### 3.217 $$\int \text{csch}(3 x) \sinh (x) \, dx$$

Optimal. Leaf size=15 $\frac{\tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right )}{\sqrt{3}}$

[Out]

ArcTan[Tanh[x]/Sqrt[3]]/Sqrt[3]

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Rubi [A]  time = 0.0362462, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {203} $\frac{\tan ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right )}{\sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[3*x]*Sinh[x],x]

[Out]

ArcTan[Tanh[x]/Sqrt[3]]/Sqrt[3]

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])

Rubi steps

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

Mathematica [C]  time = 0.0241455, size = 44, normalized size = 2.93 $-\frac{1}{4} e^{2 x} \left (e^{2 x} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};e^{6 x}\right )-2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};e^{6 x}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[3*x]*Sinh[x],x]

[Out]

-(E^(2*x)*(-2*Hypergeometric2F1[1/3, 1, 4/3, E^(6*x)] + E^(2*x)*Hypergeometric2F1[2/3, 1, 5/3, E^(6*x)]))/4

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Maple [C]  time = 0.033, size = 40, normalized size = 2.7 \begin{align*}{\frac{i}{6}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) -{\frac{i}{6}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(3*x)*sinh(x),x)

[Out]

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

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Maxima [B]  time = 1.53863, size = 53, normalized size = 3.53 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (-x\right )} + 1\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (-x\right )} - 1\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(3*x)*sinh(x),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.12863, size = 115, normalized size = 7.67 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{3 \, \sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(3*x)*sinh(x),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*arctan(-1/3*(3*sqrt(3)*cosh(x) + sqrt(3)*sinh(x))/(cosh(x) - sinh(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{csch}{\left (3 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(3*x)*sinh(x),x)

[Out]

Integral(sinh(x)*csch(3*x), x)

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Giac [A]  time = 1.20302, size = 26, normalized size = 1.73 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (2 \, x\right )} + 1\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(3*x)*sinh(x),x, algorithm="giac")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(2*x) + 1))