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

Optimal. Leaf size=431 \[ -\frac {3 i (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}+\frac {f^2 \log (\cos (c+d x))}{3 a d^3}+\frac {3 i f (e+f x) \text {Li}_2\left (-i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 i f (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{4 a d^2}-\frac {3 f^2 \text {Li}_3\left (-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {3 f^2 \text {Li}_3\left (i e^{i (c+d x)}\right )}{4 a d^3}-\frac {3 f (e+f x) \sec (c+d x)}{4 a d^2}-\frac {f^2 \sec ^2(c+d x)}{12 a d^3}-\frac {f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac {(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac {f (e+f x) \tan (c+d x)}{3 a d^2}+\frac {f^2 \sec (c+d x) \tan (c+d x)}{12 a d^3}+\frac {3 (e+f x)^2 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x) \sec ^2(c+d x) \tan (c+d x)}{6 a d^2}+\frac {(e+f x)^2 \sec ^3(c+d x) \tan (c+d x)}{4 a d} \]

[Out]

3/4*I*f*(f*x+e)*polylog(2,-I*exp(I*(d*x+c)))/a/d^2+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*x+e)^2*arctan(exp(I*(d*x+c)))/a/d-3/4*f^2*polyl
og(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

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Rubi [A]
time = 0.26, antiderivative size = 431, normalized size of antiderivative = 1.00, 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 = {4627, 4271, 3853, 3855, 4266, 2611, 2320, 6724, 4494, 4270, 4269, 3556} \begin {gather*} -\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 {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 i (e+f x)^2 \text {ArcTan}\left (e^{i (c+d x)}\right )}{4 a d}-\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 {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} \end {gather*}

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 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*} \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 ) \text {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 ) \text {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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1577\) vs. \(2(431)=862\).
time = 8.49, size = 1577, normalized size = 3.66 \begin {gather*} -\frac {(\cos (c)+i \sin (c)) \left (-3 i d^2 e^2 x-4 i f^2 x-3 i d^2 e f x^2-i d^2 f^2 x^3+3 d e^2 \log (-\cos (c+d x)-i (-1+\sin (c+d x)))+\frac {4 f^2 \log (-\cos (c+d x)-i (-1+\sin (c+d x)))}{d}+6 d e f x \log (1+i \cos (c+d x)-\sin (c+d x))+3 d f^2 x^2 \log (1+i \cos (c+d x)-\sin (c+d x))-3 i d e^2 \log (-\cos (c+d x)-i (-1+\sin (c+d x))) (\cos (c)-i \sin (c))-\frac {4 i f^2 \log (-\cos (c+d x)-i (-1+\sin (c+d x))) (\cos (c)-i \sin (c))}{d}-6 i d e f x \log (1+i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c))-3 i d f^2 x^2 \log (1+i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c))+\frac {6 f^2 \text {Li}_3(-i \cos (c+d x)+\sin (c+d x)) (\cos (c)+i (-1+\sin (c))) (\cos (c)-i \sin (c))}{d}+6 f (e+f x) \text {Li}_2(-i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i \sin (c)) (-1-i \cos (c)+\sin (c))\right )}{8 a d^2 (\cos (c)+i (-1+\sin (c)))}-\frac {i (\cos (c)+i \sin (c)) \left (9 d^3 e^2 x+28 d f^2 x+9 d^3 e f x^2+3 d^3 f^2 x^3+18 i d^2 e f x \log (1-i \cos (c+d x)+\sin (c+d x))+9 i d^2 f^2 x^2 \log (1-i \cos (c+d x)+\sin (c+d x))+9 i d^2 e^2 \log (\cos (c+d x)+i (1+\sin (c+d x)))+28 i f^2 \log (\cos (c+d x)+i (1+\sin (c+d x)))-18 d^2 e f x \log (1-i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i \sin (c))-9 d^2 f^2 x^2 \log (1-i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i \sin (c))-9 d^2 e^2 \log (\cos (c+d x)+i (1+\sin (c+d x))) (\cos (c)-i \sin (c))-28 f^2 \log (\cos (c+d x)+i (1+\sin (c+d x))) (\cos (c)-i \sin (c))-18 f^2 \text {Li}_3(i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))+18 d f (e+f x) \text {Li}_2(i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i \sin (c)) (\cos (c)+i (1+\sin (c)))\right )}{24 a d^3 (\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 )} \end {gather*}

Warning: Unable to verify antiderivative.

[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*I)*d^2*e^2*x - (4*I)*f^2*x - (3*I)*d^2*e*f*x^2 - I*d^2*f^2*x^3 + 3*d*e^2*Log[-C
os[c + d*x] - I*(-1 + Sin[c + d*x])] + (4*f^2*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])])/d + 6*d*e*f*x*Log[1
+ I*Cos[c + d*x] - Sin[c + d*x]] + 3*d*f^2*x^2*Log[1 + I*Cos[c + d*x] - Sin[c + d*x]] - (3*I)*d*e^2*Log[-Cos[c
+ d*x] - I*(-1 + Sin[c + d*x])]*(Cos[c] - I*Sin[c]) - ((4*I)*f^2*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])]*(
Cos[c] - I*Sin[c]))/d - (6*I)*d*e*f*x*Log[1 + I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[c]) - (3*I)*d*f^2
*x^2*Log[1 + I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[c]) + (6*f^2*PolyLog[3, (-I)*Cos[c + d*x] + Sin[c
+ d*x]]*(Cos[c] + I*(-1 + Sin[c]))*(Cos[c] - I*Sin[c]))/d + 6*f*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] + Sin[c
+ d*x]]*(Cos[c] - I*Sin[c])*(-1 - I*Cos[c] + Sin[c])))/(a*d^2*(Cos[c] + I*(-1 + Sin[c]))) - ((I/24)*(Cos[c] +
I*Sin[c])*(9*d^3*e^2*x + 28*d*f^2*x + 9*d^3*e*f*x^2 + 3*d^3*f^2*x^3 + (18*I)*d^2*e*f*x*Log[1 - I*Cos[c + d*x]
+ Sin[c + d*x]] + (9*I)*d^2*f^2*x^2*Log[1 - I*Cos[c + d*x] + Sin[c + d*x]] + (9*I)*d^2*e^2*Log[Cos[c + d*x] +
I*(1 + Sin[c + d*x])] + (28*I)*f^2*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])] - 18*d^2*e*f*x*Log[1 - I*Cos[c +
d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c]) - 9*d^2*f^2*x^2*Log[1 - I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Si
n[c]) - 9*d^2*e^2*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])]*(Cos[c] - I*Sin[c]) - 28*f^2*Log[Cos[c + d*x] + I*(
1 + Sin[c + d*x])]*(Cos[c] - I*Sin[c]) - 18*f^2*PolyLog[3, I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin
[c])) + 18*d*f*(e + f*x)*PolyLog[2, I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c]
))))/(a*d^3*(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*(C
os[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] - S
in[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] + Si
n[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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1118 vs. \(2 (382 ) = 764\).
time = 0.36, size = 1119, normalized size = 2.60

method	result	size
risch	\(\text {Expression too large to display}\)	\(1119\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1546 vs. \(2 (383) = 766\).
time = 0.48, size = 1546, normalized size = 3.59 \begin {gather*} \text {Too large to display} \end {gather*}

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*f*x*e - 16*(d*f^2*x + d*f*e)*cos(d*x + c)^3 - 2*(9*d^2*f^2*x^2 + 18*d^2*f*x*e + 9
*d^2*e^2 + 2*f^2)*cos(d*x + c)^2 + 6*d^2*e^2 - 28*(d*f^2*x + d*f*e)*cos(d*x + c) - 18*((I*d*f^2*x + I*d*f*e)*c
os(d*x + c)^2*sin(d*x + c) + (I*d*f^2*x + I*d*f*e)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) + sin(d*x + c)) - 18*(
(I*d*f^2*x + I*d*f*e)*cos(d*x + c)^2*sin(d*x + c) + (I*d*f^2*x + I*d*f*e)*cos(d*x + c)^2)*dilog(I*cos(d*x + c)
- sin(d*x + c)) - 18*((-I*d*f^2*x - I*d*f*e)*cos(d*x + c)^2*sin(d*x + c) + (-I*d*f^2*x - I*d*f*e)*cos(d*x + c
)^2)*dilog(-I*cos(d*x + c) + sin(d*x + c)) - 18*((-I*d*f^2*x - I*d*f*e)*cos(d*x + c)^2*sin(d*x + c) + (-I*d*f^
2*x - I*d*f*e)*cos(d*x + c)^2)*dilog(-I*cos(d*x + c) - sin(d*x + c)) - ((18*c*d*f*e - (9*c^2 + 28)*f^2 - 9*d^2
*e^2)*cos(d*x + c)^2*sin(d*x + c) + (18*c*d*f*e - (9*c^2 + 28)*f^2 - 9*d^2*e^2)*cos(d*x + c)^2)*log(cos(d*x +
c) + I*sin(d*x + c) + I) + 3*((6*c*d*f*e - (3*c^2 + 4)*f^2 - 3*d^2*e^2)*cos(d*x + c)^2*sin(d*x + c) + (6*c*d*f
*e - (3*c^2 + 4)*f^2 - 3*d^2*e^2)*cos(d*x + c)^2)*log(cos(d*x + c) - I*sin(d*x + c) + I) + 9*((d^2*f^2*x^2 - c
^2*f^2 + 2*(d^2*f*x + c*d*f)*e)*cos(d*x + c)^2*sin(d*x + c) + (d^2*f^2*x^2 - c^2*f^2 + 2*(d^2*f*x + c*d*f)*e)*
cos(d*x + c)^2)*log(I*cos(d*x + c) + sin(d*x + c) + 1) - 9*((d^2*f^2*x^2 - c^2*f^2 + 2*(d^2*f*x + c*d*f)*e)*co
s(d*x + c)^2*sin(d*x + c) + (d^2*f^2*x^2 - c^2*f^2 + 2*(d^2*f*x + c*d*f)*e)*cos(d*x + c)^2)*log(I*cos(d*x + c)
- sin(d*x + c) + 1) + 9*((d^2*f^2*x^2 - c^2*f^2 + 2*(d^2*f*x + c*d*f)*e)*cos(d*x + c)^2*sin(d*x + c) + (d^2*f
^2*x^2 - c^2*f^2 + 2*(d^2*f*x + c*d*f)*e)*cos(d*x + c)^2)*log(-I*cos(d*x + c) + sin(d*x + c) + 1) - 9*((d^2*f^
2*x^2 - c^2*f^2 + 2*(d^2*f*x + c*d*f)*e)*cos(d*x + c)^2*sin(d*x + c) + (d^2*f^2*x^2 - c^2*f^2 + 2*(d^2*f*x + c
*d*f)*e)*cos(d*x + c)^2)*log(-I*cos(d*x + c) - sin(d*x + c) + 1) - ((18*c*d*f*e - (9*c^2 + 28)*f^2 - 9*d^2*e^2
)*cos(d*x + c)^2*sin(d*x + c) + (18*c*d*f*e - (9*c^2 + 28)*f^2 - 9*d^2*e^2)*cos(d*x + c)^2)*log(-cos(d*x + c)
+ I*sin(d*x + c) + I) + 3*((6*c*d*f*e - (3*c^2 + 4)*f^2 - 3*d^2*e^2)*cos(d*x + c)^2*sin(d*x + c) + (6*c*d*f*e
- (3*c^2 + 4)*f^2 - 3*d^2*e^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*co
s(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*f*x*e + 9*d^2*e^2 - 10*(d
*f^2*x + d*f*e)*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

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]

(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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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