3.13 \(\int (c+d x) \tanh ^3(e+f x) \, dx\)
Optimal. Leaf size=100 \[ \frac{d \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac{(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac{(c+d x) \tanh ^2(e+f x)}{2 f}-\frac{(c+d x)^2}{2 d}-\frac{d \tanh (e+f x)}{2 f^2}+\frac{d x}{2 f} \]
[Out]
(d*x)/(2*f) - (c + d*x)^2/(2*d) + ((c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (d*PolyLog[2, -E^(2*(e + f*x))])/(2
*f^2) - (d*Tanh[e + f*x])/(2*f^2) - ((c + d*x)*Tanh[e + f*x]^2)/(2*f)
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Rubi [A] time = 0.137082, antiderivative size = 100, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used =
{3720, 3473, 8, 3718, 2190, 2279, 2391} \[ \frac{d \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac{(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac{(c+d x) \tanh ^2(e+f x)}{2 f}-\frac{(c+d x)^2}{2 d}-\frac{d \tanh (e+f x)}{2 f^2}+\frac{d x}{2 f} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)*Tanh[e + f*x]^3,x]
[Out]
(d*x)/(2*f) - (c + d*x)^2/(2*d) + ((c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (d*PolyLog[2, -E^(2*(e + f*x))])/(2
*f^2) - (d*Tanh[e + f*x])/(2*f^2) - ((c + d*x)*Tanh[e + f*x]^2)/(2*f)
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 3473
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
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 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 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]
Rubi steps
\begin{align*} \int (c+d x) \tanh ^3(e+f x) \, dx &=-\frac{(c+d x) \tanh ^2(e+f x)}{2 f}+\frac{d \int \tanh ^2(e+f x) \, dx}{2 f}+\int (c+d x) \tanh (e+f x) \, dx\\ &=-\frac{(c+d x)^2}{2 d}-\frac{d \tanh (e+f x)}{2 f^2}-\frac{(c+d x) \tanh ^2(e+f x)}{2 f}+2 \int \frac{e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx+\frac{d \int 1 \, dx}{2 f}\\ &=\frac{d x}{2 f}-\frac{(c+d x)^2}{2 d}+\frac{(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac{d \tanh (e+f x)}{2 f^2}-\frac{(c+d x) \tanh ^2(e+f x)}{2 f}-\frac{d \int \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac{d x}{2 f}-\frac{(c+d x)^2}{2 d}+\frac{(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac{d \tanh (e+f x)}{2 f^2}-\frac{(c+d x) \tanh ^2(e+f x)}{2 f}-\frac{d \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}\\ &=\frac{d x}{2 f}-\frac{(c+d x)^2}{2 d}+\frac{(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac{d \text{Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac{d \tanh (e+f x)}{2 f^2}-\frac{(c+d x) \tanh ^2(e+f x)}{2 f}\\ \end{align*}
Mathematica [C] time = 6.13654, size = 264, normalized size = 2.64 \[ -\frac{d \text{csch}(e) \text{sech}(e) \left (-f^2 x^2 e^{-\tanh ^{-1}(\coth (e))}+\frac{i \coth (e) \left (i \text{PolyLog}\left (2,e^{2 i \left (i \tanh ^{-1}(\coth (e))+i f x\right )}\right )-f x \left (-\pi +2 i \tanh ^{-1}(\coth (e))\right )-2 \left (i \tanh ^{-1}(\coth (e))+i f x\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}(\coth (e))+i f x\right )}\right )+2 i \tanh ^{-1}(\coth (e)) \log \left (i \sinh \left (\tanh ^{-1}(\coth (e))+f x\right )\right )-\pi \log \left (e^{2 f x}+1\right )+\pi \log (\cosh (f x))\right )}{\sqrt{1-\coth ^2(e)}}\right )}{2 f^2 \sqrt{\text{csch}^2(e) \left (\sinh ^2(e)-\cosh ^2(e)\right )}}-\frac{c \tanh ^2(e+f x)}{2 f}+\frac{c \log (\cosh (e+f x))}{f}-\frac{d \text{sech}(e) \sinh (f x) \text{sech}(e+f x)}{2 f^2}+\frac{d x \text{sech}^2(e+f x)}{2 f}+\frac{1}{2} d x^2 \tanh (e) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*x)*Tanh[e + f*x]^3,x]
[Out]
(c*Log[Cosh[e + f*x]])/f + (d*x*Sech[e + f*x]^2)/(2*f) - (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*ArcTanh[Coth[e]]))] + Pi*Log[Cosh[f*x]] + (2*I)*ArcTanh[Coth[e]]*Log[I*Sinh[f*x + ArcTanh[Co
th[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[Cs
ch[e]^2*(-Cosh[e]^2 + Sinh[e]^2)]) - (d*Sech[e]*Sech[e + f*x]*Sinh[f*x])/(2*f^2) + (d*x^2*Tanh[e])/2 - (c*Tanh
[e + f*x]^2)/(2*f)
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Maple [A] time = 0.039, size = 166, normalized size = 1.7 \begin{align*} -{\frac{d{x}^{2}}{2}}+cx+{\frac{2\,dfx{{\rm e}^{2\,fx+2\,e}}+2\,cf{{\rm e}^{2\,fx+2\,e}}+d{{\rm e}^{2\,fx+2\,e}}+d}{{f}^{2} \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) ^{2}}}+{\frac{c\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) }{f}}-2\,{\frac{c\ln \left ({{\rm e}^{fx+e}} \right ) }{f}}-2\,{\frac{dex}{f}}-{\frac{d{e}^{2}}{{f}^{2}}}+{\frac{d\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) x}{f}}+{\frac{d{\it polylog} \left ( 2,-{{\rm e}^{2\,fx+2\,e}} \right ) }{2\,{f}^{2}}}+2\,{\frac{de\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)*tanh(f*x+e)^3,x)
[Out]
-1/2*d*x^2+c*x+(2*d*f*x*exp(2*f*x+2*e)+2*c*f*exp(2*f*x+2*e)+d*exp(2*f*x+2*e)+d)/f^2/(exp(2*f*x+2*e)+1)^2+1/f*c
*ln(exp(2*f*x+2*e)+1)-2/f*c*ln(exp(f*x+e))-2/f*d*e*x-1/f^2*d*e^2+1/f*d*ln(exp(2*f*x+2*e)+1)*x+1/2*d*polylog(2,
-exp(2*f*x+2*e))/f^2+2/f^2*d*e*ln(exp(f*x+e))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} c{\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{1}{2} \, d{\left (\frac{f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} + f^{2} x^{2} + 2 \,{\left (f^{2} x^{2} e^{\left (2 \, e\right )} + 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 2}{f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}} - 4 \, \int \frac{x}{e^{\left (2 \, f x + 2 \, e\right )} + 1}\,{d x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="maxima")
[Out]
c*(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)))
+ 1/2*d*((f^2*x^2*e^(4*f*x + 4*e) + f^2*x^2 + 2*(f^2*x^2*e^(2*e) + 2*f*x*e^(2*e) + e^(2*e))*e^(2*f*x) + 2)/(f
^2*e^(4*f*x + 4*e) + 2*f^2*e^(2*f*x + 2*e) + f^2) - 4*integrate(x/(e^(2*f*x + 2*e) + 1), x))
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Fricas [C] time = 2.42079, size = 3710, normalized size = 37.1 \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)*tanh(f*x+e)^3,x, algorithm="fricas")
[Out]
-1/2*(d*f^2*x^2 + (d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*cosh(f*x + e)^4 + 4*(d*f^2*x^2 + 2*c*f^2*x - 2*d
*e^2 + 4*c*e*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*sinh(f*x + e)^4 +
2*c*f^2*x - 2*d*e^2 + 4*c*e*f + 2*(d*f^2*x^2 - 2*d*e^2 + 2*(2*c*e - c)*f + 2*(c*f^2 - d*f)*x - d)*cosh(f*x + e
)^2 + 2*(d*f^2*x^2 - 2*d*e^2 + 3*(d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*cosh(f*x + e)^2 + 2*(2*c*e - c)*f
+ 2*(c*f^2 - d*f)*x - d)*sinh(f*x + e)^2 - 2*(d*cosh(f*x + e)^4 + 4*d*cosh(f*x + e)*sinh(f*x + e)^3 + d*sinh(
f*x + e)^4 + 2*d*cosh(f*x + e)^2 + 2*(3*d*cosh(f*x + e)^2 + d)*sinh(f*x + e)^2 + 4*(d*cosh(f*x + e)^3 + d*cosh
(f*x + e))*sinh(f*x + e) + d)*dilog(I*cosh(f*x + e) + I*sinh(f*x + e)) - 2*(d*cosh(f*x + e)^4 + 4*d*cosh(f*x +
e)*sinh(f*x + e)^3 + d*sinh(f*x + e)^4 + 2*d*cosh(f*x + e)^2 + 2*(3*d*cosh(f*x + e)^2 + d)*sinh(f*x + e)^2 +
4*(d*cosh(f*x + e)^3 + d*cosh(f*x + e))*sinh(f*x + e) + d)*dilog(-I*cosh(f*x + e) - I*sinh(f*x + e)) + 2*((d*e
- c*f)*cosh(f*x + e)^4 + 4*(d*e - c*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*e - c*f)*sinh(f*x + e)^4 + 2*(d*e -
c*f)*cosh(f*x + e)^2 + 2*(3*(d*e - c*f)*cosh(f*x + e)^2 + d*e - c*f)*sinh(f*x + e)^2 + d*e - c*f + 4*((d*e -
c*f)*cosh(f*x + e)^3 + (d*e - c*f)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + I) + 2*((
d*e - c*f)*cosh(f*x + e)^4 + 4*(d*e - c*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*e - c*f)*sinh(f*x + e)^4 + 2*(d*
e - c*f)*cosh(f*x + e)^2 + 2*(3*(d*e - c*f)*cosh(f*x + e)^2 + d*e - c*f)*sinh(f*x + e)^2 + d*e - c*f + 4*((d*e
- c*f)*cosh(f*x + e)^3 + (d*e - c*f)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) - I) - 2
*((d*f*x + d*e)*cosh(f*x + e)^4 + 4*(d*f*x + d*e)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*f*x + d*e)*sinh(f*x + e)^
4 + d*f*x + 2*(d*f*x + d*e)*cosh(f*x + e)^2 + 2*(d*f*x + 3*(d*f*x + d*e)*cosh(f*x + e)^2 + d*e)*sinh(f*x + e)^
2 + d*e + 4*((d*f*x + d*e)*cosh(f*x + e)^3 + (d*f*x + d*e)*cosh(f*x + e))*sinh(f*x + e))*log(I*cosh(f*x + e) +
I*sinh(f*x + e) + 1) - 2*((d*f*x + d*e)*cosh(f*x + e)^4 + 4*(d*f*x + d*e)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*
f*x + d*e)*sinh(f*x + e)^4 + d*f*x + 2*(d*f*x + d*e)*cosh(f*x + e)^2 + 2*(d*f*x + 3*(d*f*x + d*e)*cosh(f*x + e
)^2 + d*e)*sinh(f*x + e)^2 + d*e + 4*((d*f*x + d*e)*cosh(f*x + e)^3 + (d*f*x + d*e)*cosh(f*x + e))*sinh(f*x +
e))*log(-I*cosh(f*x + e) - I*sinh(f*x + e) + 1) + 4*((d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*cosh(f*x + e)
^3 + (d*f^2*x^2 - 2*d*e^2 + 2*(2*c*e - c)*f + 2*(c*f^2 - d*f)*x - d)*cosh(f*x + e))*sinh(f*x + e) - 2*d)/(f^2*
cosh(f*x + e)^4 + 4*f^2*cosh(f*x + e)*sinh(f*x + e)^3 + f^2*sinh(f*x + e)^4 + 2*f^2*cosh(f*x + e)^2 + 2*(3*f^2
*cosh(f*x + e)^2 + f^2)*sinh(f*x + e)^2 + f^2 + 4*(f^2*cosh(f*x + e)^3 + f^2*cosh(f*x + e))*sinh(f*x + e))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right ) \tanh ^{3}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*tanh(f*x+e)**3,x)
[Out]
Integral((c + d*x)*tanh(e + f*x)**3, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \tanh \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="giac")
[Out]
integrate((d*x + c)*tanh(f*x + e)^3, x)