3.3.7 \(\int \coth (3 x) \sinh (x) \, dx\) [207]

Optimal. Leaf size=20 \[ -\frac {\text {ArcTan}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{\sqrt {3}}+\sinh (x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {396, 209} \begin {gather*} \sinh (x)-\frac {\text {ArcTan}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 \sinh (x)}{\sqrt {3}}\right )}{\sqrt {3}}+\sinh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(16)=32\).
time = 0.99, size = 51, normalized size = 2.55

method	result	size
default	\(-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {\sqrt {3}\, \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {3}\right )}{3}-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}-\frac {\sqrt {3}\, \arctan \left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {3}}{3}\right )}{3}\)	\(51\)
risch	\(\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {3}\, {\mathrm e}^{x}-1\right )}{6}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {3}\, {\mathrm e}^{x}-1\right )}{6}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (16) = 32\).
time = 0.47, size = 49, normalized size = 2.45 \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 ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(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)) - 1/2*e^(-x) +
1/2*e^x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (16) = 32\).
time = 0.35, size = 118, normalized size = 5.90 \begin {gather*} -\frac {2 \, {\left (\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right )\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} \cosh \left (x\right ) + \frac {1}{3} \, \sqrt {3} \sinh \left (x\right )\right ) - 2 \, {\left (\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right )\right )} \arctan \left (-\frac {\sqrt {3} \cosh \left (x\right )^{2} + 2 \, \sqrt {3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {3} \sinh \left (x\right )^{2} + 2 \, \sqrt {3}}{3 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) - 3 \, \cosh \left (x\right )^{2} - 6 \, \cosh \left (x\right ) \sinh \left (x\right ) - 3 \, \sinh \left (x\right )^{2} + 3}{6 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*(sqrt(3)*cosh(x) + sqrt(3)*sinh(x))*arctan(1/3*sqrt(3)*cosh(x) + 1/3*sqrt(3)*sinh(x)) - 2*(sqrt(3)*cos
h(x) + sqrt(3)*sinh(x))*arctan(-1/3*(sqrt(3)*cosh(x)^2 + 2*sqrt(3)*cosh(x)*sinh(x) + sqrt(3)*sinh(x)^2 + 2*sqr
t(3))/(cosh(x) - sinh(x))) - 3*cosh(x)^2 - 6*cosh(x)*sinh(x) - 3*sinh(x)^2 + 3)/(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 )} \coth {\left (3 x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(x)*coth(3*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
time = 0.39, size = 36, normalized size = 1.80 \begin {gather*} -\frac {1}{6} \, \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}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.08, size = 47, normalized size = 2.35 \begin {gather*} \frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,{\mathrm {e}}^x}{3}+\frac {\sqrt {3}\,{\mathrm {e}}^{3\,x}}{3}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^x}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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