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

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.548333, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3722, 3718, 2190, 2531, 6609, 2282, 6589, 3720, 32} \[ -\frac{3 a b d^2 (c+d x) \text{PolyLog}\left (3,-e^{2 (e+f x)}\right )}{f^3}+\frac{3 a b d (c+d x)^2 \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac{3 a b d^3 \text{PolyLog}\left (4,-e^{2 (e+f x)}\right )}{2 f^4}+\frac{3 b^2 d^2 (c+d x) \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}-\frac{3 b^2 d^3 \text{PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^4}+\frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac{a b (c+d x)^4}{2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac{b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac{b^2 (c+d x)^3}{f}+\frac{b^2 (c+d x)^4}{4 d} \]

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 3722

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 3718

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2531

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 6609

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]

Rule 2282

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 6589

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 3720

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 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]

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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 7.55792, size = 508, normalized size = 1.83 \[ \frac{1}{8} \left (4 b \left (-\frac{3 d^2 \left (2 f x \text{PolyLog}\left (2,-e^{-2 (e+f x)}\right )+\text{PolyLog}\left (3,-e^{-2 (e+f x)}\right )\right ) (2 a c f+b d)}{f^4}-\frac{6 c d \text{PolyLog}\left (2,-e^{-2 (e+f x)}\right ) (a c f+b d)}{f^3}-\frac{3 a d^3 \left (2 f^2 x^2 \text{PolyLog}\left (2,-e^{-2 (e+f x)}\right )+2 f x \text{PolyLog}\left (3,-e^{-2 (e+f x)}\right )+\text{PolyLog}\left (4,-e^{-2 (e+f x)}\right )\right )}{f^4}-\frac{2 c^2 \left (2 f x-\log \left (e^{2 (e+f x)}+1\right )\right ) (2 a c f+3 b d)}{f^2}+\frac{6 d^2 x^2 \log \left (e^{-2 (e+f x)}+1\right ) (2 a c f+b d)}{f^2}+\frac{12 c d x \log \left (e^{-2 (e+f x)}+1\right ) (a c f+b d)}{f^2}+\frac{2 a (c+d x)^4}{d \left (e^{2 e}+1\right )}+\frac{4 a d^3 x^3 \log \left (e^{-2 (e+f x)}+1\right )}{f}+\frac{4 b (c+d x)^3}{\left (e^{2 e}+1\right ) f}\right )+\frac{\text{sech}(e) \text{sech}(e+f x) \left (f x \left (a^2+b^2\right ) \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)+f x \left (a^2+b^2\right ) \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)+2 a b f x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \sinh (2 e+f x)-2 b \sinh (f x) \left (a f x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )+4 b (c+d x)^3\right )\right )}{f}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.178, size = 873, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.66118, size = 844, normalized size = 3.05 \begin{align*} \text{result too large to display} \end{align*}

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*e^(2*e) + b^2*d^3*f*e^(2*e))*x^4 + 4*(2*a*b*c*d^
2*f*e^(2*e) + b^2*c*d^2*f*e^(2*e))*x^3)*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]  time = 3.47157, size = 8259, normalized size = 29.82 \begin{align*} \text{result too large to display} \end{align*}

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*b*d^3*e^4 + 4*(a^2 - 2*a*b + b^2)*c^3*f^4*x - 8*b^2*d^3*e^3 - 8*(2*a*b*c^3*e - b^2*c^3)*f^3 + 24*(a*b
*c^2*d*e^2 - b^2*c^2*d*e)*f^2 + ((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*a*b*d^3*e^4 - 16*a*b*c^3*e*f^3 - 8*b^2*d^
3*e^3 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2*f^4)*x^3 + 24*(a*b*c^2*d*e^2 - b^2*c^2*d*e)*f^2 - 6*(4*b^
2*c*d^2*f^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 - 8*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e^2)*f - 4*(6*b^2*c^2*d*f^
3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x)*cosh(f*x + e)^2 + 2*((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*a*b*d^3*e^4 - 16*
a*b*c^3*e*f^3 - 8*b^2*d^3*e^3 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2*f^4)*x^3 + 24*(a*b*c^2*d*e^2 - b^
2*c^2*d*e)*f^2 - 6*(4*b^2*c*d^2*f^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 - 8*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e^
2)*f - 4*(6*b^2*c^2*d*f^3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x)*cosh(f*x + e)*sinh(f*x + e) + ((a^2 - 2*a*b + b^2)
*d^3*f^4*x^4 + 4*a*b*d^3*e^4 - 16*a*b*c^3*e*f^3 - 8*b^2*d^3*e^3 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2
*f^4)*x^3 + 24*(a*b*c^2*d*e^2 - b^2*c^2*d*e)*f^2 - 6*(4*b^2*c*d^2*f^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 - 8
*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e^2)*f - 4*(6*b^2*c^2*d*f^3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x)*sinh(f*x + e)^2
- 8*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e^2)*f + 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 + e)^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 + e)*sinh(f*x + e) + (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 + e)^2 + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*
x)*dilog(I*cosh(f*x + e) + I*sinh(f*x + e)) + 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 + e)^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 + e)*sinh(f*x + e) + (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 + e)^2 + (2*a*b*c*d^2*f^2 + b^2*d^3*f
)*x)*dilog(-I*cosh(f*x + e) - I*sinh(f*x + e)) - 4*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c
^2*d*e - b^2*c^2*d)*f^2 + (2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 -
6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*cosh(f*x + e)^2 + 2*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*
a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*cosh(f*x + e)*sinh(f*x + e) + (2*a*b*d^3*e^3
- 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*sinh
(f*x + e)^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*log(cosh(f*x + e) + sinh(f*x + e) + I) - 4*(2*a*b*d^3*e^3 - 2
*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 + (2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*
e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*cosh(f*x + e)^2 + 2*(2*a*b*d^3*e^
3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*cos
h(f*x + e)*sinh(f*x + e) + (2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2
- 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*sinh(f*x + e)^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*log(cosh(f*x + e) +
sinh(f*x + e) - I) + 4*(2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2
*f^3 + b^2*d^3*f^2)*x^2 + (2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*
d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*cosh(f*x
+ e)^2 + 2*(2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*
d^3*f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*cosh(f*x + e)*sinh(f*x
+ e) + (2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*
f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*sinh(f*x + e)^2 - 6*(a*b*c
*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log(I*cosh(f*x + e) + I*sinh(f*x + e) + 1) +
4*(2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x
^2 + (2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2
)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*cosh(f*x + e)^2 + 2*(2*a*b*d^
3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b
*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*cosh(f*x + e)*sinh(f*x + e) + (2*a*b*d^3*f^
3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b*c*d
^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*sinh(f*x + e)^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*
e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log(-I*cosh(f*x + e) - I*sinh(f*x + e) + 1) + 48*(a*b*d^3*cosh(f*x
+ e)^2 + 2*a*b*d^3*cosh(f*x + e)*sinh(f*x + e) + a*b*d^3*sinh(f*x + e)^2 + a*b*d^3)*polylog(4, I*cosh(f*x + e
) + I*sinh(f*x + e)) + 48*(a*b*d^3*cosh(f*x + e)^2 + 2*a*b*d^3*cosh(f*x + e)*sinh(f*x + e) + a*b*d^3*sinh(f*x
+ e)^2 + a*b*d^3)*polylog(4, -I*cosh(f*x + e) - I*sinh(f*x + e)) - 24*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3
+ (2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*cosh(f*x + e)^2 + 2*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*cos
h(f*x + e)*sinh(f*x + e) + (2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*sinh(f*x + e)^2)*polylog(3, I*cosh(f*x +
e) + I*sinh(f*x + e)) - 24*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3 + (2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3
)*cosh(f*x + e)^2 + 2*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*cosh(f*x + e)*sinh(f*x + e) + (2*a*b*d^3*f*x +
2*a*b*c*d^2*f + b^2*d^3)*sinh(f*x + e)^2)*polylog(3, -I*cosh(f*x + e) - I*sinh(f*x + e)))/(f^4*cosh(f*x + e)^
2 + 2*f^4*cosh(f*x + e)*sinh(f*x + e) + f^4*sinh(f*x + e)^2 + f^4)

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

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., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{3}{\left (b \tanh \left (f x + e\right ) + a\right )}^{2}\,{d x} \end{align*}

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)