3.11 \(\int (c+d x)^3 \tanh ^3(e+f x) \, dx\)

Optimal. Leaf size=237 \[ -\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 d^2 (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^3}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}+\frac {(c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x)^3 \tanh ^2(e+f x)}{2 f}-\frac {3 d (c+d x)^2}{2 f^2}+\frac {(c+d x)^3}{2 f}-\frac {(c+d x)^4}{4 d}+\frac {3 d^3 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 d^3 \text {Li}_4\left (-e^{2 (e+f x)}\right )}{4 f^4} \]

[Out]

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

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Rubi [A]  time = 0.39, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3720, 3718, 2190, 2279, 2391, 32, 2531, 6609, 2282, 6589} \[ -\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 d^3 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 d^3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4}+\frac {3 d^2 (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^3}-\frac {3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}+\frac {(c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x)^3 \tanh ^2(e+f x)}{2 f}-\frac {3 d (c+d x)^2}{2 f^2}+\frac {(c+d x)^3}{2 f}-\frac {(c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Tanh[e + f*x]^3,x]

[Out]

(-3*d*(c + d*x)^2)/(2*f^2) + (c + d*x)^3/(2*f) - (c + d*x)^4/(4*d) + (3*d^2*(c + d*x)*Log[1 + E^(2*(e + f*x))]
)/f^3 + ((c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (3*d^3*PolyLog[2, -E^(2*(e + f*x))])/(2*f^4) + (3*d*(c + d*
x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) + (3*d^3*P
olyLog[4, -E^(2*(e + f*x))])/(4*f^4) - (3*d*(c + d*x)^2*Tanh[e + f*x])/(2*f^2) - ((c + d*x)^3*Tanh[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 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 2279

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 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 2391

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 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 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 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 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 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]

Rubi steps

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

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Mathematica [C]  time = 12.75, size = 801, normalized size = 3.38 \[ \frac {\text {sech}(e) (\cosh (e) \log (\cosh (e) \cosh (f x)+\sinh (e) \sinh (f x))-f x \sinh (e)) c^3}{f \left (\cosh ^2(e)-\sinh ^2(e)\right )}+\frac {3 d \text {csch}(e) \left (e^{-\tanh ^{-1}(\coth (e))} f^2 x^2-\frac {i \coth (e) \left (-f x \left (2 i \tanh ^{-1}(\coth (e))-\pi \right )-\pi \log \left (1+e^{2 f x}\right )-2 \left (i f x+i \tanh ^{-1}(\coth (e))\right ) \log \left (1-e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )+\pi \log (\cosh (f x))+2 i \tanh ^{-1}(\coth (e)) \log \left (i \sinh \left (f x+\tanh ^{-1}(\coth (e))\right )\right )+i \text {Li}_2\left (e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )\right )}{\sqrt {1-\coth ^2(e)}}\right ) \text {sech}(e) c^2}{2 f^2 \sqrt {\text {csch}^2(e) \left (\sinh ^2(e)-\cosh ^2(e)\right )}}+\frac {d^2 e^{-e} \left (2 f^2 \left (2 f x+3 \left (1+e^{2 e}\right ) \log \left (1+e^{-2 (e+f x)}\right )\right ) x^2-6 \left (1+e^{2 e}\right ) f \text {Li}_2\left (-e^{-2 (e+f x)}\right ) x-3 \left (1+e^{2 e}\right ) \text {Li}_3\left (-e^{-2 (e+f x)}\right )\right ) \text {sech}(e) c}{4 f^3}+\frac {3 d^2 \text {sech}(e) (\cosh (e) \log (\cosh (e) \cosh (f x)+\sinh (e) \sinh (f x))-f x \sinh (e)) c}{f^3 \left (\cosh ^2(e)-\sinh ^2(e)\right )}+\frac {(c+d x)^3 \text {sech}^2(e+f x)}{2 f}+\frac {1}{8} d^3 e^e \left (2 e^{-2 e} x^4+\frac {4 \left (1+e^{-2 e}\right ) \log \left (1+e^{-2 (e+f x)}\right ) x^3}{f}-\frac {3 e^{-2 e} \left (1+e^{2 e}\right ) \left (2 f^2 \text {Li}_2\left (-e^{-2 (e+f x)}\right ) x^2+2 f \text {Li}_3\left (-e^{-2 (e+f x)}\right ) x+\text {Li}_4\left (-e^{-2 (e+f x)}\right )\right )}{f^4}\right ) \text {sech}(e)-\frac {3 \text {sech}(e) \text {sech}(e+f x) \left (x^2 \sinh (f x) d^3+2 c x \sinh (f x) d^2+c^2 \sinh (f x) d\right )}{2 f^2}+\frac {1}{4} x \left (4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right ) \tanh (e)+\frac {3 d^3 \text {csch}(e) \left (e^{-\tanh ^{-1}(\coth (e))} f^2 x^2-\frac {i \coth (e) \left (-f x \left (2 i \tanh ^{-1}(\coth (e))-\pi \right )-\pi \log \left (1+e^{2 f x}\right )-2 \left (i f x+i \tanh ^{-1}(\coth (e))\right ) \log \left (1-e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )+\pi \log (\cosh (f x))+2 i \tanh ^{-1}(\coth (e)) \log \left (i \sinh \left (f x+\tanh ^{-1}(\coth (e))\right )\right )+i \text {Li}_2\left (e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )\right )}{\sqrt {1-\coth ^2(e)}}\right ) \text {sech}(e)}{2 f^4 \sqrt {\text {csch}^2(e) \left (\sinh ^2(e)-\cosh ^2(e)\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*d^2*(2*f^2*x^2*(2*f*x + 3*(1 + E^(2*e))*Log[1 + E^(-2*(e + f*x))]) - 6*(1 + E^(2*e))*f*x*PolyLog[2, -E^(-2*
(e + f*x))] - 3*(1 + E^(2*e))*PolyLog[3, -E^(-2*(e + f*x))])*Sech[e])/(4*E^e*f^3) + (d^3*E^e*((2*x^4)/E^(2*e)
+ (4*(1 + E^(-2*e))*x^3*Log[1 + E^(-2*(e + f*x))])/f - (3*(1 + E^(2*e))*(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))]))/(E^(2*e)*f^4))*Sech[e])/8 + ((c + d
*x)^3*Sech[e + f*x]^2)/(2*f) + (3*c*d^2*Sech[e]*(Cosh[e]*Log[Cosh[e]*Cosh[f*x] + Sinh[e]*Sinh[f*x]] - f*x*Sinh
[e]))/(f^3*(Cosh[e]^2 - Sinh[e]^2)) + (c^3*Sech[e]*(Cosh[e]*Log[Cosh[e]*Cosh[f*x] + Sinh[e]*Sinh[f*x]] - f*x*S
inh[e]))/(f*(Cosh[e]^2 - Sinh[e]^2)) + (3*d^3*Csch[e]*((f^2*x^2)/E^ArcTanh[Coth[e]] - (I*Coth[e]*(-(f*x*(-Pi +
(2*I)*ArcTanh[Coth[e]])) - Pi*Log[1 + E^(2*f*x)] - 2*(I*f*x + I*ArcTanh[Coth[e]])*Log[1 - E^((2*I)*(I*f*x + I
*ArcTanh[Coth[e]]))] + Pi*Log[Cosh[f*x]] + (2*I)*ArcTanh[Coth[e]]*Log[I*Sinh[f*x + ArcTanh[Coth[e]]]] + I*Poly
Log[2, E^((2*I)*(I*f*x + I*ArcTanh[Coth[e]]))]))/Sqrt[1 - Coth[e]^2])*Sech[e])/(2*f^4*Sqrt[Csch[e]^2*(-Cosh[e]
^2 + Sinh[e]^2)]) + (3*c^2*d*Csch[e]*((f^2*x^2)/E^ArcTanh[Coth[e]] - (I*Coth[e]*(-(f*x*(-Pi + (2*I)*ArcTanh[Co
th[e]])) - Pi*Log[1 + E^(2*f*x)] - 2*(I*f*x + I*ArcTanh[Coth[e]])*Log[1 - E^((2*I)*(I*f*x + I*ArcTanh[Coth[e]]
))] + Pi*Log[Cosh[f*x]] + (2*I)*ArcTanh[Coth[e]]*Log[I*Sinh[f*x + ArcTanh[Coth[e]]]] + I*PolyLog[2, E^((2*I)*(
I*f*x + I*ArcTanh[Coth[e]]))]))/Sqrt[1 - Coth[e]^2])*Sech[e])/(2*f^2*Sqrt[Csch[e]^2*(-Cosh[e]^2 + Sinh[e]^2)])
- (3*Sech[e]*Sech[e + f*x]*(c^2*d*Sinh[f*x] + 2*c*d^2*x*Sinh[f*x] + d^3*x^2*Sinh[f*x]))/(2*f^2) + (x*(4*c^3 +
6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Tanh[e])/4

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fricas [C]  time = 0.62, size = 5569, normalized size = 23.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 6*c^2*d*f^4*x^2 + 4*c^3*f^4*x - 2*d^3*e^4 + 8*c^3*e*f^3 - 12*d^3*e^2 + (
d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 - 2*d^3*e^4 - 12*c^2*d*e^2*f^2 + 8*c^3*e*f^3 - 12*d^3*e^2 + 6*(c^2*d*f^4 + 2*d^3
*f^2)*x^2 + 8*(c*d^2*e^3 + 3*c*d^2*e)*f + 4*(c^3*f^4 + 6*c*d^2*f^2)*x)*cosh(f*x + e)^4 + 4*(d^3*f^4*x^4 + 4*c*
d^2*f^4*x^3 - 2*d^3*e^4 - 12*c^2*d*e^2*f^2 + 8*c^3*e*f^3 - 12*d^3*e^2 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 + 8*(c*d
^2*e^3 + 3*c*d^2*e)*f + 4*(c^3*f^4 + 6*c*d^2*f^2)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*f^4*x^4 + 4*c*d^2*f^
4*x^3 - 2*d^3*e^4 - 12*c^2*d*e^2*f^2 + 8*c^3*e*f^3 - 12*d^3*e^2 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 + 8*(c*d^2*e^3
+ 3*c*d^2*e)*f + 4*(c^3*f^4 + 6*c*d^2*f^2)*x)*sinh(f*x + e)^4 - 12*(c^2*d*e^2 + c^2*d)*f^2 + 2*(d^3*f^4*x^4 -
2*d^3*e^4 - 12*d^3*e^2 + 4*(2*c^3*e - c^3)*f^3 + 4*(c*d^2*f^4 - d^3*f^3)*x^3 - 6*(2*c^2*d*e^2 + c^2*d)*f^2 +
6*(c^2*d*f^4 - 2*c*d^2*f^3 + d^3*f^2)*x^2 + 8*(c*d^2*e^3 + 3*c*d^2*e)*f + 4*(c^3*f^4 - 3*c^2*d*f^3 + 3*c*d^2*f
^2)*x)*cosh(f*x + e)^2 + 2*(d^3*f^4*x^4 - 2*d^3*e^4 - 12*d^3*e^2 + 4*(2*c^3*e - c^3)*f^3 + 4*(c*d^2*f^4 - d^3*
f^3)*x^3 - 6*(2*c^2*d*e^2 + c^2*d)*f^2 + 6*(c^2*d*f^4 - 2*c*d^2*f^3 + d^3*f^2)*x^2 + 3*(d^3*f^4*x^4 + 4*c*d^2*
f^4*x^3 - 2*d^3*e^4 - 12*c^2*d*e^2*f^2 + 8*c^3*e*f^3 - 12*d^3*e^2 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 + 8*(c*d^2*e
^3 + 3*c*d^2*e)*f + 4*(c^3*f^4 + 6*c*d^2*f^2)*x)*cosh(f*x + e)^2 + 8*(c*d^2*e^3 + 3*c*d^2*e)*f + 4*(c^3*f^4 -
3*c^2*d*f^3 + 3*c*d^2*f^2)*x)*sinh(f*x + e)^2 + 8*(c*d^2*e^3 + 3*c*d^2*e)*f - 12*(d^3*f^2*x^2 + 2*c*d^2*f^2*x
+ c^2*d*f^2 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^4 + 4*(d^3*f^2*x^2 + 2*c*d^2*f^2*x
+ c^2*d*f^2 + d^3)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*sinh(f*x +
e)^4 + d^3 + 2*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^2 + 2*(d^3*f^2*x^2 + 2*c*d^2*f^2
*x + c^2*d*f^2 + d^3 + 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*
((d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 +
d^3)*cosh(f*x + e))*sinh(f*x + e))*dilog(I*cosh(f*x + e) + I*sinh(f*x + e)) - 12*(d^3*f^2*x^2 + 2*c*d^2*f^2*x
+ c^2*d*f^2 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^4 + 4*(d^3*f^2*x^2 + 2*c*d^2*f^2*x
+ c^2*d*f^2 + d^3)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*sinh(f*x +
e)^4 + d^3 + 2*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^2 + 2*(d^3*f^2*x^2 + 2*c*d^2*f^2
*x + c^2*d*f^2 + d^3 + 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*
((d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + e)^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 +
d^3)*cosh(f*x + e))*sinh(f*x + e))*dilog(-I*cosh(f*x + e) - I*sinh(f*x + e)) + 4*(d^3*e^3 + 3*c^2*d*e*f^2 - c^
3*f^3 + (d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)^4 + 4*(d^3*e^3 +
3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*e^3 + 3*c^2
*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*sinh(f*x + e)^4 + 3*d^3*e + 2*(d^3*e^3 + 3*c^2*d*e*f^2
- c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)^2 + 2*(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3
*e + 3*(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)^2 - 3*(c*d^2*e^2
+ c*d^2)*f)*sinh(f*x + e)^2 - 3*(c*d^2*e^2 + c*d^2)*f + 4*((d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c
*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)^3 + (d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)
*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + I) + 4*(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 +
(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)^4 + 4*(d^3*e^3 + 3*c^2*
d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*e^3 + 3*c^2*d*e*f^
2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*sinh(f*x + e)^4 + 3*d^3*e + 2*(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*
f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)^2 + 2*(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e + 3*
(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f*x + e)^2 - 3*(c*d^2*e^2 + c*d^2
)*f)*sinh(f*x + e)^2 - 3*(c*d^2*e^2 + c*d^2)*f + 4*((d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^
2 + c*d^2)*f)*cosh(f*x + e)^3 + (d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 3*d^3*e - 3*(c*d^2*e^2 + c*d^2)*f)*cosh(f
*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) - I) - 4*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 -
3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d
^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*cosh(f*x + e)^4 + 4*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f +
3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*f^3*x^3 + 3*c*d^2*f^3
*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*sinh(f*x + e)^4 + 3*d^3*e
+ 2*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)
*x)*cosh(f*x + e)^2 + 2*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3
*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)
*cosh(f*x + e)^2 + 3*(c^2*d*f^3 + d^3*f)*x)*sinh(f*x + e)^2 + 3*(c^2*d*f^3 + d^3*f)*x + 4*((d^3*f^3*x^3 + 3*c*
d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*cosh(f*x + e)^3 + (
d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*c
osh(f*x + e))*sinh(f*x + e))*log(I*cosh(f*x + e) + I*sinh(f*x + e) + 1) - 4*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d
^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*
f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*cosh(f*x + e)^4 + 4*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^
2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*f^3*x^3 + 3*
c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*sinh(f*x + e)^4 +
3*d^3*e + 2*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3
+ d^3*f)*x)*cosh(f*x + e)^2 + 2*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*
d^3*e + 3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 +
d^3*f)*x)*cosh(f*x + e)^2 + 3*(c^2*d*f^3 + d^3*f)*x)*sinh(f*x + e)^2 + 3*(c^2*d*f^3 + d^3*f)*x + 4*((d^3*f^3*x
^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^3*f)*x)*cosh(f*x +
e)^3 + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*d^3*e + 3*(c^2*d*f^3 + d^
3*f)*x)*cosh(f*x + e))*sinh(f*x + e))*log(-I*cosh(f*x + e) - I*sinh(f*x + e) + 1) - 24*(d^3*cosh(f*x + e)^4 +
4*d^3*cosh(f*x + e)*sinh(f*x + e)^3 + d^3*sinh(f*x + e)^4 + 2*d^3*cosh(f*x + e)^2 + d^3 + 2*(3*d^3*cosh(f*x +
e)^2 + d^3)*sinh(f*x + e)^2 + 4*(d^3*cosh(f*x + e)^3 + d^3*cosh(f*x + e))*sinh(f*x + e))*polylog(4, I*cosh(f*x
+ e) + I*sinh(f*x + e)) - 24*(d^3*cosh(f*x + e)^4 + 4*d^3*cosh(f*x + e)*sinh(f*x + e)^3 + d^3*sinh(f*x + e)^4
+ 2*d^3*cosh(f*x + e)^2 + d^3 + 2*(3*d^3*cosh(f*x + e)^2 + d^3)*sinh(f*x + e)^2 + 4*(d^3*cosh(f*x + e)^3 + d^
3*cosh(f*x + e))*sinh(f*x + e))*polylog(4, -I*cosh(f*x + e) - I*sinh(f*x + e)) + 24*(d^3*f*x + (d^3*f*x + c*d^
2*f)*cosh(f*x + e)^4 + 4*(d^3*f*x + c*d^2*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*f*x + c*d^2*f)*sinh(f*x + e)
^4 + c*d^2*f + 2*(d^3*f*x + c*d^2*f)*cosh(f*x + e)^2 + 2*(d^3*f*x + c*d^2*f + 3*(d^3*f*x + c*d^2*f)*cosh(f*x +
e)^2)*sinh(f*x + e)^2 + 4*((d^3*f*x + c*d^2*f)*cosh(f*x + e)^3 + (d^3*f*x + c*d^2*f)*cosh(f*x + e))*sinh(f*x
+ e))*polylog(3, I*cosh(f*x + e) + I*sinh(f*x + e)) + 24*(d^3*f*x + (d^3*f*x + c*d^2*f)*cosh(f*x + e)^4 + 4*(d
^3*f*x + c*d^2*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^3*f*x + c*d^2*f)*sinh(f*x + e)^4 + c*d^2*f + 2*(d^3*f*x +
c*d^2*f)*cosh(f*x + e)^2 + 2*(d^3*f*x + c*d^2*f + 3*(d^3*f*x + c*d^2*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*
((d^3*f*x + c*d^2*f)*cosh(f*x + e)^3 + (d^3*f*x + c*d^2*f)*cosh(f*x + e))*sinh(f*x + e))*polylog(3, -I*cosh(f*
x + e) - I*sinh(f*x + e)) + 4*((d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 - 2*d^3*e^4 - 12*c^2*d*e^2*f^2 + 8*c^3*e*f^3 - 1
2*d^3*e^2 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 + 8*(c*d^2*e^3 + 3*c*d^2*e)*f + 4*(c^3*f^4 + 6*c*d^2*f^2)*x)*cosh(f*
x + e)^3 + (d^3*f^4*x^4 - 2*d^3*e^4 - 12*d^3*e^2 + 4*(2*c^3*e - c^3)*f^3 + 4*(c*d^2*f^4 - d^3*f^3)*x^3 - 6*(2*
c^2*d*e^2 + c^2*d)*f^2 + 6*(c^2*d*f^4 - 2*c*d^2*f^3 + d^3*f^2)*x^2 + 8*(c*d^2*e^3 + 3*c*d^2*e)*f + 4*(c^3*f^4
- 3*c^2*d*f^3 + 3*c*d^2*f^2)*x)*cosh(f*x + e))*sinh(f*x + e))/(f^4*cosh(f*x + e)^4 + 4*f^4*cosh(f*x + e)*sinh(
f*x + e)^3 + f^4*sinh(f*x + e)^4 + 2*f^4*cosh(f*x + e)^2 + f^4 + 2*(3*f^4*cosh(f*x + e)^2 + f^4)*sinh(f*x + e)
^2 + 4*(f^4*cosh(f*x + e)^3 + f^4*cosh(f*x + e))*sinh(f*x + e))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \tanh \left (f x + e\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*tanh(f*x+e)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*tanh(f*x + e)^3, x)

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maple [B]  time = 0.31, size = 677, normalized size = 2.86 \[ -c \,d^{2} x^{3}-\frac {6 d \,c^{2} e x}{f}+\frac {6 d^{2} c \,e^{2} x}{f^{2}}+\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{2 f x +2 e}+1\right ) x^{2}}{f}+\frac {3 c^{2} d \ln \left ({\mathrm e}^{2 f x +2 e}+1\right ) x}{f}-\frac {6 c \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {6 c^{2} d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}-\frac {3 d \,c^{2} e^{2}}{f^{2}}-\frac {2 d^{3} e^{3} x}{f^{3}}+\frac {2 d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {d^{3} \ln \left ({\mathrm e}^{2 f x +2 e}+1\right ) x^{3}}{f}-\frac {3 c \,d^{2} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}+\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{2 f^{2}}-\frac {3 d^{3} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right ) x}{2 f^{3}}-\frac {6 d^{3} e x}{f^{3}}-\frac {6 c \,d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {6 d^{3} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{2 f x +2 e}+1\right ) x}{f^{3}}+\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{2 f x +2 e}+1\right )}{f^{3}}+\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{4}}+\frac {4 d^{2} c \,e^{3}}{f^{3}}-\frac {3 e^{2} d^{3}}{f^{4}}-\frac {3 d^{3} x^{2}}{f^{2}}+\frac {3 c^{2} d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}+\frac {2 d^{3} f \,x^{3} {\mathrm e}^{2 f x +2 e}+6 c \,d^{2} f \,x^{2} {\mathrm e}^{2 f x +2 e}+6 c^{2} d f x \,{\mathrm e}^{2 f x +2 e}+3 d^{3} x^{2} {\mathrm e}^{2 f x +2 e}+2 c^{3} f \,{\mathrm e}^{2 f x +2 e}+6 c \,d^{2} x \,{\mathrm e}^{2 f x +2 e}+3 c^{2} d \,{\mathrm e}^{2 f x +2 e}+3 d^{3} x^{2}+6 c \,d^{2} x +3 c^{2} d}{f^{2} \left ({\mathrm e}^{2 f x +2 e}+1\right )^{2}}+\frac {3 d^{3} \polylog \left (4, -{\mathrm e}^{2 f x +2 e}\right )}{4 f^{4}}-\frac {3 c^{2} d \,x^{2}}{2}-\frac {3 d^{3} e^{4}}{2 f^{4}}+\frac {c^{3} \ln \left ({\mathrm e}^{2 f x +2 e}+1\right )}{f}-\frac {2 c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {d^{3} x^{4}}{4}+c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.57, size = 595, normalized size = 2.51 \[ c^{3} {\left (x + \frac {e}{f} + \frac {\log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{f} + \frac {2 \, e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}}\right )} - \frac {6 \, c d^{2} x}{f^{2}} + \frac {3 \, {\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 )} c d^{2}}{2 \, f^{3}} + \frac {3 \, c d^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{f^{3}} + \frac {d^{3} f^{2} x^{4} + 4 \, c d^{2} f^{2} x^{3} + 24 \, c d^{2} x + 12 \, c^{2} d + 6 \, {\left (c^{2} d f^{2} + 2 \, d^{3}\right )} x^{2} + {\left (d^{3} f^{2} x^{4} e^{\left (4 \, e\right )} + 4 \, c d^{2} f^{2} x^{3} e^{\left (4 \, e\right )} + 6 \, c^{2} d f^{2} x^{2} e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left (d^{3} f^{2} x^{4} e^{\left (2 \, e\right )} + 4 \, {\left (c d^{2} f^{2} e^{\left (2 \, e\right )} + d^{3} f e^{\left (2 \, e\right )}\right )} x^{3} + 6 \, c^{2} d e^{\left (2 \, e\right )} + 6 \, {\left (c^{2} d f^{2} e^{\left (2 \, e\right )} + 2 \, c d^{2} f e^{\left (2 \, e\right )} + d^{3} e^{\left (2 \, e\right )}\right )} x^{2} + 12 \, {\left (c^{2} d f e^{\left (2 \, e\right )} + c d^{2} e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}{4 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} + \frac {{\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 )} d^{3}}{3 \, f^{4}} + \frac {3 \, {\left (c^{2} d f^{2} + d^{3}\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 )}}{2 \, f^{4}} - \frac {d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} + 6 \, {\left (c^{2} d f^{2} + d^{3}\right )} f^{2} x^{2}}{2 \, f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(x + e/f + log(e^(-2*f*x - 2*e) + 1)/f + 2*e^(-2*f*x - 2*e)/(f*(2*e^(-2*f*x - 2*e) + e^(-4*f*x - 4*e) + 1)
)) - 6*c*d^2*x/f^2 + 3/2*(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)))*c*d^2/f^3 + 3*c*d^2*log(e^(2*f*x + 2*e) + 1)/f^3 + 1/4*(d^3*f^2*x^4 + 4*c*d^2*f^2*x^3 + 24*c*d^
2*x + 12*c^2*d + 6*(c^2*d*f^2 + 2*d^3)*x^2 + (d^3*f^2*x^4*e^(4*e) + 4*c*d^2*f^2*x^3*e^(4*e) + 6*c^2*d*f^2*x^2*
e^(4*e))*e^(4*f*x) + 2*(d^3*f^2*x^4*e^(2*e) + 4*(c*d^2*f^2*e^(2*e) + d^3*f*e^(2*e))*x^3 + 6*c^2*d*e^(2*e) + 6*
(c^2*d*f^2*e^(2*e) + 2*c*d^2*f*e^(2*e) + d^3*e^(2*e))*x^2 + 12*(c^2*d*f*e^(2*e) + c*d^2*e^(2*e))*x)*e^(2*f*x))
/(f^2*e^(4*f*x + 4*e) + 2*f^2*e^(2*f*x + 2*e) + f^2) + 1/3*(4*f^3*x^3*log(e^(2*f*x + 2*e) + 1) + 6*f^2*x^2*dil
og(-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)))*d^3/f^4 + 3/2*(c^2
*d*f^2 + d^3)*(2*f*x*log(e^(2*f*x + 2*e) + 1) + dilog(-e^(2*f*x + 2*e)))/f^4 - 1/2*(d^3*f^4*x^4 + 4*c*d^2*f^4*
x^3 + 6*(c^2*d*f^2 + d^3)*f^2*x^2)/f^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tanh}\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^3*(c + d*x)^3,x)

[Out]

int(tanh(e + f*x)^3*(c + d*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{3} \tanh ^{3}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x)**3*tanh(e + f*x)**3, x)

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