3.218 \(\int \text {csch}(4 x) \sinh (x) \, dx\)

Optimal. Leaf size=26 \[ \frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tan ^{-1}(\sinh (x)) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1093, 203} \[ \frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tan ^{-1}(\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csch[4*x]*Sinh[x],x]

[Out]

-ArcTan[Sinh[x]]/4 + ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \text {csch}(4 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{4+12 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{4+8 x^2} \, dx,x,\sinh (x)\right )-2 \operatorname {Subst}\left (\int \frac {1}{8+8 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{4} \tan ^{-1}(\sinh (x))+\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tan ^{-1}(\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[4*x]*Sinh[x],x]

[Out]

-1/4*ArcTan[Sinh[x]] + ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2])

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fricas [B]  time = 0.45, size = 76, normalized size = 2.92 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \cosh \relax (x) + \frac {1}{2} \, \sqrt {2} \sinh \relax (x)\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}}{2 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) - \frac {1}{2} \, \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(4*x)*sinh(x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.12, size = 44, normalized size = 1.69 \[ -\frac {1}{8} \, \pi + \frac {1}{8} \, \sqrt {2} {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac {1}{4} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(4*x)*sinh(x),x, algorithm="giac")

[Out]

-1/8*pi + 1/8*sqrt(2)*(pi + 2*arctan(1/2*sqrt(2)*(e^(2*x) - 1)*e^(-x))) - 1/4*arctan(1/2*(e^(2*x) - 1)*e^(-x))

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maple [C]  time = 0.22, size = 62, normalized size = 2.38 \[ \frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{8}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(4*x)*sinh(x),x)

[Out]

1/4*I*ln(exp(x)-I)-1/4*I*ln(exp(x)+I)+1/8*I*2^(1/2)*ln(exp(2*x)+I*2^(1/2)*exp(x)-1)-1/8*I*2^(1/2)*ln(exp(2*x)-
I*2^(1/2)*exp(x)-1)

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maxima [B]  time = 0.44, size = 50, normalized size = 1.92 \[ -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) + \frac {1}{2} \, \arctan \left (e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(4*x)*sinh(x),x, algorithm="maxima")

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^(-x))) - 1/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^(-x)))
+ 1/2*arctan(e^(-x))

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mupad [B]  time = 0.07, size = 42, normalized size = 1.62 \[ \frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {\sqrt {2}\,{\mathrm {e}}^{3\,x}}{2}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}\right )\right )}{8}-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/sinh(4*x),x)

[Out]

(2^(1/2)*(2*atan((2^(1/2)*exp(x))/2 + (2^(1/2)*exp(3*x))/2) + 2*atan((2^(1/2)*exp(x))/2)))/8 - atan(exp(x))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {csch}{\left (4 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(4*x)*sinh(x),x)

[Out]

Integral(sinh(x)*csch(4*x), x)

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