3.3.17 \(\int \text {csch}(3 x) \sinh (x) \, dx\) [217]

Optimal. Leaf size=15 \[ \frac {\text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

1/3*arctan(1/3*tanh(x)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\tanh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 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 &=\text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 44, normalized size = 2.93 \begin {gather*} -\frac {1}{4} e^{2 x} \left (-2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{6 x}\right )+e^{2 x} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};e^{6 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(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)]))

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Maple [C] Result contains complex when optimal does not.
time = 0.88, size = 40, normalized size = 2.67

method	result	size
risch	\(\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(3*x)*sinh(x),x,method=_RETURNVERBOSE)

[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] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (13) = 26\).
time = 0.48, size = 39, normalized size = 2.60 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (13) = 26\).
time = 0.36, size = 31, normalized size = 2.07 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh {\left (x \right )} \operatorname {csch}{\left (3 x \right )}\, dx \end {gather*}

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 = 0.39, size = 19, normalized size = 1.27 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (2 \, x\right )} + 1\right )}\right ) \end {gather*}

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))

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Mupad [B]
time = 1.46, size = 19, normalized size = 1.27 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,{\mathrm {e}}^{2\,x}+1\right )}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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