\(\int \frac {(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [282]

Optimal result

Integrand size = 28, antiderivative size = 431 \[ \int \frac {(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 i (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 f^2 \text {arctanh}(\sin (c+d x))}{6 a d^3}+\frac {f^2 \log (\cos (c+d x))}{3 a d^3}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {3 f (e+f x) \sec (c+d x)}{4 a d^2}-\frac {f^2 \sec ^2(c+d x)}{12 a d^3}-\frac {f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac {(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac {f (e+f x) \tan (c+d x)}{3 a d^2}+\frac {f^2 \sec (c+d x) \tan (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x) \sec ^2(c+d x) \tan (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d} \]

[Out]

-3/4*I*(f*x+e)^2*arctan(exp(I*(d*x+c)))/a/d+5/6*f^2*arctanh(sin(d*x+c))/a/d^3+1/3*f^2*ln(cos(d*x+c))/a/d^3-3/4
*I*f*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^2+3/4*I*f*(f*x+e)*polylog(2,-I*exp(I*(d*x+c)))/a/d^2-3/4*f^2*poly
log(3,-I*exp(I*(d*x+c)))/a/d^3+3/4*f^2*polylog(3,I*exp(I*(d*x+c)))/a/d^3-3/4*f*(f*x+e)*sec(d*x+c)/a/d^2-1/12*f
^2*sec(d*x+c)^2/a/d^3-1/6*f*(f*x+e)*sec(d*x+c)^3/a/d^2-1/4*(f*x+e)^2*sec(d*x+c)^4/a/d+1/3*f*(f*x+e)*tan(d*x+c)
/a/d^2+1/12*f^2*sec(d*x+c)*tan(d*x+c)/a/d^3+3/8*(f*x+e)^2*sec(d*x+c)*tan(d*x+c)/a/d+1/6*f*(f*x+e)*sec(d*x+c)^2
*tan(d*x+c)/a/d^2+1/4*(f*x+e)^2*sec(d*x+c)^3*tan(d*x+c)/a/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4627, 4271, 3853, 3855, 4266, 2611, 2320, 6724, 4494, 4270, 4269, 3556} \[ \int \frac {(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 i (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 f^2 \text {arctanh}(\sin (c+d x))}{6 a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {f^2 \sec ^2(c+d x)}{12 a d^3}+\frac {f^2 \log (\cos (c+d x))}{3 a d^3}+\frac {f^2 \tan (c+d x) \sec (c+d x)}{12 a d^3}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{4 a d^2}+\frac {f (e+f x) \tan (c+d x)}{3 a d^2}-\frac {f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac {3 f (e+f x) \sec (c+d x)}{4 a d^2}+\frac {f (e+f x) \tan (c+d x) \sec ^2(c+d x)}{6 a d^2}-\frac {(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac {(e+f x)^2 \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac {3 (e+f x)^2 \tan (c+d x) \sec (c+d x)}{8 a d} \]

[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 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 3853

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 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 4270

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] &&
GtQ[n, 1] && NeQ[n, 2]

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)^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 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 f^2 \text {arctanh}(\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 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 f^2 \text {arctanh}(\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) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,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 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right ) \, dx}{4 a d^2}+\frac {\left (3 i f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) \, dx}{4 a d^2} \\ & = -\frac {3 i (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 f^2 \text {arctanh}(\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) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^3}+\frac {\left (3 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^3} \\ & = -\frac {3 i (e+f x)^2 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 f^2 \text {arctanh}(\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) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1579\) vs. \(2(431)=862\).

Time = 9.56 (sec) , antiderivative size = 1579, normalized size of antiderivative = 3.66 \[ \int \frac {(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {(\cos (c)+i \sin (c)) \left (3 d^2 e^2 x \cos (c)+4 f^2 x \cos (c)+3 d^2 e f x^2 \cos (c)+6 e f \operatorname {PolyLog}(2,i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i (-1+\sin (c)))+6 f^2 x \operatorname {PolyLog}(2,i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i (-1+\sin (c)))+d^2 f^2 x^3 (\cos (c)-i \sin (c))+\frac {\left (3 d^2 e^2+4 f^2\right ) \log (-\cos (c+d x)-i (-1+\sin (c+d x))) (\cos (c)+i (-1+\sin (c))) (\cos (c)-i \sin (c))}{d}+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))+\frac {6 f^2 \operatorname {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^2 x \sin (c)-4 i f^2 x \sin (c)-3 i d^2 e f x^2 \sin (c)+\left (3 d^2 e^2+4 f^2\right ) x (\cos (c)-i \sin (c)) (-1-i \cos (c)+\sin (c))\right )}{8 a d^2 (\cos (c)+i (-1+\sin (c)))}-\frac {(\cos (c)+i \sin (c)) \left (9 d^2 e^2 x \cos (c)+28 f^2 x \cos (c)+9 d^2 e f x^2 \cos (c)+3 d^2 f^2 x^3 \cos (c)-9 i d^2 e^2 x \sin (c)-28 i f^2 x \sin (c)-9 i d^2 e f x^2 \sin (c)-3 i d^2 f^2 x^3 \sin (c)+18 e f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))+18 f^2 x \operatorname {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 \sin (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)))-\frac {\left (9 d^2 e^2+28 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 {18 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 (9 d^2 e^2+28 f^2\right ) x (i \cos (c)+\sin (c)) (\cos (c)+i (1+\sin (c)))\right )}{24 a d^2 (\cos (c)+i (1+\sin (c)))}+\frac {\frac {3 e^2 x \cos (c)}{4 a}+\frac {3 i e^2 x \sin (c)}{4 a}}{1+\cos (2 c)+i \sin (2 c)}+\frac {\frac {3 e f x^2 \cos (c)}{4 a}+\frac {3 i e f x^2 \sin (c)}{4 a}}{1+\cos (2 c)+i \sin (2 c)}+\frac {\frac {f^2 x^3 \cos (c)}{4 a}+\frac {i f^2 x^3 \sin (c)}{4 a}}{1+\cos (2 c)+i \sin (2 c)}+\frac {e^2+2 e f x+f^2 x^2}{8 a d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {-e f \sin \left (\frac {d x}{2}\right )-f^2 x \sin \left (\frac {d x}{2}\right )}{2 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 )}+\frac {-e^2-2 e f x-f^2 x^2}{8 a d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4}+\frac {e f \sin \left (\frac {d x}{2}\right )+f^2 x \sin \left (\frac {d x}{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 )^3}+\frac {-3 d^2 e^2 \cos \left (\frac {c}{2}\right )-d e f \cos \left (\frac {c}{2}\right )-f^2 \cos \left (\frac {c}{2}\right )-6 d^2 e f x \cos \left (\frac {c}{2}\right )-d f^2 x \cos \left (\frac {c}{2}\right )-3 d^2 f^2 x^2 \cos \left (\frac {c}{2}\right )-3 d^2 e^2 \sin \left (\frac {c}{2}\right )+d e f \sin \left (\frac {c}{2}\right )-f^2 \sin \left (\frac {c}{2}\right )-6 d^2 e f x \sin \left (\frac {c}{2}\right )+d f^2 x \sin \left (\frac {c}{2}\right )-3 d^2 f^2 x^2 \sin \left (\frac {c}{2}\right )}{12 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 )^2}+\frac {7 \left (e f \sin \left (\frac {d x}{2}\right )+f^2 x \sin \left (\frac {d x}{2}\right )\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 )} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (382 ) = 764\).

Time = 0.80 (sec) , antiderivative size = 1035, normalized size of antiderivative = 2.40

method	result	size
risch	\(\text {Expression too large to display}\)	\(1035\)

[In]

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

[Out]

-1/12*I*(18*d^2*e*f*x*exp(5*I*(d*x+c))+9*d^2*f^2*x^2*exp(5*I*(d*x+c))+12*exp(3*I*(d*x+c))*d^2*e*f*x+18*exp(I*(
d*x+c))*d^2*e*f*x-36*I*d^2*e*f*x*exp(2*I*(d*x+c))+36*I*d^2*e*f*x*exp(4*I*(d*x+c))+9*d^2*e^2*exp(5*I*(d*x+c))+2
*I*d*f^2*x*exp(I*(d*x+c))+2*I*d*e*f*exp(I*(d*x+c))+6*exp(3*I*(d*x+c))*d^2*f^2*x^2+9*exp(I*(d*x+c))*d^2*f^2*x^2
+36*exp(4*I*(d*x+c))*d*f^2*x+44*exp(2*I*(d*x+c))*d*f^2*x+36*exp(4*I*(d*x+c))*d*e*f+44*exp(2*I*(d*x+c))*d*e*f+1
8*I*d^2*e^2*exp(4*I*(d*x+c))-18*I*d^2*e^2*exp(2*I*(d*x+c))+4*f^2*exp(3*I*(d*x+c))+2*f^2*exp(I*(d*x+c))+6*exp(3
*I*(d*x+c))*d^2*e^2+9*exp(I*(d*x+c))*d^2*e^2+18*I*d^2*f^2*x^2*exp(4*I*(d*x+c))-18*I*d*f^2*x*exp(5*I*(d*x+c))-1
8*I*d*e*f*exp(5*I*(d*x+c))+8*d*f^2*x-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))+8*d*e*f+2*f^2*exp(5*I*(d*x+c)))/(exp(I*(d*x+c))+I)^4/d^3/(-I+exp(I*(d*x+c)))^2/a-3/4/a/d^2*
ln(1+I*exp(I*(d*x+c)))*c*e*f+3/8/d/a*f^2*ln(1-I*exp(I*(d*x+c)))*x^2-3/4*I/a/d^2*e*f*polylog(2,I*exp(I*(d*x+c))
)+3/2*I/a/d^2*e*f*c*arctan(exp(I*(d*x+c)))+1/3/a/d^3*f^2*ln(1+exp(2*I*(d*x+c)))-3/8/a/d*ln(1+I*exp(I*(d*x+c)))
*f^2*x^2+3/4/d/a*e*f*ln(1-I*exp(I*(d*x+c)))*x+3/4*f^2*polylog(3,I*exp(I*(d*x+c)))/a/d^3-3/4*f^2*polylog(3,-I*e
xp(I*(d*x+c)))/a/d^3+3/4/d^2/a*e*f*ln(1-I*exp(I*(d*x+c)))*c-2/3/a/d^3*f^2*ln(exp(I*(d*x+c)))+3/4*I/a/d^2*e*f*p
olylog(2,-I*exp(I*(d*x+c)))+3/8/a/d^3*f^2*ln(1+I*exp(I*(d*x+c)))*c^2-3/4*I/a/d^3*f^2*c^2*arctan(exp(I*(d*x+c))
)-3/8/d^3/a*c^2*f^2*ln(1-I*exp(I*(d*x+c)))-5/3*I/a/d^3*f^2*arctan(exp(I*(d*x+c)))-3/4*I/a/d*e^2*arctan(exp(I*(
d*x+c)))-3/4*I/a/d^2*f^2*polylog(2,I*exp(I*(d*x+c)))*x-3/4/a/d*ln(1+I*exp(I*(d*x+c)))*e*f*x+3/4*I/a/d^2*f^2*po
lylog(2,-I*exp(I*(d*x+c)))*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. 1517 vs. \(2 (373) = 746\).

Time = 0.37 (sec) , antiderivative size = 1517, normalized size of antiderivative = 3.52 \[ \int \frac {(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[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 + I*d*e*f)*c
os(d*x + c)^2*sin(d*x + c) + (I*d*f^2*x + 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 + I*d*e*f)*cos(d*x + c)^2*sin(d*x + c) + (I*d*f^2*x + 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 - I*d*e*f)*cos(d*x + c)^2*sin(d*x + c) + (-I*d*f^2*x - 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 - I*d*e*f)*cos(d*x + c)^2*sin(d*x + c) + (-I*d*f^
2*x - 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)*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) + 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)*lo
g(-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) - 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)) + 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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}