3.220 \(\int \text {csch}(6 x) \sinh (x) \, dx\)

Optimal. Leaf size=36 \[ \frac {1}{6} \tan ^{-1}(\sinh (x))+\frac {1}{6} \tan ^{-1}(2 \sinh (x))-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

[Out]

1/6*arctan(sinh(x))+1/6*arctan(2*sinh(x))-1/6*arctan(2/3*sinh(x)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {12, 2057, 203} \[ \frac {1}{6} \tan ^{-1}(\sinh (x))+\frac {1}{6} \tan ^{-1}(2 \sinh (x))-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[6*x]*Sinh[x],x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rubi steps

\begin {align*} \int \text {csch}(6 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{2 \left (3+19 x^2+32 x^4+16 x^6\right )} \, dx,x,\sinh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{3+19 x^2+32 x^4+16 x^6} \, dx,x,\sinh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{3 \left (1+x^2\right )}+\frac {2}{3 \left (1+4 x^2\right )}-\frac {2}{3+4 x^2}\right ) \, dx,x,\sinh (x)\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\sinh (x)\right )-\operatorname {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sinh (x)\right )\\ &=\frac {1}{6} \tan ^{-1}(\sinh (x))+\frac {1}{6} \tan ^{-1}(2 \sinh (x))-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 30, normalized size = 0.83 \[ \frac {1}{6} \left (\tan ^{-1}(\sinh (x))+\tan ^{-1}(2 \sinh (x))-\sqrt {3} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[6*x]*Sinh[x],x]

[Out]

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

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fricas [B]  time = 0.44, size = 107, normalized size = 2.97 \[ -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \cosh \relax (x) + \frac {1}{3} \, \sqrt {3} \sinh \relax (x)\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} \cosh \relax (x)^{2} + 2 \, \sqrt {3} \cosh \relax (x) \sinh \relax (x) + \sqrt {3} \sinh \relax (x)^{2} + 2 \, \sqrt {3}}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) - \frac {1}{6} \, \arctan \left (-\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + \frac {1}{2} \, \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(6*x)*sinh(x),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*cosh(x) + 1/3*sqrt(3)*sinh(x)) + 1/6*sqrt(3)*arctan(-1/3*(sqrt(3)*cosh(x)^2 +
2*sqrt(3)*cosh(x)*sinh(x) + sqrt(3)*sinh(x)^2 + 2*sqrt(3))/(cosh(x) - sinh(x))) - 1/6*arctan(-(cosh(x)^2 + 2*c
osh(x)*sinh(x) + sinh(x)^2)/(cosh(x) - sinh(x))) + 1/2*arctan(cosh(x) + sinh(x))

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giac [B]  time = 0.14, size = 58, normalized size = 1.61 \[ \frac {1}{6} \, \pi - \frac {1}{12} \, \sqrt {3} {\left (\pi + 2 \, \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} + \frac {1}{6} \, \arctan \left ({\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) + \frac {1}{6} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(6*x)*sinh(x),x, algorithm="giac")

[Out]

1/6*pi - 1/12*sqrt(3)*(pi + 2*arctan(1/3*sqrt(3)*(e^(2*x) - 1)*e^(-x))) + 1/6*arctan((e^(2*x) - 1)*e^(-x)) + 1
/6*arctan(1/2*(e^(2*x) - 1)*e^(-x))

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maple [C]  time = 0.25, size = 92, normalized size = 2.56 \[ \frac {i \ln \left ({\mathrm e}^{x}+i\right )}{6}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{6}+\frac {i \ln \left ({\mathrm e}^{2 x}+i {\mathrm e}^{x}-1\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 x}-i {\mathrm e}^{x}-1\right )}{12}+\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {3}\, {\mathrm e}^{x}-1\right )}{12}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {3}\, {\mathrm e}^{x}-1\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(6*x)*sinh(x),x)

[Out]

1/6*I*ln(exp(x)+I)-1/6*I*ln(exp(x)-I)+1/12*I*ln(exp(2*x)+I*exp(x)-1)-1/12*I*ln(exp(2*x)-I*exp(x)-1)+1/12*I*3^(
1/2)*ln(exp(2*x)-I*3^(1/2)*exp(x)-1)-1/12*I*3^(1/2)*ln(exp(2*x)+I*3^(1/2)*exp(x)-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) + \frac {1}{3} \, \arctan \left (e^{x}\right ) + \int \frac {e^{\left (3 \, x\right )} + e^{x}}{6 \, {\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(6*x)*sinh(x),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x - 1)) + 1/3*arctan(e^x) +
integrate(1/6*(e^(3*x) + e^x)/(e^(4*x) - e^(2*x) + 1), x)

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mupad [B]  time = 0.20, size = 41, normalized size = 1.14 \[ \frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{3}-\frac {\mathrm {atan}\left ({\mathrm {e}}^{-x}-{\mathrm {e}}^x\right )}{6}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^x}{3}-\frac {\sqrt {3}\,{\mathrm {e}}^{-x}}{3}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/sinh(6*x),x)

[Out]

atan(exp(x))/3 - atan(exp(-x) - exp(x))/6 - (3^(1/2)*atan((3^(1/2)*exp(x))/3 - (3^(1/2)*exp(-x))/3))/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {csch}{\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(6*x)*sinh(x),x)

[Out]

Integral(sinh(x)*csch(6*x), x)

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