3.1.7 \(\int (c+d x)^2 (a+a \sec (e+f x))^2 \, dx\) [7]

Optimal. Leaf size=262 \[ -\frac {i a^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {4 i a^2 (c+d x)^2 \text {ArcTan}\left (e^{i (e+f x)}\right )}{f}+\frac {2 a^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {4 i a^2 d (c+d x) \text {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {4 i a^2 d (c+d x) \text {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {i a^2 d^2 \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac {4 a^2 d^2 \text {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {4 a^2 d^2 \text {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac {a^2 (c+d x)^2 \tan (e+f x)}{f} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4275, 4266, 2611, 2320, 6724, 4269, 3800, 2221, 2317, 2438} \begin {gather*} -\frac {4 i a^2 (c+d x)^2 \text {ArcTan}\left (e^{i (e+f x)}\right )}{f}+\frac {4 i a^2 d (c+d x) \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {4 i a^2 d (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {2 a^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {i a^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {i a^2 d^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {4 a^2 d^2 \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {4 a^2 d^2 \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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 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 4275

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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 (c+d x)^2 (a+a \sec (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a^2 (c+d x)^2 \sec (e+f x)+a^2 (c+d x)^2 \sec ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+a^2 \int (c+d x)^2 \sec ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^2 \sec (e+f x) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}-\frac {4 i a^2 (c+d x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {a^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {\left (2 a^2 d\right ) \int (c+d x) \tan (e+f x) \, dx}{f}-\frac {\left (4 a^2 d\right ) \int (c+d x) \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f}+\frac {\left (4 a^2 d\right ) \int (c+d x) \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f}\\ &=-\frac {i a^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {4 i a^2 (c+d x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {4 i a^2 d (c+d x) \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {4 i a^2 d (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {\left (4 i a^2 d^2\right ) \int \text {Li}_2\left (-i e^{i (e+f x)}\right ) \, dx}{f^2}+\frac {\left (4 i a^2 d^2\right ) \int \text {Li}_2\left (i e^{i (e+f x)}\right ) \, dx}{f^2}+\frac {\left (4 i a^2 d\right ) \int \frac {e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx}{f}\\ &=-\frac {i a^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {4 i a^2 (c+d x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {2 a^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {4 i a^2 d (c+d x) \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {4 i a^2 d (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {\left (4 a^2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^3}+\frac {\left (4 a^2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^3}-\frac {\left (2 a^2 d^2\right ) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {i a^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {4 i a^2 (c+d x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {2 a^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {4 i a^2 d (c+d x) \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {4 i a^2 d (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}-\frac {4 a^2 d^2 \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {4 a^2 d^2 \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3}+\frac {a^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {\left (i a^2 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^3}\\ &=-\frac {i a^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {4 i a^2 (c+d x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {2 a^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {4 i a^2 d (c+d x) \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {4 i a^2 d (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}-\frac {i a^2 d^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {4 a^2 d^2 \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {4 a^2 d^2 \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3}+\frac {a^2 (c+d x)^2 \tan (e+f x)}{f}\\ \end {align*}

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Mathematica [A]
time = 5.71, size = 505, normalized size = 1.93 \begin {gather*} \frac {1}{12} a^2 (1+\cos (e+f x))^2 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \left (x \left (3 c^2+3 c d x+d^2 x^2\right )+\frac {3 (c+d x)^2 \sin \left (\frac {f x}{2}\right )}{f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {3 (c+d x)^2 \sin \left (\frac {f x}{2}\right )}{f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {3 i \left (2 c d f^2 x+d^2 f^2 x^2+4 c^2 f^2 \text {ArcTan}(\cos (e+f x)+i \sin (e+f x))+8 c d f^2 x \text {ArcTan}(\cos (e+f x)+i \sin (e+f x))+4 d^2 f^2 x^2 \text {ArcTan}(\cos (e+f x)+i \sin (e+f x))+2 i c d f \log (1+\cos (2 (e+f x))+i \sin (2 (e+f x)))+2 i d^2 f x \log (1+\cos (2 (e+f x))+i \sin (2 (e+f x)))+4 d f (c+d x) \text {PolyLog}(2,i \cos (e+f x)-\sin (e+f x))-4 d f (c+d x) \text {PolyLog}(2,-i \cos (e+f x)+\sin (e+f x))+d^2 \text {PolyLog}(2,-\cos (2 (e+f x))-i \sin (2 (e+f x)))+4 i d^2 \text {PolyLog}(3,i \cos (e+f x)-\sin (e+f x))-4 i d^2 \text {PolyLog}(3,-i \cos (e+f x)+\sin (e+f x))+2 i c d f^2 x \tan (e)+i d^2 f^2 x^2 \tan (e)\right )}{f^3}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(1 + Cos[e + f*x])^2*Sec[(e + f*x)/2]^4*(x*(3*c^2 + 3*c*d*x + d^2*x^2) + (3*(c + d*x)^2*Sin[(f*x)/2])/(f*
(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (3*(c + d*x)^2*Sin[(f*x)/2])/(f*(Cos[e/2] + Sin
[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - ((3*I)*(2*c*d*f^2*x + d^2*f^2*x^2 + 4*c^2*f^2*ArcTan[Cos[e + f
*x] + I*Sin[e + f*x]] + 8*c*d*f^2*x*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 4*d^2*f^2*x^2*ArcTan[Cos[e + f*x]
+ I*Sin[e + f*x]] + (2*I)*c*d*f*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + (2*I)*d^2*f*x*Log[1 + Cos[2*(
e + f*x)] + I*Sin[2*(e + f*x)]] + 4*d*f*(c + d*x)*PolyLog[2, I*Cos[e + f*x] - Sin[e + f*x]] - 4*d*f*(c + d*x)*
PolyLog[2, (-I)*Cos[e + f*x] + Sin[e + f*x]] + d^2*PolyLog[2, -Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]] + (4*I)*
d^2*PolyLog[3, I*Cos[e + f*x] - Sin[e + f*x]] - (4*I)*d^2*PolyLog[3, (-I)*Cos[e + f*x] + Sin[e + f*x]] + (2*I)
*c*d*f^2*x*Tan[e] + I*d^2*f^2*x^2*Tan[e]))/f^3))/12

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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. 678 vs. \(2 (237 ) = 474\).
time = 0.87, size = 679, normalized size = 2.59

method	result	size
risch	\(\frac {8 i a^{2} c d e \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {4 i a^{2} d^{2} \polylog \left (2, -i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}-\frac {2 a^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{f}+\frac {2 a^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{f}+\frac {2 a^{2} e^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {2 i a^{2} d^{2} x^{2}}{f}-\frac {2 i a^{2} e^{2} d^{2}}{f^{3}}-\frac {4 i a^{2} c^{2} \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+\frac {4 a^{2} d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {2 a^{2} d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x}{f^{2}}-\frac {2 a^{2} e^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {i a^{2} d^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{3}}+\frac {4 a^{2} d^{2} \polylog \left (3, i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {4 i a^{2} d^{2} \polylog \left (2, i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}-\frac {4 i a^{2} c d \polylog \left (2, i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {4 i a^{2} c d \polylog \left (2, -i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {4 a^{2} c d \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}-\frac {4 a^{2} c d \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}+\frac {4 a^{2} c d \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}+\frac {4 a^{2} c d \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}-\frac {4 i a^{2} d^{2} e x}{f^{2}}-\frac {4 i a^{2} d^{2} e^{2} \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {2 i a^{2} \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {2 a^{2} c d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f^{2}}-\frac {4 a^{2} c d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {a^{2} c^{3}}{3 d}+a^{2} d c \,x^{2}+a^{2} c^{2} x -\frac {4 a^{2} d^{2} \polylog \left (3, -i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}\)	\(679\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1829 vs. \(2 (233) = 466\).
time = 0.69, size = 1829, normalized size = 6.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(f*x + e)*a^2*c^2 + (f*x + e)^3*a^2*d^2/f^2 + 3*(f*x + e)^2*a^2*c*d/f - 3*(f*x + e)^2*a^2*d^2*e/f^2 - 6
*(f*x + e)*a^2*c*d*e/f + 6*a^2*c^2*log(sec(f*x + e) + tan(f*x + e)) - 12*a^2*c*d*e*log(sec(f*x + e) + tan(f*x
+ e))/f + 3*(f*x + e)*a^2*d^2*e^2/f^2 + 6*a^2*d^2*e^2*log(sec(f*x + e) + tan(f*x + e))/f^2 + 3*(2*a^2*c^2*f^2
- 4*a^2*c*d*f*e + 2*a^2*d^2*e^2 - 2*((f*x + e)^2*a^2*d^2 + 2*(a^2*c*d*f - a^2*d^2*e)*(f*x + e) + ((f*x + e)^2*
a^2*d^2 + 2*(a^2*c*d*f - a^2*d^2*e)*(f*x + e))*cos(2*f*x + 2*e) + (I*(f*x + e)^2*a^2*d^2 + 2*(I*a^2*c*d*f - I*
a^2*d^2*e)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(cos(f*x + e), sin(f*x + e) + 1) - 2*((f*x + e)^2*a^2*d^2 + 2*(
a^2*c*d*f - a^2*d^2*e)*(f*x + e) + ((f*x + e)^2*a^2*d^2 + 2*(a^2*c*d*f - a^2*d^2*e)*(f*x + e))*cos(2*f*x + 2*e
) + (I*(f*x + e)^2*a^2*d^2 + 2*(I*a^2*c*d*f - I*a^2*d^2*e)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(cos(f*x + e),
-sin(f*x + e) + 1) + 2*((f*x + e)*a^2*d^2 + a^2*c*d*f - a^2*d^2*e + ((f*x + e)*a^2*d^2 + a^2*c*d*f - a^2*d^2*e
)*cos(2*f*x + 2*e) - (-I*(f*x + e)*a^2*d^2 - I*a^2*c*d*f + I*a^2*d^2*e)*sin(2*f*x + 2*e))*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e) + 1) - 2*((f*x + e)^2*a^2*d^2 + 2*(a^2*c*d*f - a^2*d^2*e)*(f*x + e))*cos(2*f*x + 2*e) -
(a^2*d^2*cos(2*f*x + 2*e) + I*a^2*d^2*sin(2*f*x + 2*e) + a^2*d^2)*dilog(-e^(2*I*f*x + 2*I*e)) - 4*((f*x + e)*
a^2*d^2 + a^2*c*d*f - a^2*d^2*e + ((f*x + e)*a^2*d^2 + a^2*c*d*f - a^2*d^2*e)*cos(2*f*x + 2*e) + (I*(f*x + e)*
a^2*d^2 + I*a^2*c*d*f - I*a^2*d^2*e)*sin(2*f*x + 2*e))*dilog(I*e^(I*f*x + I*e)) + 4*((f*x + e)*a^2*d^2 + a^2*c
*d*f - a^2*d^2*e + ((f*x + e)*a^2*d^2 + a^2*c*d*f - a^2*d^2*e)*cos(2*f*x + 2*e) - (-I*(f*x + e)*a^2*d^2 - I*a^
2*c*d*f + I*a^2*d^2*e)*sin(2*f*x + 2*e))*dilog(-I*e^(I*f*x + I*e)) + (-I*(f*x + e)*a^2*d^2 - I*a^2*c*d*f + I*a
^2*d^2*e + (-I*(f*x + e)*a^2*d^2 - I*a^2*c*d*f + I*a^2*d^2*e)*cos(2*f*x + 2*e) + ((f*x + e)*a^2*d^2 + a^2*c*d*
f - a^2*d^2*e)*sin(2*f*x + 2*e))*log(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1) + (-I*(
f*x + e)^2*a^2*d^2 - 2*(I*a^2*c*d*f - I*a^2*d^2*e)*(f*x + e) + (-I*(f*x + e)^2*a^2*d^2 - 2*(I*a^2*c*d*f - I*a^
2*d^2*e)*(f*x + e))*cos(2*f*x + 2*e) + ((f*x + e)^2*a^2*d^2 + 2*(a^2*c*d*f - a^2*d^2*e)*(f*x + e))*sin(2*f*x +
2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*sin(f*x + e) + 1) + (I*(f*x + e)^2*a^2*d^2 - 2*(-I*a^2*c*d*f +
I*a^2*d^2*e)*(f*x + e) + (I*(f*x + e)^2*a^2*d^2 - 2*(-I*a^2*c*d*f + I*a^2*d^2*e)*(f*x + e))*cos(2*f*x + 2*e) -
((f*x + e)^2*a^2*d^2 + 2*(a^2*c*d*f - a^2*d^2*e)*(f*x + e))*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x +
e)^2 - 2*sin(f*x + e) + 1) - 4*(I*a^2*d^2*cos(2*f*x + 2*e) - a^2*d^2*sin(2*f*x + 2*e) + I*a^2*d^2)*polylog(3,
I*e^(I*f*x + I*e)) - 4*(-I*a^2*d^2*cos(2*f*x + 2*e) + a^2*d^2*sin(2*f*x + 2*e) - I*a^2*d^2)*polylog(3, -I*e^(I
*f*x + I*e)) - 2*(I*(f*x + e)^2*a^2*d^2 + 2*(I*a^2*c*d*f - I*a^2*d^2*e)*(f*x + e))*sin(2*f*x + 2*e))/(-I*f^2*c
os(2*f*x + 2*e) + f^2*sin(2*f*x + 2*e) - I*f^2))/f

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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. 1151 vs. \(2 (233) = 466\).
time = 3.41, size = 1151, normalized size = 4.39 \begin {gather*} -\frac {6 \, a^{2} d^{2} \cos \left (f x + e\right ) {\rm polylog}\left (3, i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) - 6 \, a^{2} d^{2} \cos \left (f x + e\right ) {\rm polylog}\left (3, i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + 6 \, a^{2} d^{2} \cos \left (f x + e\right ) {\rm polylog}\left (3, -i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) - 6 \, a^{2} d^{2} \cos \left (f x + e\right ) {\rm polylog}\left (3, -i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + 3 \, {\left (2 i \, a^{2} d^{2} f x + 2 i \, a^{2} c d f - i \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) {\rm Li}_2\left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 3 \, {\left (2 i \, a^{2} d^{2} f x + 2 i \, a^{2} c d f + i \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) {\rm Li}_2\left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + 3 \, {\left (-2 i \, a^{2} d^{2} f x - 2 i \, a^{2} c d f + i \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 3 \, {\left (-2 i \, a^{2} d^{2} f x - 2 i \, a^{2} c d f - i \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 3 \, {\left (a^{2} c^{2} f^{2} + a^{2} c d f + a^{2} d^{2} e^{2} - {\left (2 \, a^{2} c d f + a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) + 3 \, {\left (a^{2} c^{2} f^{2} - a^{2} c d f + a^{2} d^{2} e^{2} - {\left (2 \, a^{2} c d f - a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) - 3 \, {\left (a^{2} d^{2} f^{2} x^{2} - a^{2} d^{2} e^{2} + {\left (2 \, a^{2} c d f^{2} + a^{2} d^{2} f\right )} x + {\left (2 \, a^{2} c d f + a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (a^{2} d^{2} f^{2} x^{2} - a^{2} d^{2} e^{2} + {\left (2 \, a^{2} c d f^{2} - a^{2} d^{2} f\right )} x + {\left (2 \, a^{2} c d f - a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (a^{2} d^{2} f^{2} x^{2} - a^{2} d^{2} e^{2} + {\left (2 \, a^{2} c d f^{2} + a^{2} d^{2} f\right )} x + {\left (2 \, a^{2} c d f + a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (a^{2} d^{2} f^{2} x^{2} - a^{2} d^{2} e^{2} + {\left (2 \, a^{2} c d f^{2} - a^{2} d^{2} f\right )} x + {\left (2 \, a^{2} c d f - a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (a^{2} c^{2} f^{2} + a^{2} c d f + a^{2} d^{2} e^{2} - {\left (2 \, a^{2} c d f + a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) + 3 \, {\left (a^{2} c^{2} f^{2} - a^{2} c d f + a^{2} d^{2} e^{2} - {\left (2 \, a^{2} c d f - a^{2} d^{2}\right )} e\right )} \cos \left (f x + e\right ) \log \left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) - {\left (a^{2} d^{2} f^{3} x^{3} + 3 \, a^{2} c d f^{3} x^{2} + 3 \, a^{2} c^{2} f^{3} x\right )} \cos \left (f x + e\right ) - 3 \, {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} \sin \left (f x + e\right )}{3 \, f^{3} \cos \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*a^2*d^2*cos(f*x + e)*polylog(3, I*cos(f*x + e) + sin(f*x + e)) - 6*a^2*d^2*cos(f*x + e)*polylog(3, I*c
os(f*x + e) - sin(f*x + e)) + 6*a^2*d^2*cos(f*x + e)*polylog(3, -I*cos(f*x + e) + sin(f*x + e)) - 6*a^2*d^2*co
s(f*x + e)*polylog(3, -I*cos(f*x + e) - sin(f*x + e)) + 3*(2*I*a^2*d^2*f*x + 2*I*a^2*c*d*f - I*a^2*d^2)*cos(f*
x + e)*dilog(I*cos(f*x + e) + sin(f*x + e)) + 3*(2*I*a^2*d^2*f*x + 2*I*a^2*c*d*f + I*a^2*d^2)*cos(f*x + e)*dil
og(I*cos(f*x + e) - sin(f*x + e)) + 3*(-2*I*a^2*d^2*f*x - 2*I*a^2*c*d*f + I*a^2*d^2)*cos(f*x + e)*dilog(-I*cos
(f*x + e) + sin(f*x + e)) + 3*(-2*I*a^2*d^2*f*x - 2*I*a^2*c*d*f - I*a^2*d^2)*cos(f*x + e)*dilog(-I*cos(f*x + e
) - sin(f*x + e)) - 3*(a^2*c^2*f^2 + a^2*c*d*f + a^2*d^2*e^2 - (2*a^2*c*d*f + a^2*d^2)*e)*cos(f*x + e)*log(cos
(f*x + e) + I*sin(f*x + e) + I) + 3*(a^2*c^2*f^2 - a^2*c*d*f + a^2*d^2*e^2 - (2*a^2*c*d*f - a^2*d^2)*e)*cos(f*
x + e)*log(cos(f*x + e) - I*sin(f*x + e) + I) - 3*(a^2*d^2*f^2*x^2 - a^2*d^2*e^2 + (2*a^2*c*d*f^2 + a^2*d^2*f)
*x + (2*a^2*c*d*f + a^2*d^2)*e)*cos(f*x + e)*log(I*cos(f*x + e) + sin(f*x + e) + 1) + 3*(a^2*d^2*f^2*x^2 - a^2
*d^2*e^2 + (2*a^2*c*d*f^2 - a^2*d^2*f)*x + (2*a^2*c*d*f - a^2*d^2)*e)*cos(f*x + e)*log(I*cos(f*x + e) - sin(f*
x + e) + 1) - 3*(a^2*d^2*f^2*x^2 - a^2*d^2*e^2 + (2*a^2*c*d*f^2 + a^2*d^2*f)*x + (2*a^2*c*d*f + a^2*d^2)*e)*co
s(f*x + e)*log(-I*cos(f*x + e) + sin(f*x + e) + 1) + 3*(a^2*d^2*f^2*x^2 - a^2*d^2*e^2 + (2*a^2*c*d*f^2 - a^2*d
^2*f)*x + (2*a^2*c*d*f - a^2*d^2)*e)*cos(f*x + e)*log(-I*cos(f*x + e) - sin(f*x + e) + 1) - 3*(a^2*c^2*f^2 + a
^2*c*d*f + a^2*d^2*e^2 - (2*a^2*c*d*f + a^2*d^2)*e)*cos(f*x + e)*log(-cos(f*x + e) + I*sin(f*x + e) + I) + 3*(
a^2*c^2*f^2 - a^2*c*d*f + a^2*d^2*e^2 - (2*a^2*c*d*f - a^2*d^2)*e)*cos(f*x + e)*log(-cos(f*x + e) - I*sin(f*x
+ e) + I) - (a^2*d^2*f^3*x^3 + 3*a^2*c*d*f^3*x^2 + 3*a^2*c^2*f^3*x)*cos(f*x + e) - 3*(a^2*d^2*f^2*x^2 + 2*a^2*
c*d*f^2*x + a^2*c^2*f^2)*sin(f*x + e))/(f^3*cos(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int c^{2}\, dx + \int 2 c^{2} \sec {\left (e + f x \right )}\, dx + \int c^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int d^{2} x^{2}\, dx + \int 2 c d x\, dx + \int 2 d^{2} x^{2} \sec {\left (e + f x \right )}\, dx + \int d^{2} x^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int 4 c d x \sec {\left (e + f x \right )}\, dx + \int 2 c d x \sec ^{2}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(Integral(c**2, x) + Integral(2*c**2*sec(e + f*x), x) + Integral(c**2*sec(e + f*x)**2, x) + Integral(d**2
*x**2, x) + Integral(2*c*d*x, x) + Integral(2*d**2*x**2*sec(e + f*x), x) + Integral(d**2*x**2*sec(e + f*x)**2,
x) + Integral(4*c*d*x*sec(e + f*x), x) + Integral(2*c*d*x*sec(e + f*x)**2, x))

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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