3.581 \(\int \sinh ^4(x) \tanh (x) \, dx\)

Optimal. Leaf size=18 \[ \frac{\cosh ^4(x)}{4}-\cosh ^2(x)+\log (\cosh (x)) \]

[Out]

-Cosh[x]^2 + Cosh[x]^4/4 + Log[Cosh[x]]

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Rubi [A]  time = 0.0424013, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{\cosh ^4(x)}{4}-\cosh ^2(x)+\log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]  Int[Sinh[x]^4*Tanh[x],x]

[Out]

-Cosh[x]^2 + Cosh[x]^4/4 + Log[Cosh[x]]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(tanh(x)**5/sech(x)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 0.00704923, size = 20, normalized size = 1.11 \[ -\frac{3}{8} \cosh (2 x)+\frac{1}{32} \cosh (4 x)+\log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Sinh[x]^4*Tanh[x],x]

[Out]

(-3*Cosh[2*x])/8 + Cosh[4*x]/32 + Log[Cosh[x]]

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Maple [A]  time = 0.018, size = 17, normalized size = 0.9 \[{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{4}}-{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( \cosh \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tanh(x)^5/sech(x)^4,x)

[Out]

1/4*sinh(x)^4-1/2*sinh(x)^2+ln(cosh(x))

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Maxima [A]  time = 1.5143, size = 47, normalized size = 2.61 \[ -\frac{1}{64} \,{\left (12 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} + x - \frac{3}{16} \, e^{\left (-2 \, x\right )} + \frac{1}{64} \, e^{\left (-4 \, x\right )} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tanh(x)^5/sech(x)^4,x, algorithm="maxima")

[Out]

-1/64*(12*e^(-2*x) - 1)*e^(4*x) + x - 3/16*e^(-2*x) + 1/64*e^(-4*x) + log(e^(-2*
x) + 1)

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Fricas [A]  time = 0.2254, size = 347, normalized size = 19.28 \[ \frac{\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \,{\left (7 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 12 \, \cosh \left (x\right )^{6} + 8 \,{\left (7 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 64 \, x \cosh \left (x\right )^{4} + 2 \,{\left (35 \, \cosh \left (x\right )^{4} - 90 \, \cosh \left (x\right )^{2} - 32 \, x\right )} \sinh \left (x\right )^{4} + 8 \,{\left (7 \, \cosh \left (x\right )^{5} - 30 \, \cosh \left (x\right )^{3} - 32 \, x \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \,{\left (7 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} - 96 \, x \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{2} - 12 \, \cosh \left (x\right )^{2} + 64 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 8 \,{\left (\cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} - 32 \, x \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{64 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tanh(x)^5/sech(x)^4,x, algorithm="fricas")

[Out]

1/64*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 3)*sinh(x)^
6 - 12*cosh(x)^6 + 8*(7*cosh(x)^3 - 9*cosh(x))*sinh(x)^5 - 64*x*cosh(x)^4 + 2*(3
5*cosh(x)^4 - 90*cosh(x)^2 - 32*x)*sinh(x)^4 + 8*(7*cosh(x)^5 - 30*cosh(x)^3 - 3
2*x*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 45*cosh(x)^4 - 96*x*cosh(x)^2 - 3)*sin
h(x)^2 - 12*cosh(x)^2 + 64*(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x
)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)*log(2*cosh(x)/(cosh(x) - sinh(x))) + 8*(c
osh(x)^7 - 9*cosh(x)^5 - 32*x*cosh(x)^3 - 3*cosh(x))*sinh(x) + 1)/(cosh(x)^4 + 4
*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tanh ^{5}{\left (x \right )}}{\operatorname{sech}^{4}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tanh(x)**5/sech(x)**4,x)

[Out]

Integral(tanh(x)**5/sech(x)**4, x)

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GIAC/XCAS [A]  time = 0.207452, size = 58, normalized size = 3.22 \[ \frac{1}{64} \,{\left (48 \, e^{\left (4 \, x\right )} - 12 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} - x + \frac{1}{64} \, e^{\left (4 \, x\right )} - \frac{3}{16} \, e^{\left (2 \, x\right )} +{\rm ln}\left (e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tanh(x)^5/sech(x)^4,x, algorithm="giac")

[Out]

1/64*(48*e^(4*x) - 12*e^(2*x) + 1)*e^(-4*x) - x + 1/64*e^(4*x) - 3/16*e^(2*x) +
ln(e^(2*x) + 1)