3.592 \(\int \frac{\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt{\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{6} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{\sinh (2 x)}}\right )+\frac{\cosh (x)}{\sqrt{\sinh (2 x)}}+\sqrt{2} \tan ^{-1}\left (\text{sech}(x) \sqrt{\sinh (x) \cosh (x)}\right )-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left (\text{sech}(x) \sqrt{\sinh (x) \cosh (x)}\right ) \]

[Out]

Sqrt[2]*ArcTan[Sech[x]*Sqrt[Cosh[x]*Sinh[x]]] + ArcTan[Sinh[x]/Sqrt[Sinh[2*x]]]/
6 - (Sqrt[2]*ArcTanh[Sech[x]*Sqrt[Cosh[x]*Sinh[x]]])/3 + Cosh[x]/Sqrt[Sinh[2*x]]

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Rubi [A]  time = 1.58, antiderivative size = 102, normalized size of antiderivative = 1.48, number of steps used = 8, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{2 \sinh (x) \tanh ^{-1}\left (\sqrt{\tanh (x)}\right )}{3 \sqrt{\sinh (2 x)} \sqrt{\tanh (x)}}+\frac{\cosh (x)}{\sqrt{\sinh (2 x)}}+\frac{2 \sinh (x) \tan ^{-1}\left (\sqrt{\tanh (x)}\right )}{\sqrt{\sinh (2 x)} \sqrt{\tanh (x)}}+\frac{\sinh (x) \tan ^{-1}\left (\frac{\sqrt{\tanh (x)}}{\sqrt{2}}\right )}{3 \sqrt{2} \sqrt{\sinh (2 x)} \sqrt{\tanh (x)}} \]

Antiderivative was successfully verified.

[In]  Int[(Cosh[x]*(-Cosh[2*x] + Tanh[x]))/(Sqrt[Sinh[2*x]]*(Sinh[x]^2 + Sinh[2*x])),x]

[Out]

Cosh[x]/Sqrt[Sinh[2*x]] + (2*ArcTan[Sqrt[Tanh[x]]]*Sinh[x])/(Sqrt[Sinh[2*x]]*Sqr
t[Tanh[x]]) + (ArcTan[Sqrt[Tanh[x]]/Sqrt[2]]*Sinh[x])/(3*Sqrt[2]*Sqrt[Sinh[2*x]]
*Sqrt[Tanh[x]]) - (2*ArcTanh[Sqrt[Tanh[x]]]*Sinh[x])/(3*Sqrt[Sinh[2*x]]*Sqrt[Tan
h[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(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)**2+sinh(2*x))/sinh(2*x)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 31.186, size = 487, normalized size = 7.06 \[ -\frac{\sqrt{\sinh (2 x)} \coth (x) (\tanh (x)-\cosh (2 x))}{-2 \sinh (x)+\cosh (x)+\cosh (3 x)}+\frac{\cosh (x) (\tanh (x)-\cosh (2 x)) \left (\frac{16 (-1)^{5/12} \sinh ^{\frac{3}{2}}(2 x) \sqrt{\tanh ^3\left (\frac{x}{2}\right )+\tanh \left (\frac{x}{2}\right )} \left (2 \left (\sqrt [3]{-1}-1\right ) \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )+\left (3-3 i \sqrt{3}\right ) \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )+i \left (\sqrt{3}+i\right ) \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )+2 \left (\sqrt [3]{-1}-1\right ) \Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )\right )}{3 \left (\sqrt{3}-i\right ) (\cosh (x)+1)^3 \sqrt{\tanh \left (\frac{x}{2}\right )} \sqrt{\tanh ^2\left (\frac{x}{2}\right )+1} \left (\frac{\sinh (2 x)}{(\cosh (x)+1)^2}\right )^{3/2}}-\frac{6 \sqrt [4]{-1} \sqrt{\sinh (2 x)} \sqrt{\tanh \left (\frac{x}{2}\right )} \sqrt{\tanh ^3\left (\frac{x}{2}\right )+\tanh \left (\frac{x}{2}\right )} \sqrt{\coth ^2\left (\frac{x}{2}\right )+1} \left (F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{\tanh \left (\frac{x}{2}\right )}}\right )\right |-1\right )-\Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{\tanh \left (\frac{x}{2}\right )}}\right )\right |-1\right )-\Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{\tanh \left (\frac{x}{2}\right )}}\right )\right |-1\right )\right )}{(\cosh (x)+1) \left (\tanh ^2\left (\frac{x}{2}\right )+1\right ) \sqrt{\frac{\sinh (2 x)}{(\cosh (x)+1)^2}}}\right )}{2 (-2 \sinh (x)+\cosh (x)+\cosh (3 x))} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(Cosh[x]*(-Cosh[2*x] + Tanh[x]))/(Sqrt[Sinh[2*x]]*(Sinh[x]^2 + Sinh[2*x])),x]

[Out]

-((Coth[x]*Sqrt[Sinh[2*x]]*(-Cosh[2*x] + Tanh[x]))/(Cosh[x] + Cosh[3*x] - 2*Sinh
[x])) + (Cosh[x]*((-6*(-1)^(1/4)*Sqrt[1 + Coth[x/2]^2]*(EllipticF[I*ArcSinh[(-1)
^(1/4)/Sqrt[Tanh[x/2]]], -1] - EllipticPi[-(-1)^(1/6), I*ArcSinh[(-1)^(1/4)/Sqrt
[Tanh[x/2]]], -1] - EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(1/4)/Sqrt[Tanh[x/2]]
], -1])*Sqrt[Sinh[2*x]]*Sqrt[Tanh[x/2]]*Sqrt[Tanh[x/2] + Tanh[x/2]^3])/((1 + Cos
h[x])*Sqrt[Sinh[2*x]/(1 + Cosh[x])^2]*(1 + Tanh[x/2]^2)) + (16*(-1)^(5/12)*((3 -
(3*I)*Sqrt[3])*EllipticPi[-I, I*ArcSinh[(-1)^(1/4)*Sqrt[Tanh[x/2]]], -1] + 2*(-
1 + (-1)^(1/3))*EllipticPi[I, ArcSin[(-1)^(3/4)*Sqrt[Tanh[x/2]]], -1] + I*(I + S
qrt[3])*EllipticPi[-(-1)^(1/6), I*ArcSinh[(-1)^(1/4)*Sqrt[Tanh[x/2]]], -1] + 2*(
-1 + (-1)^(1/3))*EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(1/4)*Sqrt[Tanh[x/2]]],
-1])*Sinh[2*x]^(3/2)*Sqrt[Tanh[x/2] + Tanh[x/2]^3])/(3*(-I + Sqrt[3])*(1 + Cosh[
x])^3*(Sinh[2*x]/(1 + Cosh[x])^2)^(3/2)*Sqrt[Tanh[x/2]]*Sqrt[1 + Tanh[x/2]^2]))*
(-Cosh[2*x] + Tanh[x]))/(2*(Cosh[x] + Cosh[3*x] - 2*Sinh[x]))

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Maple [C]  time = 0.299, size = 609, normalized size = 8.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/2),x)

[Out]

-1/12*((tanh(1/2*x)^2+1)*tanh(1/2*x)/(tanh(1/2*x)^2-1)^2)^(1/2)*(tanh(1/2*x)^2-1
)*(12*I*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1
/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((tanh
(1/2*x)^2+1)*tanh(1/2*x))^(1/2)-9*I*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tan
h(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticF((-I*(tanh(1/2*x)+I))^(1/2),1/
2*2^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)-4*I*(-I*(tanh(1/2*x)+I))^(1/2)*
2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1
/2*x)+I))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)-12*
(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(
1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((tanh(1/2*x)^
2+1)*tanh(1/2*x))^(1/2)-4*(-I*(tanh(1/2*x)+I))^(1/2)*2^(1/2)*(-I*(-tanh(1/2*x)+I
))^(1/2)*(I*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),1/2+1/2*I,1
/2*2^(1/2))*((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)-I*2^(1/2)*sum(_alpha*(I*_alpha
+1+I)*(-I*(tanh(1/2*x)+I))^(1/2)*(-I*(-tanh(1/2*x)+I))^(1/2)*(I*tanh(1/2*x))^(1/
2)/((tanh(1/2*x)^2+1)*tanh(1/2*x))^(1/2)*EllipticPi((-I*(tanh(1/2*x)+I))^(1/2),_
alpha+1-I,1/2*2^(1/2)),_alpha=RootOf(_Z^2+_Z+1))*((tanh(1/2*x)^2+1)*tanh(1/2*x))
^(1/2)*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2)+6*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2)*tan
h(1/2*x)^2+6*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2))/(tanh(1/2*x)^2+1)/tanh(1/2*x)/(t
anh(1/2*x)^3+tanh(1/2*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (\cosh \left (2 \, x\right ) - \tanh \left (x\right )\right )} \cosh \left (x\right )}{{\left (\sinh \left (x\right )^{2} + \sinh \left (2 \, x\right )\right )} \sqrt{\sinh \left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(cosh(2*x) - tanh(x))*cosh(x)/((sinh(x)^2 + sinh(2*x))*sqrt(sinh(2*x))),x, algorithm="maxima")

[Out]

-integrate((cosh(2*x) - tanh(x))*cosh(x)/((sinh(x)^2 + sinh(2*x))*sqrt(sinh(2*x)
)), x)

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Fricas [A]  time = 0.24017, size = 405, normalized size = 5.87 \[ -\frac{{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \arctan \left (\frac{\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} + 3 \, \sqrt{2}}{8 \, \sqrt{\frac{\cosh \left (x\right ) \sinh \left (x\right )}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}\right ) + 6 \,{\left (\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} - \sqrt{2}\right )} \arctan \left (\frac{1}{2 \, \sqrt{\frac{\cosh \left (x\right ) \sinh \left (x\right )}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}\right ) -{\left (\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} - \sqrt{2}\right )} \log \left (2 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 12 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 2 \, \sinh \left (x\right )^{4} - 4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt{\frac{\cosh \left (x\right ) \sinh \left (x\right )}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 1\right ) - 12 \, \sqrt{2} \sqrt{\frac{\cosh \left (x\right ) \sinh \left (x\right )}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{12 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(cosh(2*x) - tanh(x))*cosh(x)/((sinh(x)^2 + sinh(2*x))*sqrt(sinh(2*x))),x, algorithm="fricas")

[Out]

-1/12*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*arctan(1/8*(sqrt(2)*cosh(
x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + 3*sqrt(2))/sqrt(cosh(x)*s
inh(x)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))) + 6*(sqrt(2)*cosh(x)^2 + 2*
sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 - sqrt(2))*arctan(1/2/sqrt(cosh(x)*s
inh(x)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))) - (sqrt(2)*cosh(x)^2 + 2*sq
rt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 - sqrt(2))*log(2*cosh(x)^4 + 8*cosh(x)
^3*sinh(x) + 12*cosh(x)^2*sinh(x)^2 + 8*cosh(x)*sinh(x)^3 + 2*sinh(x)^4 - 4*(cos
h(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(cosh(x)*sinh(x)/(cosh(x)^2 - 2*cosh
(x)*sinh(x) + sinh(x)^2)) - 1) - 12*sqrt(2)*sqrt(cosh(x)*sinh(x)/(cosh(x)^2 - 2*
cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 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(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)**2+sinh(2*x))/sinh(2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (\cosh \left (2 \, x\right ) - \tanh \left (x\right )\right )} \cosh \left (x\right )}{{\left (\sinh \left (x\right )^{2} + \sinh \left (2 \, x\right )\right )} \sqrt{\sinh \left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(cosh(2*x) - tanh(x))*cosh(x)/((sinh(x)^2 + sinh(2*x))*sqrt(sinh(2*x))),x, algorithm="giac")

[Out]

integrate(-(cosh(2*x) - tanh(x))*cosh(x)/((sinh(x)^2 + sinh(2*x))*sqrt(sinh(2*x)
)), x)