3.1.11 \(\int (c+d x)^3 \tanh ^3(e+f x) \, dx\) [11]

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

[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 = 0.27, 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 = {3801, 3799, 2221, 2317, 2438, 32, 2611, 6744, 2320, 6724} \begin {gather*} -\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} \end {gather*}

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 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 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 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 \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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 10.68, size = 817, normalized size = 3.45 \begin {gather*} \frac {c d^2 e^{-e} \left (-2 f^2 x^2 \left (2 e^{2 e} f x-3 \left (1+e^{2 e}\right ) \log \left (1+e^{2 (e+f x)}\right )\right )+6 \left (1+e^{2 e}\right ) f x \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )-3 \left (1+e^{2 e}\right ) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )\right ) \text {sech}(e)}{4 f^3}+\frac {1}{4} d^3 e^e \left (-x^4+\left (1+e^{-2 e}\right ) x^4-\frac {e^{-2 e} \left (1+e^{2 e}\right ) \left (2 f^4 x^4-4 f^3 x^3 \log \left (1+e^{2 (e+f x)}\right )-6 f^2 x^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )+6 f x \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )-3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )\right )}{2 f^4}\right ) \text {sech}(e)+\frac {(c+d x)^3 \text {sech}^2(e+f x)}{2 f}+\frac {3 c d^2 \text {sech}(e) (\cosh (e) \log (\cosh (e) \cosh (f x)+\sinh (e) \sinh (f x))-f x \sinh (e))}{f^3 \left (\cosh ^2(e)-\sinh ^2(e)\right )}+\frac {c^3 \text {sech}(e) (\cosh (e) \log (\cosh (e) \cosh (f x)+\sinh (e) \sinh (f x))-f x \sinh (e))}{f \left (\cosh ^2(e)-\sinh ^2(e)\right )}+\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 (-\pi +2 i \tanh ^{-1}(\coth (e))\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 {PolyLog}\left (2,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 (-\cosh ^2(e)+\sinh ^2(e)\right )}}+\frac {3 c^2 d \text {csch}(e) \left (e^{-\tanh ^{-1}(\coth (e))} f^2 x^2-\frac {i \coth (e) \left (-f x \left (-\pi +2 i \tanh ^{-1}(\coth (e))\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 {PolyLog}\left (2,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^2 \sqrt {\text {csch}^2(e) \left (-\cosh ^2(e)+\sinh ^2(e)\right )}}-\frac {3 \text {sech}(e) \text {sech}(e+f x) \left (c^2 d \sinh (f x)+2 c d^2 x \sinh (f x)+d^3 x^2 \sinh (f x)\right )}{2 f^2}+\frac {1}{4} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \tanh (e) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*d^2*(-2*f^2*x^2*(2*E^(2*e)*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*(-x^4 + (1
+ E^(-2*e))*x^4 - ((1 + E^(2*e))*(2*f^4*x^4 - 4*f^3*x^3*Log[1 + E^(2*(e + f*x))] - 6*f^2*x^2*PolyLog[2, -E^(2*
(e + f*x))] + 6*f*x*PolyLog[3, -E^(2*(e + f*x))] - 3*PolyLog[4, -E^(2*(e + f*x))]))/(2*E^(2*e)*f^4))*Sech[e])/
4 + ((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*Sinh[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*PolyLog[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[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*ArcTa
nh[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 + S
inh[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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(684\) vs. \(2(219)=438\).
time = 2.02, size = 685, normalized size = 2.89

method	result	size
risch	\(\frac {3 d^{2} c \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}-d^{2} c \,x^{3}+\frac {3 d^{3} \polylog \left (4, -{\mathrm e}^{2 f x +2 e}\right )}{4 f^{4}}-\frac {6 c^{2} d e x}{f}-\frac {3 c^{2} d \,e^{2}}{f^{2}}-\frac {3 d^{3} e^{4}}{2 f^{4}}-\frac {2 c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {c^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}+c^{3} x +\frac {6 c \,e^{2} d^{2} x}{f^{2}}+\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{4}}+\frac {3 d \,c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {6 d e \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {6 d^{2} e^{2} c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {3 d^{2} c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f}-\frac {2 d^{3} e^{3} x}{f^{3}}-\frac {3 d^{3} x^{2}}{f^{2}}-\frac {3 e^{2} d^{3}}{f^{4}}-\frac {3 d^{3} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right ) x}{2 f^{3}}+\frac {3 d \,c^{2} \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 (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}-\frac {3 d \,c^{2} x^{2}}{2}+\frac {3 c \,d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}+\frac {6 d^{3} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {2 d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}-\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} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f^{3}}+\frac {d^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{3}}{f}-\frac {6 d^{3} e x}{f^{3}}-\frac {6 c \,d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {4 c \,d^{2} e^{3}}{f^{3}}-\frac {d^{3} x^{4}}{4}+\frac {c^{4}}{4 d}\)	\(685\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (225) = 450\).
time = 0.57, size = 603, normalized size = 2.54 \begin {gather*} 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} + d^{3} f\right )} x^{3} e^{\left (2 \, e\right )} + 6 \, c^{2} d e^{\left (2 \, e\right )} + 6 \, {\left (c^{2} d f^{2} + 2 \, c d^{2} f + d^{3}\right )} x^{2} e^{\left (2 \, e\right )} + 12 \, {\left (c^{2} d f + c d^{2}\right )} x e^{\left (2 \, e\right )}\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}} \end {gather*}

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 + d^3*f)*x^3*e^(2*e) + 6*c^2*d*e^(2*e) + 6*(c^2*d*f
^2 + 2*c*d^2*f + d^3)*x^2*e^(2*e) + 12*(c^2*d*f + c*d^2)*x*e^(2*e))*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*dilog(-e^(2*f*x + 2*e)) - 6*f*x*pol
ylog(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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Fricas [C] Result contains complex when optimal does not.
time = 0.55, size = 9360, normalized size = 39.49 \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*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 + 8*c*d^2*f*cosh(1)^3 - 2*d^3*cosh(1)^4 -
2*d^3*sinh(1)^4 - 12*c^2*d*f^2 + (d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 8*c*d^2*f*cosh(1)^3 - 2*d^3*cosh(1)^4 - 2*d^
3*sinh(1)^4 + 8*(c*d^2*f - d^3*cosh(1))*sinh(1)^3 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 - 12*(c^2*d*f^2 + d^3)*cosh(
1)^2 - 12*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3)*sinh(1)^2 + 4*(c^3*f^4 + 6*c*d^2*f^2)*x + 8*(c
^3*f^3 + 3*c*d^2*f)*cosh(1) + 8*(c^3*f^3 + 3*c*d^2*f*cosh(1)^2 - d^3*cosh(1)^3 + 3*c*d^2*f - 3*(c^2*d*f^2 + d^
3)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 8*c*d^2*f*cosh(1)^3
- 2*d^3*cosh(1)^4 - 2*d^3*sinh(1)^4 + 8*(c*d^2*f - d^3*cosh(1))*sinh(1)^3 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 - 12
*(c^2*d*f^2 + d^3)*cosh(1)^2 - 12*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3)*sinh(1)^2 + 4*(c^3*f^4
+ 6*c*d^2*f^2)*x + 8*(c^3*f^3 + 3*c*d^2*f)*cosh(1) + 8*(c^3*f^3 + 3*c*d^2*f*cosh(1)^2 - d^3*cosh(1)^3 + 3*c*d
^2*f - 3*(c^2*d*f^2 + d^3)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (
d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 8*c*d^2*f*cosh(1)^3 - 2*d^3*cosh(1)^4 - 2*d^3*sinh(1)^4 + 8*(c*d^2*f - d^3*cos
h(1))*sinh(1)^3 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 - 12*(c^2*d*f^2 + d^3)*cosh(1)^2 - 12*(c^2*d*f^2 - 2*c*d^2*f*c
osh(1) + d^3*cosh(1)^2 + d^3)*sinh(1)^2 + 4*(c^3*f^4 + 6*c*d^2*f^2)*x + 8*(c^3*f^3 + 3*c*d^2*f)*cosh(1) + 8*(c
^3*f^3 + 3*c*d^2*f*cosh(1)^2 - d^3*cosh(1)^3 + 3*c*d^2*f - 3*(c^2*d*f^2 + d^3)*cosh(1))*sinh(1))*sinh(f*x + co
sh(1) + sinh(1))^4 + 8*(c*d^2*f - d^3*cosh(1))*sinh(1)^3 - 12*(c^2*d*f^2 + d^3)*cosh(1)^2 + 2*(d^3*f^4*x^4 + 8
*c*d^2*f*cosh(1)^3 - 2*d^3*cosh(1)^4 - 2*d^3*sinh(1)^4 - 4*c^3*f^3 - 6*c^2*d*f^2 + 4*(c*d^2*f^4 - d^3*f^3)*x^3
+ 8*(c*d^2*f - d^3*cosh(1))*sinh(1)^3 + 6*(c^2*d*f^4 - 2*c*d^2*f^3 + d^3*f^2)*x^2 - 12*(c^2*d*f^2 + d^3)*cosh
(1)^2 - 12*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3)*sinh(1)^2 + 4*(c^3*f^4 - 3*c^2*d*f^3 + 3*c*d^
2*f^2)*x + 8*(c^3*f^3 + 3*c*d^2*f)*cosh(1) + 8*(c^3*f^3 + 3*c*d^2*f*cosh(1)^2 - d^3*cosh(1)^3 + 3*c*d^2*f - 3*
(c^2*d*f^2 + d^3)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 12*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*
cosh(1)^2 + d^3)*sinh(1)^2 + 2*(d^3*f^4*x^4 + 8*c*d^2*f*cosh(1)^3 - 2*d^3*cosh(1)^4 - 2*d^3*sinh(1)^4 - 4*c^3*
f^3 - 6*c^2*d*f^2 + 4*(c*d^2*f^4 - d^3*f^3)*x^3 + 8*(c*d^2*f - d^3*cosh(1))*sinh(1)^3 + 6*(c^2*d*f^4 - 2*c*d^2
*f^3 + d^3*f^2)*x^2 - 12*(c^2*d*f^2 + d^3)*cosh(1)^2 + 3*(d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 8*c*d^2*f*cosh(1)^3
- 2*d^3*cosh(1)^4 - 2*d^3*sinh(1)^4 + 8*(c*d^2*f - d^3*cosh(1))*sinh(1)^3 + 6*(c^2*d*f^4 + 2*d^3*f^2)*x^2 - 12
*(c^2*d*f^2 + d^3)*cosh(1)^2 - 12*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3)*sinh(1)^2 + 4*(c^3*f^4
+ 6*c*d^2*f^2)*x + 8*(c^3*f^3 + 3*c*d^2*f)*cosh(1) + 8*(c^3*f^3 + 3*c*d^2*f*cosh(1)^2 - d^3*cosh(1)^3 + 3*c*d
^2*f - 3*(c^2*d*f^2 + d^3)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 12*(c^2*d*f^2 - 2*c*d^2*f*cosh(
1) + d^3*cosh(1)^2 + d^3)*sinh(1)^2 + 4*(c^3*f^4 - 3*c^2*d*f^3 + 3*c*d^2*f^2)*x + 8*(c^3*f^3 + 3*c*d^2*f)*cosh
(1) + 8*(c^3*f^3 + 3*c*d^2*f*cosh(1)^2 - d^3*cosh(1)^3 + 3*c*d^2*f - 3*(c^2*d*f^2 + d^3)*cosh(1))*sinh(1))*sin
h(f*x + cosh(1) + sinh(1))^2 + 8*(c^3*f^3 + 3*c*d^2*f)*cosh(1) - 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 + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (d^3*f^2*x^2 + 2*c*d^2*
f^2*x + c^2*d*f^2 + d^3)*sinh(f*x + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))^2)*sinh(f*x + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f
^2 + d^3)*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*dilog(I*cosh(f*x + cosh(1) + sinh(1))
+ I*sinh(f*x + cosh(1) + sinh(1))) - 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 + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*sin
h(f*x + cosh(1) + sinh(1))^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 + cosh(1) + si
nh(1))^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 + cosh(1) + sinh(1))^2)*sinh(f*x + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + d^3)*cosh(f*x + cosh(1)
+ sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*dilog(-I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x + cosh(1) + si
nh(1))) - 4*(c^3*f^3 + 3*c*d^2*f*cosh(1)^2 - d^...

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

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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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*tanh(f*x+e)^3,x, algorithm="giac")

[Out]

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

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

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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