3.1.50 \(\int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx\) [50]

Optimal. Leaf size=436 \[ \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 i a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}-\frac {a b^2 (c+d x)^3}{d}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}-\frac {3 i a b^2 d^2 \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a^2 b d (c+d x) \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {i b^3 d (c+d x) \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f} \]

[Out]

b^3*c*d*x/f+1/2*b^3*d^2*x^2/f-1/3*I*b^3*(d*x+c)^3/d+1/3*a^3*(d*x+c)^3/d-3*I*a*b^2*d^2*polylog(2,-exp(2*I*(f*x+
e)))/f^3-a*b^2*(d*x+c)^3/d+3*I*a^2*b*d*(d*x+c)*polylog(2,-exp(2*I*(f*x+e)))/f^2+6*a*b^2*d*(d*x+c)*ln(1+exp(2*I
*(f*x+e)))/f^2-3*a^2*b*(d*x+c)^2*ln(1+exp(2*I*(f*x+e)))/f+b^3*(d*x+c)^2*ln(1+exp(2*I*(f*x+e)))/f-b^3*d^2*ln(co
s(f*x+e))/f^3-3*I*a*b^2*(d*x+c)^2/f+I*a^2*b*(d*x+c)^3/d-I*b^3*d*(d*x+c)*polylog(2,-exp(2*I*(f*x+e)))/f^2-3/2*a
^2*b*d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3+1/2*b^3*d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3-b^3*d*(d*x+c)*tan(f*x+e
)/f^2+3*a*b^2*(d*x+c)^2*tan(f*x+e)/f+1/2*b^3*(d*x+c)^2*tan(f*x+e)^2/f

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Rubi [A]
time = 0.44, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3803, 3800, 2221, 2611, 2320, 6724, 3801, 2317, 2438, 32, 3556} \begin {gather*} \frac {a^3 (c+d x)^3}{3 d}+\frac {3 i a^2 b d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {3 i a b^2 (c+d x)^2}{f}-\frac {a b^2 (c+d x)^3}{d}-\frac {3 i a b^2 d^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {i b^3 d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {b^3 d^2 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*(a + b*Tan[e + f*x])^3,x]

[Out]

(b^3*c*d*x)/f + (b^3*d^2*x^2)/(2*f) - ((3*I)*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) + (I*a^2*b*(c + d*
x)^3)/d - (a*b^2*(c + d*x)^3)/d - ((I/3)*b^3*(c + d*x)^3)/d + (6*a*b^2*d*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))
])/f^2 - (3*a^2*b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])
/f - (b^3*d^2*Log[Cos[e + f*x]])/f^3 - ((3*I)*a*b^2*d^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a^2*b*d
*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (I*b^3*d*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 -
(3*a^2*b*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) + (b^3*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) -
(b^3*d*(c + d*x)*Tan[e + f*x])/f^2 + (3*a*b^2*(c + d*x)^2*Tan[e + f*x])/f + (b^3*(c + d*x)^2*Tan[e + f*x]^2)/(
2*f)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 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 3801

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

Rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[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+b \tan (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \tan (e+f x)+3 a b^2 (c+d x)^2 \tan ^2(e+f x)+b^3 (c+d x)^2 \tan ^3(e+f x)\right ) \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int (c+d x)^2 \tan (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \tan ^2(e+f x) \, dx+b^3 \int (c+d x)^2 \tan ^3(e+f x) \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}-\left (6 i a^2 b\right ) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{1+e^{2 i (e+f x)}} \, dx-\left (3 a b^2\right ) \int (c+d x)^2 \, dx-b^3 \int (c+d x)^2 \tan (e+f x) \, dx-\frac {\left (6 a b^2 d\right ) \int (c+d x) \tan (e+f x) \, dx}{f}-\frac {\left (b^3 d\right ) \int (c+d x) \tan ^2(e+f x) \, dx}{f}\\ &=-\frac {3 i a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}-\frac {a b^2 (c+d x)^3}{d}-\frac {i b^3 (c+d x)^3}{3 d}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\left (2 i b^3\right ) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{1+e^{2 i (e+f x)}} \, dx+\frac {\left (b^3 d^2\right ) \int \tan (e+f x) \, dx}{f^2}+\frac {\left (6 a^2 b d\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}+\frac {\left (12 i a b^2 d\right ) \int \frac {e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \, dx}{f}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 i a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}-\frac {a b^2 (c+d x)^3}{d}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac {3 i a^2 b d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}-\frac {\left (3 i a^2 b d^2\right ) \int \text {Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 a b^2 d^2\right ) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac {\left (2 b^3 d\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 i a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}-\frac {a b^2 (c+d x)^3}{d}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac {3 i a^2 b d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {i b^3 d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}-\frac {\left (3 a^2 b d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}+\frac {\left (3 i a b^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 {\left (i b^3 d^2\right ) \int \text {Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 i a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}-\frac {a b^2 (c+d x)^3}{d}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}-\frac {3 i a b^2 d^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a^2 b d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {i b^3 d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\frac {\left (b^3 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}\\ &=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 i a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}-\frac {a b^2 (c+d x)^3}{d}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}-\frac {3 i a b^2 d^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a^2 b d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {i b^3 d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}\\ \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. \(1860\) vs. \(2(436)=872\).
time = 7.37, size = 1860, normalized size = 4.27 \begin {gather*} \frac {a^2 b d^2 e^{-i e} \left (2 i f^2 x^2 \left (2 e^{2 i e} f x+3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{2 i (e+f x)}\right )\right )+6 i \left (1+e^{2 i e}\right ) f x \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )-3 \left (1+e^{2 i e}\right ) \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )\right ) \sec (e)}{4 f^3}-\frac {b^3 d^2 e^{-i e} \left (2 i f^2 x^2 \left (2 e^{2 i e} f x+3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{2 i (e+f x)}\right )\right )+6 i \left (1+e^{2 i e}\right ) f x \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )-3 \left (1+e^{2 i e}\right ) \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )\right ) \sec (e)}{12 f^3}-\frac {b^3 d^2 \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f^3 \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {6 a b^2 c d \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f^2 \left (\cos ^2(e)+\sin ^2(e)\right )}-\frac {3 a^2 b c^2 \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {b^3 c^2 \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {3 a b^2 d^2 \csc (e) \left (e^{-i \text {ArcTan}(\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \text {ArcTan}(\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\text {ArcTan}(\cot (e))) \log \left (1-e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )+\pi \log (\cos (f x))-2 \text {ArcTan}(\cot (e)) \log (\sin (f x-\text {ArcTan}(\cot (e))))+i \text {PolyLog}\left (2,e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )\right )}{\sqrt {1+\cot ^2(e)}}\right ) \sec (e)}{f^3 \sqrt {\csc ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}-\frac {3 a^2 b c d \csc (e) \left (e^{-i \text {ArcTan}(\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \text {ArcTan}(\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\text {ArcTan}(\cot (e))) \log \left (1-e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )+\pi \log (\cos (f x))-2 \text {ArcTan}(\cot (e)) \log (\sin (f x-\text {ArcTan}(\cot (e))))+i \text {PolyLog}\left (2,e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )\right )}{\sqrt {1+\cot ^2(e)}}\right ) \sec (e)}{f^2 \sqrt {\csc ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}+\frac {b^3 c d \csc (e) \left (e^{-i \text {ArcTan}(\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \text {ArcTan}(\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\text {ArcTan}(\cot (e))) \log \left (1-e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )+\pi \log (\cos (f x))-2 \text {ArcTan}(\cot (e)) \log (\sin (f x-\text {ArcTan}(\cot (e))))+i \text {PolyLog}\left (2,e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )\right )}{\sqrt {1+\cot ^2(e)}}\right ) \sec (e)}{f^2 \sqrt {\csc ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}+\frac {\sec (e) \sec ^2(e+f x) \left (6 b^3 c^2 f \cos (e)+12 b^3 c d f x \cos (e)+6 a^3 c^2 f^2 x \cos (e)-18 a b^2 c^2 f^2 x \cos (e)+6 b^3 d^2 f x^2 \cos (e)+6 a^3 c d f^2 x^2 \cos (e)-18 a b^2 c d f^2 x^2 \cos (e)+2 a^3 d^2 f^2 x^3 \cos (e)-6 a b^2 d^2 f^2 x^3 \cos (e)+3 a^3 c^2 f^2 x \cos (e+2 f x)-9 a b^2 c^2 f^2 x \cos (e+2 f x)+3 a^3 c d f^2 x^2 \cos (e+2 f x)-9 a b^2 c d f^2 x^2 \cos (e+2 f x)+a^3 d^2 f^2 x^3 \cos (e+2 f x)-3 a b^2 d^2 f^2 x^3 \cos (e+2 f x)+3 a^3 c^2 f^2 x \cos (3 e+2 f x)-9 a b^2 c^2 f^2 x \cos (3 e+2 f x)+3 a^3 c d f^2 x^2 \cos (3 e+2 f x)-9 a b^2 c d f^2 x^2 \cos (3 e+2 f x)+a^3 d^2 f^2 x^3 \cos (3 e+2 f x)-3 a b^2 d^2 f^2 x^3 \cos (3 e+2 f x)+6 b^3 c d \sin (e)-18 a b^2 c^2 f \sin (e)+6 b^3 d^2 x \sin (e)-36 a b^2 c d f x \sin (e)+18 a^2 b c^2 f^2 x \sin (e)-6 b^3 c^2 f^2 x \sin (e)-18 a b^2 d^2 f x^2 \sin (e)+18 a^2 b c d f^2 x^2 \sin (e)-6 b^3 c d f^2 x^2 \sin (e)+6 a^2 b d^2 f^2 x^3 \sin (e)-2 b^3 d^2 f^2 x^3 \sin (e)-6 b^3 c d \sin (e+2 f x)+18 a b^2 c^2 f \sin (e+2 f x)-6 b^3 d^2 x \sin (e+2 f x)+36 a b^2 c d f x \sin (e+2 f x)-9 a^2 b c^2 f^2 x \sin (e+2 f x)+3 b^3 c^2 f^2 x \sin (e+2 f x)+18 a b^2 d^2 f x^2 \sin (e+2 f x)-9 a^2 b c d f^2 x^2 \sin (e+2 f x)+3 b^3 c d f^2 x^2 \sin (e+2 f x)-3 a^2 b d^2 f^2 x^3 \sin (e+2 f x)+b^3 d^2 f^2 x^3 \sin (e+2 f x)+9 a^2 b c^2 f^2 x \sin (3 e+2 f x)-3 b^3 c^2 f^2 x \sin (3 e+2 f x)+9 a^2 b c d f^2 x^2 \sin (3 e+2 f x)-3 b^3 c d f^2 x^2 \sin (3 e+2 f x)+3 a^2 b d^2 f^2 x^3 \sin (3 e+2 f x)-b^3 d^2 f^2 x^3 \sin (3 e+2 f x)\right )}{12 f^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x)^2*(a + b*Tan[e + f*x])^3,x]

[Out]

(a^2*b*d^2*((2*I)*f^2*x^2*(2*E^((2*I)*e)*f*x + (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((2*I)*(e + f*x))]) + (6*I)*(
1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((2*I)*(e + f*x))] - 3*(1 + E^((2*I)*e))*PolyLog[3, -E^((2*I)*(e + f*x))])*
Sec[e])/(4*E^(I*e)*f^3) - (b^3*d^2*((2*I)*f^2*x^2*(2*E^((2*I)*e)*f*x + (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((2*I
)*(e + f*x))]) + (6*I)*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((2*I)*(e + f*x))] - 3*(1 + E^((2*I)*e))*PolyLog[3,
-E^((2*I)*(e + f*x))])*Sec[e])/(12*E^(I*e)*f^3) - (b^3*d^2*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*
x]] + f*x*Sin[e]))/(f^3*(Cos[e]^2 + Sin[e]^2)) + (6*a*b^2*c*d*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[
f*x]] + f*x*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) - (3*a^2*b*c^2*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Si
n[f*x]] + f*x*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (b^3*c^2*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*
x]] + f*x*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (3*a*b^2*d^2*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(
I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x -
ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I
)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^3*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a^2*b
*c*d*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f
*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]
]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e]
)/(f^2*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (b^3*c*d*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*
x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - Arc
Tan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f
*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^2*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (Sec[e]*Sec[
e + f*x]^2*(6*b^3*c^2*f*Cos[e] + 12*b^3*c*d*f*x*Cos[e] + 6*a^3*c^2*f^2*x*Cos[e] - 18*a*b^2*c^2*f^2*x*Cos[e] +
6*b^3*d^2*f*x^2*Cos[e] + 6*a^3*c*d*f^2*x^2*Cos[e] - 18*a*b^2*c*d*f^2*x^2*Cos[e] + 2*a^3*d^2*f^2*x^3*Cos[e] - 6
*a*b^2*d^2*f^2*x^3*Cos[e] + 3*a^3*c^2*f^2*x*Cos[e + 2*f*x] - 9*a*b^2*c^2*f^2*x*Cos[e + 2*f*x] + 3*a^3*c*d*f^2*
x^2*Cos[e + 2*f*x] - 9*a*b^2*c*d*f^2*x^2*Cos[e + 2*f*x] + a^3*d^2*f^2*x^3*Cos[e + 2*f*x] - 3*a*b^2*d^2*f^2*x^3
*Cos[e + 2*f*x] + 3*a^3*c^2*f^2*x*Cos[3*e + 2*f*x] - 9*a*b^2*c^2*f^2*x*Cos[3*e + 2*f*x] + 3*a^3*c*d*f^2*x^2*Co
s[3*e + 2*f*x] - 9*a*b^2*c*d*f^2*x^2*Cos[3*e + 2*f*x] + a^3*d^2*f^2*x^3*Cos[3*e + 2*f*x] - 3*a*b^2*d^2*f^2*x^3
*Cos[3*e + 2*f*x] + 6*b^3*c*d*Sin[e] - 18*a*b^2*c^2*f*Sin[e] + 6*b^3*d^2*x*Sin[e] - 36*a*b^2*c*d*f*x*Sin[e] +
18*a^2*b*c^2*f^2*x*Sin[e] - 6*b^3*c^2*f^2*x*Sin[e] - 18*a*b^2*d^2*f*x^2*Sin[e] + 18*a^2*b*c*d*f^2*x^2*Sin[e] -
6*b^3*c*d*f^2*x^2*Sin[e] + 6*a^2*b*d^2*f^2*x^3*Sin[e] - 2*b^3*d^2*f^2*x^3*Sin[e] - 6*b^3*c*d*Sin[e + 2*f*x] +
18*a*b^2*c^2*f*Sin[e + 2*f*x] - 6*b^3*d^2*x*Sin[e + 2*f*x] + 36*a*b^2*c*d*f*x*Sin[e + 2*f*x] - 9*a^2*b*c^2*f^
2*x*Sin[e + 2*f*x] + 3*b^3*c^2*f^2*x*Sin[e + 2*f*x] + 18*a*b^2*d^2*f*x^2*Sin[e + 2*f*x] - 9*a^2*b*c*d*f^2*x^2*
Sin[e + 2*f*x] + 3*b^3*c*d*f^2*x^2*Sin[e + 2*f*x] - 3*a^2*b*d^2*f^2*x^3*Sin[e + 2*f*x] + b^3*d^2*f^2*x^3*Sin[e
+ 2*f*x] + 9*a^2*b*c^2*f^2*x*Sin[3*e + 2*f*x] - 3*b^3*c^2*f^2*x*Sin[3*e + 2*f*x] + 9*a^2*b*c*d*f^2*x^2*Sin[3*
e + 2*f*x] - 3*b^3*c*d*f^2*x^2*Sin[3*e + 2*f*x] + 3*a^2*b*d^2*f^2*x^3*Sin[3*e + 2*f*x] - b^3*d^2*f^2*x^3*Sin[3
*e + 2*f*x]))/(12*f^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. 1137 vs. \(2 (402 ) = 804\).
time = 0.38, size = 1138, normalized size = 2.61

method	result	size
risch	\(\text {Expression too large to display}\)	\(1138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2*(a+b*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

6/f^3*b*a^2*d^2*e^2*ln(exp(I*(f*x+e)))-12/f^2*b*a^2*c*d*e*ln(exp(I*(f*x+e)))-2/f^3*b^3*d^2*e^2*ln(exp(I*(f*x+e
)))-3/f*b*a^2*c^2*ln(exp(2*I*(f*x+e))+1)+4/3*I/f^3*b^3*d^2*e^3-1/3*I*b^3*d^2*x^3-2/f*b^3*c^2*ln(exp(I*(f*x+e))
)+1/f*b^3*c^2*ln(exp(2*I*(f*x+e))+1)-1/f^3*b^3*d^2*ln(exp(2*I*(f*x+e))+1)+2/f^3*b^3*d^2*ln(exp(I*(f*x+e)))+2*b
^2*(3*I*a*d^2*f*x^2*exp(2*I*(f*x+e))+6*I*a*c*d*f*x*exp(2*I*(f*x+e))+b*d^2*f*x^2*exp(2*I*(f*x+e))+3*I*a*c^2*f*e
xp(2*I*(f*x+e))+3*I*a*d^2*f*x^2-I*b*d^2*x*exp(2*I*(f*x+e))+2*b*c*d*f*x*exp(2*I*(f*x+e))+6*I*a*c*d*f*x-I*b*c*d*
exp(2*I*(f*x+e))+b*c^2*f*exp(2*I*(f*x+e))+3*I*a*c^2*f-I*b*d^2*x-I*b*c*d)/f^2/(exp(2*I*(f*x+e))+1)^2-4*I/f^3*b*
a^2*d^2*e^3+3*I*d*a^2*b*c*x^2+d*a^3*c*x^2+a^3*c^2*x-d^2*a*b^2*x^3-3*a*b^2*c^2*x-1/d*a*b^2*c^3+I*b^3*c^2*x+1/3*
I/d*b^3*c^3-I*d*b^3*c*x^2-3*d*a*b^2*c*x^2+I*d^2*a^2*b*x^3-3*I*a^2*b*c^2*x-I/d*a^2*b*c^3+1/3*d^2*a^3*x^3+1/3/d*
a^3*c^3+12/f^3*b^2*a*d^2*e*ln(exp(I*(f*x+e)))+6/f^2*b^2*a*c*d*ln(exp(2*I*(f*x+e))+1)-12/f^2*b^2*a*c*d*ln(exp(I
*(f*x+e)))+4/f^2*b^3*c*d*e*ln(exp(I*(f*x+e)))+2/f*b^3*ln(exp(2*I*(f*x+e))+1)*c*d*x+6/f^2*b^2*ln(exp(2*I*(f*x+e
))+1)*a*d^2*x-3/f*b*ln(exp(2*I*(f*x+e))+1)*a^2*d^2*x^2+2*I/f^2*b^3*d^2*e^2*x-6*I/f^3*b^2*a*d^2*e^2-I/f^2*b^3*c
*d*polylog(2,-exp(2*I*(f*x+e)))-2*I/f^2*b^3*c*d*e^2-6*I/f*b^2*a*d^2*x^2-I/f^2*b^3*polylog(2,-exp(2*I*(f*x+e)))
*d^2*x-6/f*b*ln(exp(2*I*(f*x+e))+1)*a^2*c*d*x-12*I/f^2*b^2*a*d^2*e*x-4*I/f*b^3*c*d*e*x+3*I/f^2*b*a^2*c*d*polyl
og(2,-exp(2*I*(f*x+e)))-6*I/f^2*b*a^2*d^2*e^2*x+6*I/f^2*b*a^2*c*d*e^2+3*I/f^2*b*polylog(2,-exp(2*I*(f*x+e)))*a
^2*d^2*x+12*I/f*b*a^2*c*d*e*x+1/f*b^3*ln(exp(2*I*(f*x+e))+1)*d^2*x^2+6/f*b*a^2*c^2*ln(exp(I*(f*x+e)))+1/2*b^3*
d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3-3/2*a^2*b*d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3-3*I*a*b^2*d^2*polylog(2,-e
xp(2*I*(f*x+e)))/f^3

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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. 3327 vs. \(2 (405) = 810\).
time = 2.52, size = 3327, normalized size = 7.63 \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+b*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

1/3*(3*(f*x + e)*a^3*c^2 + (f*x + e)^3*a^3*d^2/f^2 + 3*(f*x + e)^2*a^3*c*d/f - 3*(f*x + e)^2*a^3*d^2*e/f^2 - 6
*(f*x + e)*a^3*c*d*e/f + 9*a^2*b*c^2*log(sec(f*x + e)) - 18*a^2*b*c*d*e*log(sec(f*x + e))/f + 3*(f*x + e)*a^3*
d^2*e^2/f^2 + 9*a^2*b*d^2*e^2*log(sec(f*x + e))/f^2 + 3*(36*a*b^2*c^2*f^2 + 2*(3*a^2*b + 3*I*a*b^2 - b^3)*(f*x
+ e)^3*d^2 - 12*(6*a*b^2*e + b^3)*c*d*f + 6*((3*a^2*b + 3*I*a*b^2 - b^3)*c*d*f - (3*a^2*b*e + 3*I*a*b^2*e - b
^3*e)*d^2)*(f*x + e)^2 + 12*(3*a*b^2*e^2 + b^3*e)*d^2 - 6*((-3*I*a*b^2 + b^3)*c^2*f^2 + 2*(3*I*a*b^2*e - b^3*e
)*c*d*f + (-3*I*a*b^2*e^2 + b^3*e^2)*d^2)*(f*x + e) + 6*(b^3*c^2*f^2 - (3*a^2*b - b^3)*(f*x + e)^2*d^2 - 2*(b^
3*e - 3*a*b^2)*c*d*f + (b^3*(e^2 - 1) - 6*a*b^2*e)*d^2 - 2*((3*a^2*b - b^3)*c*d*f - (3*a^2*b*e - b^3*e + 3*a*b
^2)*d^2)*(f*x + e) + (b^3*c^2*f^2 - (3*a^2*b - b^3)*(f*x + e)^2*d^2 - 2*(b^3*e - 3*a*b^2)*c*d*f + (b^3*(e^2 -
1) - 6*a*b^2*e)*d^2 - 2*((3*a^2*b - b^3)*c*d*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^2)*(f*x + e))*cos(4*f*x + 4*e
) + 2*(b^3*c^2*f^2 - (3*a^2*b - b^3)*(f*x + e)^2*d^2 - 2*(b^3*e - 3*a*b^2)*c*d*f + (b^3*(e^2 - 1) - 6*a*b^2*e)
*d^2 - 2*((3*a^2*b - b^3)*c*d*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^2)*(f*x + e))*cos(2*f*x + 2*e) - (-I*b^3*c^2
*f^2 + (3*I*a^2*b - I*b^3)*(f*x + e)^2*d^2 + 2*(I*b^3*e - 3*I*a*b^2)*c*d*f + (b^3*(-I*e^2 + I) + 6*I*a*b^2*e)*
d^2 + 2*((3*I*a^2*b - I*b^3)*c*d*f + (-3*I*a^2*b*e + I*b^3*e - 3*I*a*b^2)*d^2)*(f*x + e))*sin(4*f*x + 4*e) - 2
*(-I*b^3*c^2*f^2 + (3*I*a^2*b - I*b^3)*(f*x + e)^2*d^2 + 2*(I*b^3*e - 3*I*a*b^2)*c*d*f + (b^3*(-I*e^2 + I) + 6
*I*a*b^2*e)*d^2 + 2*((3*I*a^2*b - I*b^3)*c*d*f + (-3*I*a^2*b*e + I*b^3*e - 3*I*a*b^2)*d^2)*(f*x + e))*sin(2*f*
x + 2*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1) + 2*((3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^3*d^2 + 3
*((3*a^2*b + 3*I*a*b^2 - b^3)*c*d*f - (3*a*b^2*(I*e + 2) + 3*a^2*b*e - b^3*e)*d^2)*(f*x + e)^2 - 3*((-3*I*a*b^
2 + b^3)*c^2*f^2 + 2*(3*a*b^2*(I*e + 2) - b^3*e)*c*d*f + (b^3*(e^2 - 2) + 3*a*b^2*(-I*e^2 - 4*e))*d^2)*(f*x +
e))*cos(4*f*x + 4*e) + 4*((3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^3*d^2 + 3*(3*a*b^2 - I*b^3)*c^2*f^2 - 3*(b^3*(
-2*I*e + 1) + 6*a*b^2*e)*c*d*f + 3*((3*a^2*b + 3*I*a*b^2 - b^3)*c*d*f + (b^3*(e - I) - 3*a*b^2*(I*e + 1) - 3*a
^2*b*e)*d^2)*(f*x + e)^2 - 3*(b^3*(I*e^2 - e) - 3*a*b^2*e^2)*d^2 - 3*((-3*I*a*b^2 + b^3)*c^2*f^2 - 2*(b^3*(e -
I) - 3*a*b^2*(I*e + 1))*c*d*f + (b^3*(e^2 - 2*I*e - 1) + 3*a*b^2*(-I*e^2 - 2*e))*d^2)*(f*x + e))*cos(2*f*x +
2*e) + 6*((3*a^2*b - b^3)*(f*x + e)*d^2 + (3*a^2*b - b^3)*c*d*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^2 + ((3*a^2*
b - b^3)*(f*x + e)*d^2 + (3*a^2*b - b^3)*c*d*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^2)*cos(4*f*x + 4*e) + 2*((3*a
^2*b - b^3)*(f*x + e)*d^2 + (3*a^2*b - b^3)*c*d*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^2)*cos(2*f*x + 2*e) - ((-3
*I*a^2*b + I*b^3)*(f*x + e)*d^2 + (-3*I*a^2*b + I*b^3)*c*d*f + (3*I*a^2*b*e - I*b^3*e + 3*I*a*b^2)*d^2)*sin(4*
f*x + 4*e) - 2*((-3*I*a^2*b + I*b^3)*(f*x + e)*d^2 + (-3*I*a^2*b + I*b^3)*c*d*f + (3*I*a^2*b*e - I*b^3*e + 3*I
*a*b^2)*d^2)*sin(2*f*x + 2*e))*dilog(-e^(2*I*f*x + 2*I*e)) - 3*(I*b^3*c^2*f^2 + (-3*I*a^2*b + I*b^3)*(f*x + e)
^2*d^2 + 2*(-I*b^3*e + 3*I*a*b^2)*c*d*f + (b^3*(I*e^2 - I) - 6*I*a*b^2*e)*d^2 + 2*((-3*I*a^2*b + I*b^3)*c*d*f
+ (3*I*a^2*b*e - I*b^3*e + 3*I*a*b^2)*d^2)*(f*x + e) + (I*b^3*c^2*f^2 + (-3*I*a^2*b + I*b^3)*(f*x + e)^2*d^2 +
2*(-I*b^3*e + 3*I*a*b^2)*c*d*f + (b^3*(I*e^2 - I) - 6*I*a*b^2*e)*d^2 + 2*((-3*I*a^2*b + I*b^3)*c*d*f + (3*I*a
^2*b*e - I*b^3*e + 3*I*a*b^2)*d^2)*(f*x + e))*cos(4*f*x + 4*e) + 2*(I*b^3*c^2*f^2 + (-3*I*a^2*b + I*b^3)*(f*x
+ e)^2*d^2 + 2*(-I*b^3*e + 3*I*a*b^2)*c*d*f + (b^3*(I*e^2 - I) - 6*I*a*b^2*e)*d^2 + 2*((-3*I*a^2*b + I*b^3)*c*
d*f + (3*I*a^2*b*e - I*b^3*e + 3*I*a*b^2)*d^2)*(f*x + e))*cos(2*f*x + 2*e) - (b^3*c^2*f^2 - (3*a^2*b - b^3)*(f
*x + e)^2*d^2 - 2*(b^3*e - 3*a*b^2)*c*d*f + (b^3*(e^2 - 1) - 6*a*b^2*e)*d^2 - 2*((3*a^2*b - b^3)*c*d*f - (3*a^
2*b*e - b^3*e + 3*a*b^2)*d^2)*(f*x + e))*sin(4*f*x + 4*e) - 2*(b^3*c^2*f^2 - (3*a^2*b - b^3)*(f*x + e)^2*d^2 -
2*(b^3*e - 3*a*b^2)*c*d*f + (b^3*(e^2 - 1) - 6*a*b^2*e)*d^2 - 2*((3*a^2*b - b^3)*c*d*f - (3*a^2*b*e - b^3*e +
3*a*b^2)*d^2)*(f*x + 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) - 3*((-3*I*a^2*b + I*b^3)*d^2*cos(4*f*x + 4*e) + 2*(-3*I*a^2*b + I*b^3)*d^2*cos(2*f*x + 2*e) + (3*a^2*b -
b^3)*d^2*sin(4*f*x + 4*e) + 2*(3*a^2*b - b^3)*d^2*sin(2*f*x + 2*e) + (-3*I*a^2*b + I*b^3)*d^2)*polylog(3, -e^(
2*I*f*x + 2*I*e)) - 2*((-3*I*a^2*b + 3*a*b^2 + I*b^3)*(f*x + e)^3*d^2 + 3*((-3*I*a^2*b + 3*a*b^2 + I*b^3)*c*d*
f - (3*a*b^2*(e - 2*I) - 3*I*a^2*b*e + I*b^3*e)*d^2)*(f*x + e)^2 + 3*((3*a*b^2 + I*b^3)*c^2*f^2 - 2*(3*a*b^2*(
e - 2*I) + I*b^3*e)*c*d*f + (3*a*b^2*(e^2 - 4*I*e) + b^3*(I*e^2 - 2*I))*d^2)*(f*x + e))*sin(4*f*x + 4*e) - 4*(
(-3*I*a^2*b + 3*a*b^2 + I*b^3)*(f*x + e)^3*d^2 + 3*(-3*I*a*b^2 - b^3)*c^2*f^2 + 3*(b^3*(2*e + I) + 6*I*a*b^2*e
)*c*d*f + 3*((-3*I*a^2*b + 3*a*b^2 + I*b^3)*c*d*f - (3*a*b^2*(e - I) - b^3*(-I*e - 1) - 3*I*a^2*b*e)*d^2)*(f*x
+ e)^2 - 3*(b^3*(e^2 + I*e) + 3*I*a*b^2*e^2)*d...

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Fricas [A]
time = 0.41, size = 704, normalized size = 1.61 \begin {gather*} \frac {4 \, {\left (a^{3} - 3 \, a b^{2}\right )} d^{2} f^{3} x^{3} - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (b^{3} d^{2} f^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d f^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{2} f^{2} x^{2} + 2 \, b^{3} c d f^{2} x + b^{3} c^{2} f^{2}\right )} \tan \left (f x + e\right )^{2} + 12 \, {\left (b^{3} c d f^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} f^{3}\right )} x - 6 \, {\left (-3 i \, a b^{2} d^{2} + i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} f x + i \, {\left (3 \, a^{2} b - b^{3}\right )} c d f\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 6 \, {\left (3 i \, a b^{2} d^{2} - i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} f x - i \, {\left (3 \, a^{2} b - b^{3}\right )} c d f\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 6 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} d^{2} f^{2} x^{2} - 6 \, a b^{2} c d f + b^{3} d^{2} + {\left (3 \, a^{2} b - b^{3}\right )} c^{2} f^{2} - 2 \, {\left (3 \, a b^{2} d^{2} f - {\left (3 \, a^{2} b - b^{3}\right )} c d f^{2}\right )} x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} d^{2} f^{2} x^{2} - 6 \, a b^{2} c d f + b^{3} d^{2} + {\left (3 \, a^{2} b - b^{3}\right )} c^{2} f^{2} - 2 \, {\left (3 \, a b^{2} d^{2} f - {\left (3 \, a^{2} b - b^{3}\right )} c d f^{2}\right )} x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \, {\left (3 \, a b^{2} d^{2} f^{2} x^{2} + 3 \, a b^{2} c^{2} f^{2} - b^{3} c d f + {\left (6 \, a b^{2} c d f^{2} - b^{3} d^{2} f\right )} x\right )} \tan \left (f x + e\right )}{12 \, f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*(a+b*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2*(a+b*tan(f*x+e))**3,x)

[Out]

Integral((a + b*tan(e + f*x))**3*(c + d*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+b*tan(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^2*(b*tan(f*x + e) + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(e + f*x))^3*(c + d*x)^2,x)

[Out]

int((a + b*tan(e + f*x))^3*(c + d*x)^2, x)

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