3.1.58 \(\int (c+d x)^3 (a+b \tanh (e+f x))^2 \, dx\) [58]

Optimal. Leaf size=277 \[ -\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {3 b^2 d^3 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d^3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{2 f^4}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f} \]

[Out]

-b^2*(d*x+c)^3/f+1/4*a^2*(d*x+c)^4/d-1/2*a*b*(d*x+c)^4/d+1/4*b^2*(d*x+c)^4/d+3*b^2*d*(d*x+c)^2*ln(1+exp(2*f*x+
2*e))/f^2+2*a*b*(d*x+c)^3*ln(1+exp(2*f*x+2*e))/f+3*b^2*d^2*(d*x+c)*polylog(2,-exp(2*f*x+2*e))/f^3+3*a*b*d*(d*x
+c)^2*polylog(2,-exp(2*f*x+2*e))/f^2-3/2*b^2*d^3*polylog(3,-exp(2*f*x+2*e))/f^4-3*a*b*d^2*(d*x+c)*polylog(3,-e
xp(2*f*x+2*e))/f^3+3/2*a*b*d^3*polylog(4,-exp(2*f*x+2*e))/f^4-b^2*(d*x+c)^3*tanh(f*x+e)/f

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Rubi [A]
time = 0.36, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3803, 3799, 2221, 2611, 6744, 2320, 6724, 3801, 32} \begin {gather*} \frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {a b (c+d x)^4}{2 d}+\frac {3 a b d^3 \text {Li}_4\left (-e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^2 (c+d x)^3}{f}+\frac {b^2 (c+d x)^4}{4 d}-\frac {3 b^2 d^3 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*(c + d*x)^3)/f) + (a^2*(c + d*x)^4)/(4*d) - (a*b*(c + d*x)^4)/(2*d) + (b^2*(c + d*x)^4)/(4*d) + (3*b^2*
d*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (3*b^2*d^2*(c +
d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^3 + (3*a*b*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (3*b^2*d^3*
PolyLog[3, -E^(2*(e + f*x))])/(2*f^4) - (3*a*b*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/f^3 + (3*a*b*d^3*Po
lyLog[4, -E^(2*(e + f*x))])/(2*f^4) - (b^2*(c + d*x)^3*Tanh[e + f*x])/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 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 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(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*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, 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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int (c+d x)^3 (a+b \tanh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \tanh (e+f x)+b^2 (c+d x)^3 \tanh ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \tanh (e+f x) \, dx+b^2 \int (c+d x)^3 \tanh ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f}+(4 a b) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1+e^{2 (e+f x)}} \, dx+b^2 \int (c+d x)^3 \, dx+\frac {\left (3 b^2 d\right ) \int (c+d x)^2 \tanh (e+f x) \, dx}{f}\\ &=-\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}+\frac {2 a b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {(6 a b d) \int (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}+\frac {\left (6 b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx}{f}\\ &=-\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 a b d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {\left (6 a b d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{f^3}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f}+\frac {\left (3 a b d^3\right ) \int \text {Li}_3\left (-e^{2 (e+f x)}\right ) \, dx}{f^3}-\frac {\left (3 b^2 d^3\right ) \int \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^3}\\ &=-\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{f^3}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f}+\frac {\left (3 a b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}-\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}\\ &=-\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {3 b^2 d^3 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d^3 \text {Li}_4\left (-e^{2 (e+f x)}\right )}{2 f^4}-\frac {b^2 (c+d x)^3 \tanh (e+f x)}{f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(765\) vs. \(2(277)=554\).
time = 4.94, size = 765, normalized size = 2.76 \begin {gather*} \frac {\frac {4 b e^{2 e} \left (-12 b c^2 d x-8 a c^3 f x-12 b c d^2 x^2-12 a c^2 d f x^2-4 b d^3 x^3-8 a c d^2 f x^3-2 a d^3 f x^4+4 a c^3 \log \left (1+e^{2 (e+f x)}\right )+4 a c^3 e^{-2 e} \log \left (1+e^{2 (e+f x)}\right )+\frac {6 b c^2 d \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {6 b c^2 d e^{-2 e} \log \left (1+e^{2 (e+f x)}\right )}{f}+12 a c^2 d x \log \left (1+e^{2 (e+f x)}\right )+12 a c^2 d e^{-2 e} x \log \left (1+e^{2 (e+f x)}\right )+\frac {12 b c d^2 x \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {12 b c d^2 e^{-2 e} x \log \left (1+e^{2 (e+f x)}\right )}{f}+12 a c d^2 x^2 \log \left (1+e^{2 (e+f x)}\right )+12 a c d^2 e^{-2 e} x^2 \log \left (1+e^{2 (e+f x)}\right )+\frac {6 b d^3 x^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {6 b d^3 e^{-2 e} x^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+4 a d^3 x^3 \log \left (1+e^{2 (e+f x)}\right )+4 a d^3 e^{-2 e} x^3 \log \left (1+e^{2 (e+f x)}\right )+\frac {6 d e^{-2 e} \left (1+e^{2 e}\right ) (c+d x) (b d+a f (c+d x)) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {3 d^2 e^{-2 e} \left (1+e^{2 e}\right ) (b d+2 a f (c+d x)) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a d^3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a d^3 e^{-2 e} \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{f^3}\right )}{1+e^{2 e}}+\text {sech}(e) \text {sech}(e+f x) \left (\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)+\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)-2 b \left (4 b (c+d x)^3+a f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \sinh (f x)+2 a b f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \sinh (2 e+f x)\right )}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*E^(2*e)*(-12*b*c^2*d*x - 8*a*c^3*f*x - 12*b*c*d^2*x^2 - 12*a*c^2*d*f*x^2 - 4*b*d^3*x^3 - 8*a*c*d^2*f*x^3
- 2*a*d^3*f*x^4 + 4*a*c^3*Log[1 + E^(2*(e + f*x))] + (4*a*c^3*Log[1 + E^(2*(e + f*x))])/E^(2*e) + (6*b*c^2*d*
Log[1 + E^(2*(e + f*x))])/f + (6*b*c^2*d*Log[1 + E^(2*(e + f*x))])/(E^(2*e)*f) + 12*a*c^2*d*x*Log[1 + E^(2*(e
+ f*x))] + (12*a*c^2*d*x*Log[1 + E^(2*(e + f*x))])/E^(2*e) + (12*b*c*d^2*x*Log[1 + E^(2*(e + f*x))])/f + (12*b
*c*d^2*x*Log[1 + E^(2*(e + f*x))])/(E^(2*e)*f) + 12*a*c*d^2*x^2*Log[1 + E^(2*(e + f*x))] + (12*a*c*d^2*x^2*Log
[1 + E^(2*(e + f*x))])/E^(2*e) + (6*b*d^3*x^2*Log[1 + E^(2*(e + f*x))])/f + (6*b*d^3*x^2*Log[1 + E^(2*(e + f*x
))])/(E^(2*e)*f) + 4*a*d^3*x^3*Log[1 + E^(2*(e + f*x))] + (4*a*d^3*x^3*Log[1 + E^(2*(e + f*x))])/E^(2*e) + (6*
d*(1 + E^(2*e))*(c + d*x)*(b*d + a*f*(c + d*x))*PolyLog[2, -E^(2*(e + f*x))])/(E^(2*e)*f^2) - (3*d^2*(1 + E^(2
*e))*(b*d + 2*a*f*(c + d*x))*PolyLog[3, -E^(2*(e + f*x))])/(E^(2*e)*f^3) + (3*a*d^3*PolyLog[4, -E^(2*(e + f*x)
)])/f^3 + (3*a*d^3*PolyLog[4, -E^(2*(e + f*x))])/(E^(2*e)*f^3)))/(1 + E^(2*e)) + Sech[e]*Sech[e + f*x]*((a^2 +
b^2)*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cosh[f*x] + (a^2 + b^2)*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2
*x^2 + d^3*x^3)*Cosh[2*e + f*x] - 2*b*(4*b*(c + d*x)^3 + a*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3))*Si
nh[f*x] + 2*a*b*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Sinh[2*e + f*x]))/(8*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(904\) vs. \(2(267)=534\).
time = 3.11, size = 905, normalized size = 3.27

method	result	size
risch	\(\frac {3 a b \,d^{3} \polylog \left (4, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{4}}-\frac {12 b a \,c^{2} d e x}{f}+\frac {12 b \,e^{2} a c \,d^{2} x}{f^{2}}-\frac {12 b a c \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {3 b^{2} d^{3} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{4}}+\frac {4 b^{2} e^{3} d^{3}}{f^{4}}-\frac {2 b^{2} d^{3} x^{3}}{f}+\frac {4 b a \,d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {2 b a \,d^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{3}}{f}+\frac {12 b^{2} c \,d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {6 b^{2} c \,d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}-\frac {3 b a c \,d^{2} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}-\frac {3 b a \,d^{3} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{3}}+\frac {3 b a \,d^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f^{2}}-\frac {12 b^{2} d^{2} c e x}{f^{2}}+\frac {8 b \,e^{3} a c \,d^{2}}{f^{3}}-\frac {6 b a \,c^{2} d \,e^{2}}{f^{2}}-\frac {4 b \,e^{3} a \,d^{3} x}{f^{3}}+\frac {a^{2} d^{3} x^{4}}{4}+c^{3} a^{2} x +a^{2} c \,d^{2} x^{3}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}+\frac {6 b a c \,d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f}+\frac {d^{3} b^{2} x^{4}}{4}+\frac {a^{2} c^{4}}{4 d}+\frac {b^{2} c^{4}}{4 d}+\frac {6 b \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) a \,c^{2} d x}{f}+\frac {12 b a \,c^{2} d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {6 b a c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}+2 a b \,c^{3} x +\frac {3 b^{2} c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}+\frac {3 b^{2} d^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{3}}-2 d^{2} a b c \,x^{3}-3 d a b \,c^{2} x^{2}+\frac {3 b a \,c^{2} d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}+\frac {2 b a \,c^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {6 b^{2} d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {3 b^{2} d^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f^{2}}-\frac {6 b^{2} c^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {4 b a \,c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {3 b^{2} c^{2} d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}-\frac {6 b^{2} d^{2} c \,e^{2}}{f^{3}}-\frac {6 b^{2} d^{2} c \,x^{2}}{f}-\frac {3 b \,e^{4} a \,d^{3}}{f^{4}}+\frac {6 b^{2} e^{2} d^{3} x}{f^{3}}+\frac {2 b^{2} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f \left (1+{\mathrm e}^{2 f x +2 e}\right )}-\frac {d^{3} a b \,x^{4}}{2}+d^{2} b^{2} c \,x^{3}+\frac {3 d \,b^{2} c^{2} x^{2}}{2}+b^{2} c^{3} x +\frac {a b \,c^{4}}{2 d}\)	\(905\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12/f*b*a*c^2*d*e*x+12/f^2*b*e^2*a*c*d^2*x+6/f^2*b*a*c*d^2*polylog(2,-exp(2*f*x+2*e))*x-12/f^3*b*a*c*d^2*e^2*l
n(exp(f*x+e))-3/2*b^2*d^3*polylog(3,-exp(2*f*x+2*e))/f^4+4/f^4*b^2*e^3*d^3-2/f*b^2*d^3*x^3+4/f^4*b*a*d^3*e^3*l
n(exp(f*x+e))-3/f^3*b*a*c*d^2*polylog(3,-exp(2*f*x+2*e))-3/f^3*b*a*d^3*polylog(3,-exp(2*f*x+2*e))*x+2/f*b*a*d^
3*ln(1+exp(2*f*x+2*e))*x^3+3/f^2*b*a*d^3*polylog(2,-exp(2*f*x+2*e))*x^2+12/f^3*b^2*c*d^2*e*ln(exp(f*x+e))+6/f^
2*b^2*c*d^2*ln(1+exp(2*f*x+2*e))*x+3/f^2*b*a*c^2*d*polylog(2,-exp(2*f*x+2*e))-12/f^2*b^2*d^2*c*e*x+8/f^3*b*e^3
*a*c*d^2-6/f^2*b*a*c^2*d*e^2-4/f^3*b*e^3*a*d^3*x+1/4*a^2*d^3*x^4+c^3*a^2*x+a^2*c*d^2*x^3+3/2*a^2*c^2*d*x^2+6/f
*b*a*c*d^2*ln(1+exp(2*f*x+2*e))*x^2+1/4*d^3*b^2*x^4+1/4/d*a^2*c^4+1/4/d*b^2*c^4+3/2*a*b*d^3*polylog(4,-exp(2*f
*x+2*e))/f^4+6/f*b*ln(1+exp(2*f*x+2*e))*a*c^2*d*x+12/f^2*b*a*c^2*d*e*ln(exp(f*x+e))+2*a*b*c^3*x-2*d^2*a*b*c*x^
3-3*d*a*b*c^2*x^2+2/f*b*a*c^3*ln(1+exp(2*f*x+2*e))+3/f^3*b^2*c*d^2*polylog(2,-exp(2*f*x+2*e))-6/f^4*b^2*d^3*e^
2*ln(exp(f*x+e))+3/f^2*b^2*d^3*ln(1+exp(2*f*x+2*e))*x^2+3/f^3*b^2*d^3*polylog(2,-exp(2*f*x+2*e))*x-6/f^2*b^2*c
^2*d*ln(exp(f*x+e))-4/f*b*a*c^3*ln(exp(f*x+e))+3/f^2*b^2*c^2*d*ln(1+exp(2*f*x+2*e))-6/f^3*b^2*d^2*c*e^2-6/f*b^
2*d^2*c*x^2-3/f^4*b*e^4*a*d^3+6/f^3*b^2*e^2*d^3*x+2/f*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/(1+exp(2*f*x+2*e
))-1/2*d^3*a*b*x^4+d^2*b^2*c*x^3+3/2*d*b^2*c^2*x^2+b^2*c^3*x+1/2/d*a*b*c^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (273) = 546\).
time = 0.36, size = 638, normalized size = 2.30 \begin {gather*} \frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + b^{2} c^{3} {\left (x + \frac {e}{f} - \frac {2}{f {\left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}}\right )} + a^{2} c^{3} x + \frac {3}{2} \, b^{2} c^{2} d {\left (\frac {f x^{2} + {\left (f x^{2} e^{\left (2 \, e\right )} - 4 \, x e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f e^{\left (2 \, f x + 2 \, e\right )} + f} + \frac {2 \, \log \left ({\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )} e^{\left (-2 \, e\right )}\right )}{f^{2}}\right )} + \frac {2 \, a b c^{3} \log \left (\cosh \left (f x + e\right )\right )}{f} + \frac {2 \, {\left (4 \, f^{3} x^{3} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} a b d^{3}}{3 \, f^{4}} + \frac {{\left (2 \, a b d^{3} f + b^{2} d^{3} f\right )} x^{4} + 4 \, {\left (2 \, a b c d^{2} f + {\left (c d^{2} f + 2 \, d^{3}\right )} b^{2}\right )} x^{3} + 12 \, {\left (a b c^{2} d f + 2 \, b^{2} c d^{2}\right )} x^{2} + {\left (12 \, a b c^{2} d f x^{2} e^{\left (2 \, e\right )} + {\left (2 \, a b d^{3} f + b^{2} d^{3} f\right )} x^{4} e^{\left (2 \, e\right )} + 4 \, {\left (2 \, a b c d^{2} f + b^{2} c d^{2} f\right )} x^{3} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{4 \, {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )}} + \frac {3 \, {\left (a b c^{2} d f + b^{2} c d^{2}\right )} {\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )}}{f^{3}} + \frac {3 \, {\left (2 \, a b c d^{2} f + b^{2} d^{3}\right )} {\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )}}{2 \, f^{4}} - \frac {a b d^{3} f^{4} x^{4} + 2 \, {\left (2 \, a b c d^{2} f + b^{2} d^{3}\right )} f^{3} x^{3} + 6 \, {\left (a b c^{2} d f^{2} + b^{2} c d^{2} f\right )} f^{2} x^{2}}{f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.52, size = 6280, normalized size = 22.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*(a^2 - 2*a*b + b^2)*c*d^2*f^4*x^3 + 6*(a^2 - 2*a*b + b^2)*c^2*d*f^4*x
^2 + 4*(a^2 - 2*a*b + b^2)*c^3*f^4*x + 4*a*b*d^3*cosh(1)^4 + 4*a*b*d^3*sinh(1)^4 + 8*b^2*c^3*f^3 - 8*(2*a*b*c*
d^2*f + b^2*d^3)*cosh(1)^3 - 8*(2*a*b*c*d^2*f - 2*a*b*d^3*cosh(1) + b^2*d^3)*sinh(1)^3 + 24*(a*b*c^2*d*f^2 + b
^2*c*d^2*f)*cosh(1)^2 + ((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*a*b*d^3*cosh(1)^4 + 4*a*b*d^3*sinh(1)^4 - 4*(2*b^
2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2*f^4)*x^3 - 8*(2*a*b*c*d^2*f + b^2*d^3)*cosh(1)^3 - 8*(2*a*b*c*d^2*f - 2*
a*b*d^3*cosh(1) + b^2*d^3)*sinh(1)^3 - 6*(4*b^2*c*d^2*f^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 + 24*(a*b*c^2*d
*f^2 + b^2*c*d^2*f)*cosh(1)^2 + 24*(a*b*c^2*d*f^2 + a*b*d^3*cosh(1)^2 + b^2*c*d^2*f - (2*a*b*c*d^2*f + b^2*d^3
)*cosh(1))*sinh(1)^2 - 4*(6*b^2*c^2*d*f^3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x - 8*(2*a*b*c^3*f^3 + 3*b^2*c^2*d*f^
2)*cosh(1) - 8*(2*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 + 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f + b^2*d^3)*cosh(1)^2
- 6*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 24*(a*b*c^2*d*f^2 + a*b*
d^3*cosh(1)^2 + b^2*c*d^2*f - (2*a*b*c*d^2*f + b^2*d^3)*cosh(1))*sinh(1)^2 + 2*((a^2 - 2*a*b + b^2)*d^3*f^4*x^
4 + 4*a*b*d^3*cosh(1)^4 + 4*a*b*d^3*sinh(1)^4 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2*f^4)*x^3 - 8*(2*a
*b*c*d^2*f + b^2*d^3)*cosh(1)^3 - 8*(2*a*b*c*d^2*f - 2*a*b*d^3*cosh(1) + b^2*d^3)*sinh(1)^3 - 6*(4*b^2*c*d^2*f
^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 + 24*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*cosh(1)^2 + 24*(a*b*c^2*d*f^2 + a*b
*d^3*cosh(1)^2 + b^2*c*d^2*f - (2*a*b*c*d^2*f + b^2*d^3)*cosh(1))*sinh(1)^2 - 4*(6*b^2*c^2*d*f^3 - (a^2 - 2*a*
b + b^2)*c^3*f^4)*x - 8*(2*a*b*c^3*f^3 + 3*b^2*c^2*d*f^2)*cosh(1) - 8*(2*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 + 3
*b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f + b^2*d^3)*cosh(1)^2 - 6*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*cosh(1))*sinh(1))*cos
h(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + ((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*a*b*d^3*cosh(1
)^4 + 4*a*b*d^3*sinh(1)^4 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2*f^4)*x^3 - 8*(2*a*b*c*d^2*f + b^2*d^3
)*cosh(1)^3 - 8*(2*a*b*c*d^2*f - 2*a*b*d^3*cosh(1) + b^2*d^3)*sinh(1)^3 - 6*(4*b^2*c*d^2*f^3 - (a^2 - 2*a*b +
b^2)*c^2*d*f^4)*x^2 + 24*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*cosh(1)^2 + 24*(a*b*c^2*d*f^2 + a*b*d^3*cosh(1)^2 + b^2
*c*d^2*f - (2*a*b*c*d^2*f + b^2*d^3)*cosh(1))*sinh(1)^2 - 4*(6*b^2*c^2*d*f^3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x
- 8*(2*a*b*c^3*f^3 + 3*b^2*c^2*d*f^2)*cosh(1) - 8*(2*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 + 3*b^2*c^2*d*f^2 + 3*(
2*a*b*c*d^2*f + b^2*d^3)*cosh(1)^2 - 6*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + si
nh(1))^2 - 8*(2*a*b*c^3*f^3 + 3*b^2*c^2*d*f^2)*cosh(1) + 24*(a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (
a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + cosh(1) + sinh(1))
^2 + 2*(a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + cosh(1) +
sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b
^2*d^3*f)*x)*sinh(f*x + cosh(1) + sinh(1))^2 + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*dilog(I*cosh(f*x + cosh(1) + s
inh(1)) + I*sinh(f*x + cosh(1) + sinh(1))) + 24*(a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (a*b*d^3*f^2*
x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(a*b*
d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + cosh(1) + sinh(1))*sin
h(f*x + cosh(1) + sinh(1)) + (a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)
*sinh(f*x + cosh(1) + sinh(1))^2 + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*dilog(-I*cosh(f*x + cosh(1) + sinh(1)) - I
*sinh(f*x + cosh(1) + sinh(1))) + 4*(2*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 - 2*a*b*d^3*sinh(1)^3 + 3*b^2*c^2*d*f
^2 + 3*(2*a*b*c*d^2*f + b^2*d^3)*cosh(1)^2 + (2*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 - 2*a*b*d^3*sinh(1)^3 + 3*b^
2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f + b^2*d^3)*cosh(1)^2 + 3*(2*a*b*c*d^2*f - 2*a*b*d^3*cosh(1) + b^2*d^3)*sinh(1)^
2 - 6*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*cosh(1) - 6*(a*b*c^2*d*f^2 + a*b*d^3*cosh(1)^2 + b^2*c*d^2*f - (2*a*b*c*d^
2*f + b^2*d^3)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 3*(2*a*b*c*d^2*f - 2*a*b*d^3*cosh(1) + b^2*
d^3)*sinh(1)^2 + 2*(2*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 - 2*a*b*d^3*sinh(1)^3 + 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*d
^2*f + b^2*d^3)*cosh(1)^2 + 3*(2*a*b*c*d^2*f - 2*a*b*d^3*cosh(1) + b^2*d^3)*sinh(1)^2 - 6*(a*b*c^2*d*f^2 + b^2
*c*d^2*f)*cosh(1) - 6*(a*b*c^2*d*f^2 + a*b*d^3*cosh(1)^2 + b^2*c*d^2*f - (2*a*b*c*d^2*f + b^2*d^3)*cosh(1))*si
nh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (2*a*b*c^3*f^3 - 2*a*b*d^3*cosh(1)^3 - 2*
a*b*d^3*sinh(1)^3 + 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f + b^2*d^3)*cosh(1)^2 + 3*(2*a*b*c*d^2*f - 2*a*b*d^3*cos
h(1) + b^2*d^3)*sinh(1)^2 - 6*(a*b*c^2*d*f^2 + b^2*c*d^2*f)*cosh(1) - 6*(a*b*c^2*d*f^2 + a*b*d^3*cosh(1)^2 + b
^2*c*d^2*f - (2*a*b*c*d^2*f + b^2*d^3)*cosh(1))...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tanh(e + f*x))**2*(c + d*x)**3, 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)^3*(a+b*tanh(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*(b*tanh(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+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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