3.703 \(\int \sin ^{-1}(\sinh (x)) \text{sech}^4(x) \, dx\)

Optimal. Leaf size=49 \[ -\frac{2}{3} \sin ^{-1}\left (\frac{\cosh (x)}{\sqrt{2}}\right )+\frac{1}{6} \sqrt{1-\sinh ^2(x)} \text{sech}(x)-\frac{1}{3} \tanh ^3(x) \sin ^{-1}(\sinh (x))+\tanh (x) \sin ^{-1}(\sinh (x)) \]

[Out]

(-2*ArcSin[Cosh[x]/Sqrt[2]])/3 + (Sech[x]*Sqrt[1 - Sinh[x]^2])/6 + ArcSin[Sinh[x
]]*Tanh[x] - (ArcSin[Sinh[x]]*Tanh[x]^3)/3

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Rubi [A]  time = 0.234634, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75 \[ \frac{1}{6} \sqrt{2-\cosh ^2(x)} \text{sech}(x)-\frac{2}{3} \sin ^{-1}\left (\frac{\cosh (x)}{\sqrt{2}}\right )-\frac{1}{3} \tanh ^3(x) \sin ^{-1}(\sinh (x))+\tanh (x) \sin ^{-1}(\sinh (x)) \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[Sinh[x]]*Sech[x]^4,x]

[Out]

(-2*ArcSin[Cosh[x]/Sqrt[2]])/3 + (Sqrt[2 - Cosh[x]^2]*Sech[x])/6 + ArcSin[Sinh[x
]]*Tanh[x] - (ArcSin[Sinh[x]]*Tanh[x]^3)/3

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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(asin(sinh(x))*sech(x)**4,x)

[Out]

Timed out

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Mathematica [C]  time = 0.32502, size = 66, normalized size = 1.35 \[ \frac{1}{12} \left (8 i \log \left (\sqrt{3-\cosh (2 x)}+i \sqrt{2} \cosh (x)\right )+\sqrt{6-2 \cosh (2 x)} \text{sech}(x)+4 (\cosh (2 x)+2) \tanh (x) \text{sech}^2(x) \sin ^{-1}(\sinh (x))\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSin[Sinh[x]]*Sech[x]^4,x]

[Out]

((8*I)*Log[I*Sqrt[2]*Cosh[x] + Sqrt[3 - Cosh[2*x]]] + Sqrt[6 - 2*Cosh[2*x]]*Sech
[x] + 4*ArcSin[Sinh[x]]*(2 + Cosh[2*x])*Sech[x]^2*Tanh[x])/12

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int \arcsin \left ( \sinh \left ( x \right ) \right ) \left ({\rm sech} \left (x\right ) \right ) ^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(sinh(x))*sech(x)^4,x)

[Out]

int(arcsin(sinh(x))*sech(x)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{4 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )} \arctan \left (e^{\left (2 \, x\right )} - 1, \sqrt{e^{\left (2 \, x\right )} + 2 \, e^{x} - 1} \sqrt{-e^{\left (2 \, x\right )} + 2 \, e^{x} + 1}\right ) + 16 \,{\left (e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 1\right )} \int -\frac{{\left (3 \, e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}\right )} e^{\left (\frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + 2 \, e^{x} - 1\right ) + \frac{1}{2} \, \log \left (-e^{\left (2 \, x\right )} + 2 \, e^{x} + 1\right )\right )}}{{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}{\left (e^{\left (2 \, x\right )} + 2 \, e^{x} - 1\right )}{\left (e^{\left (2 \, x\right )} - 2 \, e^{x} - 1\right )} - e^{\left (12 \, x\right )} + 6 \, e^{\left (10 \, x\right )} + e^{\left (8 \, x\right )} - 12 \, e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\,{d x}}{3 \,{\left (e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(sinh(x))*sech(x)^4,x, algorithm="maxima")

[Out]

-1/3*(4*(3*e^(2*x) + 1)*arctan2(e^(2*x) - 1, sqrt(e^(2*x) + 2*e^x - 1)*sqrt(-e^(
2*x) + 2*e^x + 1)) + 3*(e^(6*x) + 3*e^(4*x) + 3*e^(2*x) + 1)*integrate(16/3*(3*e
^(4*x) + e^(2*x))*e^(1/2*log(e^(2*x) + 2*e^x - 1) + 1/2*log(-e^(2*x) + 2*e^x + 1
))/((e^(8*x) - 4*e^(6*x) - 10*e^(4*x) - 4*e^(2*x) + 1)*e^(log(e^(2*x) + 2*e^x -
1) + log(-e^(2*x) + 2*e^x + 1)) + e^(12*x) - 6*e^(10*x) - e^(8*x) + 12*e^(6*x) -
e^(4*x) - 6*e^(2*x) + 1), x))/(e^(6*x) + 3*e^(4*x) + 3*e^(2*x) + 1)

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Fricas [A]  time = 0.248265, size = 566, normalized size = 11.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(sinh(x))*sech(x)^4,x, algorithm="fricas")

[Out]

1/6*(sqrt(2)*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*
sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)*sqrt(-(cosh(x)^2
+ sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(cosh(x)^6 + 6
*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4
*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)
^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) + 1)*arctan(1/2
*sqrt(2)*(3*cosh(x)^2 + 6*cosh(x)*sinh(x) + 3*sinh(x)^2 - 1)/sqrt(-(cosh(x)^2 +
sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))) - 8*(3*cosh(x)^2 +
6*cosh(x)*sinh(x) + 3*sinh(x)^2 + 1)*arctan(1/2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*s
inh(x) + sinh(x)^2 - 1)/sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)
*sinh(x) + sinh(x)^2))))/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cos
h(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*
(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(
x)^3 + cosh(x))*sinh(x) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(sinh(x))*sech(x)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \arcsin \left (\sinh \left (x\right )\right ) \operatorname{sech}\left (x\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(sinh(x))*sech(x)^4,x, algorithm="giac")

[Out]

integrate(arcsin(sinh(x))*sech(x)^4, x)