))*cos(4*d*x + 4*c) + 2*(-I*(d*x + c)^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 2*
(-I*(d*x + c)^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*cos(d*x + c) + (-I*(d*x + c)^2*f^3 + 2*(-I*d*e*f^2 +
I*c*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 2*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(3*d*x + 3*c) +
2*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), -sin(d*x + c) + 1) +
8*(2*(d*x + c)^3*f^3 + 3*(2*c^2 + 1)*(d*x + c)*f^3 + 6*(d*e*f^2 - c*f^3)*(d*x + c)^2)*cos(4*d*x + 4*c) + 4*(8*
I*(d*x + c)^3*f^3 - 6*I*d*e*f^2 + 3*(c^2 + 2*I*c)*f^3 + 3*(8*I*d*e*f^2 + (-8*I*c + 1)*f^3)*(d*x + c)^2 + 6*(d*
e*f^2 + (4*I*c^2 - c + I)*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 24*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(2*d*x +
2*c) + 4*(3*(d*x + c)^2*f^3 - 6*I*d*e*f^2 + (8*I*c^3 + 3*c^2 + 6*I*c)*f^3 + 6*(d*e*f^2 - (c - I)*f^3)*(d*x + c
))*cos(d*x + c) - 60*(d*e*f^2 + (d*x + c)*f^3 - c*f^3 - (d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(4*d*x + 4*c) - 2
*(I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)*cos(3*d*x + 3*c) - 2*(I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)*cos(d*x
+ c) - (I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)*sin(4*d*x + 4*c) + 2*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*sin(3*d*
x + 3*c) + 2*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*sin(d*x + c))*dilog(I*e^(I*d*x + I*c)) - 36*(d*e*f^2 + (d*x + c
)*f^3 - c*f^3 - (d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(4*d*x + 4*c) - 2*(I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)
*cos(3*d*x + 3*c) - 2*(I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)*cos(d*x + c) - (I*d*e*f^2 + I*(d*x + c)*f^3 - I*
c*f^3)*sin(4*d*x + 4*c) + 2*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*sin(3*d*x + 3*c) + 2*(d*e*f^2 + (d*x + c)*f^3 -
c*f^3)*sin(d*x + c))*dilog(-I*e^(I*d*x + I*c)) + 3*(-5*I*(d*x + c)^2*f^3 + (-5*I*c^2 - 4*I)*f^3 + 10*(-I*d*e*f
^2 + I*c*f^3)*(d*x + c) + (5*I*(d*x + c)^2*f^3 + (5*I*c^2 + 4*I)*f^3 + 10*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*cos
(4*d*x + 4*c) - 2*(5*(d*x + c)^2*f^3 + (5*c^2 + 4)*f^3 + 10*(d*e*f^2 - c*f^3)*(d*x + c))*cos(3*d*x + 3*c) - 2*
(5*(d*x + c)^2*f^3 + (5*c^2 + 4)*f^3 + 10*(d*e*f^2 - c*f^3)*(d*x + c))*cos(d*x + c) - (5*(d*x + c)^2*f^3 + (5*
c^2 + 4)*f^3 + 10*(d*e*f^2 - c*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 2*(-5*I*(d*x + c)^2*f^3 + (-5*I*c^2 - 4*I)*f
^3 + 10*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 2*(-5*I*(d*x + c)^2*f^3 + (-5*I*c^2 - 4*I)*f^3 +
10*(-I*d*e*f^2 + I*c*f^3)*(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^3 - I*c^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c) + (I*(d*x + c)^2*f^3 + I*c^2*f^3 + 2*(
I*d*e*f^2 - I*c*f^3)*(d*x + c))*cos(4*d*x + 4*c) - 2*((d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c
))*cos(3*d*x + 3*c) - 2*((d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(d*x + c) - ((d*x + c)^
2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 2*(-I*(d*x + c)^2*f^3 - I*c^2*f^3 + 2*(-I*
d*e*f^2 + I*c*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 2*(-I*(d*x + c)^2*f^3 - I*c^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*
(d*x + c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 60*(I*f^3*cos(4*d*x + 4*c
) - 2*f^3*cos(3*d*x + 3*c) - 2*f^3*cos(d*x + c) - f^3*sin(4*d*x + 4*c) - 2*I*f^3*sin(3*d*x + 3*c) - 2*I*f^3*si
n(d*x + c) - I*f^3)*polylog(3, I*e^(I*d*x + I*c)) + 36*(I*f^3*cos(4*d*x + 4*c) - 2*f^3*cos(3*d*x + 3*c) - 2*f^
3*cos(d*x + c) - f^3*sin(4*d*x + 4*c) - 2*I*f^3*sin(3*d*x + 3*c) - 2*I*f^3*sin(d*x + c) - I*f^3)*polylog(3, -I
*e^(I*d*x + I*c)) + 8*(2*I*(d*x + c)^3*f^3 + 3*(2*I*c^2 + I)*(d*x + c)*f^3 + 6*(I*d*e*f^2 - I*c*f^3)*(d*x + c)
^2)*sin(4*d*x + 4*c) - 4*(8*(d*x + c)^3*f^3 - 6*d*e*f^2 - 3*(I*c^2 - 2*c)*f^3 + 3*(8*d*e*f^2 - (8*c + I)*f^3)*
(d*x + c)^2 - 6*(I*d*e*f^2 - (4*c^2 + I*c + 1)*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 24*(I*d*e*f^2 + I*(d*x + c)*
f^3 - I*c*f^3)*sin(2*d*x + 2*c) + 4*(3*I*(d*x + c)^2*f^3 + 6*d*e*f^2 - (8*c^3 - 3*I*c^2 + 6*c)*f^3 + 6*(I*d*e*
f^2 + (-I*c - 1)*f^3)*(d*x + c))*sin(d*x + c))/(-12*I*a*d^3*cos(4*d*x + 4*c) + 24*a*d^3*cos(3*d*x + 3*c) + 24*
a*d^3*cos(d*x + c) + 12*a*d^3*sin(4*d*x + 4*c) + 24*I*a*d^3*sin(3*d*x + 3*c) + 24*I*a*d^3*sin(d*x + c) + 12*I*
a*d^3))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}