3.3.49 \(\int \cosh (x) \text {csch}(6 x) \, dx\) [249]

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

[Out]

-1/6*arctanh(cosh(x))-1/6*arctanh(2*cosh(x))+1/6*arctanh(2/3*cosh(x)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.04, 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, 2082, 213} \begin {gather*} -\frac {1}{6} \tanh ^{-1}(\cosh (x))-\frac {1}{6} \tanh ^{-1}(2 \cosh (x))+\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*ArcTanh[Cosh[x]] - ArcTanh[2*Cosh[x]]/6 + ArcTanh[(2*Cosh[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 213

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

Rule 2082

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.04, size = 91, normalized size = 2.53 \begin {gather*} \frac {1}{12} \left (2 \sqrt {3} \tanh ^{-1}\left (\frac {2-i \tanh \left (\frac {x}{2}\right )}{\sqrt {3}}\right )+2 \sqrt {3} \tanh ^{-1}\left (\frac {2+i \tanh \left (\frac {x}{2}\right )}{\sqrt {3}}\right )-2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\log (1-2 \cosh (x))-\log (1+2 \cosh (x))+2 \log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3]*ArcTanh[(2 - I*Tanh[x/2])/Sqrt[3]] + 2*Sqrt[3]*ArcTanh[(2 + I*Tanh[x/2])/Sqrt[3]] - 2*Log[Cosh[x/2]
] + Log[1 - 2*Cosh[x]] - Log[1 + 2*Cosh[x]] + 2*Log[Sinh[x/2]])/12

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(26)=52\).
time = 1.04, size = 77, normalized size = 2.14

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{x}-1\right )}{6}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{6}-\frac {\ln \left (1+{\mathrm e}^{x}+{\mathrm e}^{2 x}\right )}{12}+\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (1-{\mathrm e}^{x}+{\mathrm e}^{2 x}\right )}{12}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*csch(6*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(1/2*(e^(3*x) - e^x)/(e^(4*x) - e^(2*x) + 1), x) - 1/12*log(e^(2*x) + e^x + 1) + 1/12*log(e^(2*x) -
e^x + 1) - 1/6*log(e^x + 1) + 1/6*log(e^x - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (26) = 52\).
time = 0.43, size = 101, normalized size = 2.81 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (\frac {2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 4 \, \sqrt {3} \cosh \left (x\right ) + 5}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}\right ) - \frac {1}{12} \, \log \left (\frac {2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \frac {1}{12} \, \log \left (\frac {2 \, \cosh \left (x\right ) - 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \frac {1}{6} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{6} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*log((2*cosh(x)^2 + 2*sinh(x)^2 + 4*sqrt(3)*cosh(x) + 5)/(2*cosh(x)^2 + 2*sinh(x)^2 - 1)) - 1/12*l
og((2*cosh(x) + 1)/(cosh(x) - sinh(x))) + 1/12*log((2*cosh(x) - 1)/(cosh(x) - sinh(x))) - 1/6*log(cosh(x) + si
nh(x) + 1) + 1/6*log(cosh(x) + sinh(x) - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cosh {\left (x \right )} \operatorname {csch}{\left (6 x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*csch(6*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (26) = 52\).
time = 0.39, size = 79, normalized size = 2.19 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - e^{\left (-x\right )} - e^{x}}{\sqrt {3} + e^{\left (-x\right )} + e^{x}}\right ) - \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 1\right ) + \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 1\right ) + \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*log(-(sqrt(3) - e^(-x) - e^x)/(sqrt(3) + e^(-x) + e^x)) - 1/12*log(e^(-x) + e^x + 2) - 1/12*log(
e^(-x) + e^x + 1) + 1/12*log(e^(-x) + e^x - 1) + 1/12*log(e^(-x) + e^x - 2)

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Mupad [B]
time = 0.08, size = 91, normalized size = 2.53 \begin {gather*} \frac {\ln \left (\frac {1}{3}-\frac {{\mathrm {e}}^x}{3}\right )}{6}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{3}-\frac {1}{3}\right )}{6}-\frac {\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{36}-\frac {{\mathrm {e}}^x}{36}-\frac {1}{36}\right )}{12}+\frac {\ln \left (\frac {{\mathrm {e}}^x}{36}-\frac {{\mathrm {e}}^{2\,x}}{36}-\frac {1}{36}\right )}{12}+\frac {\sqrt {3}\,\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{12}-\frac {\sqrt {3}\,{\mathrm {e}}^x}{12}-\frac {1}{12}\right )}{12}-\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,{\mathrm {e}}^x}{12}-\frac {{\mathrm {e}}^{2\,x}}{12}-\frac {1}{12}\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(1/3 - exp(x)/3)/6 - log(- exp(x)/3 - 1/3)/6 - log(- exp(2*x)/36 - exp(x)/36 - 1/36)/12 + log(exp(x)/36 - e
xp(2*x)/36 - 1/36)/12 + (3^(1/2)*log(- exp(2*x)/12 - (3^(1/2)*exp(x))/12 - 1/12))/12 - (3^(1/2)*log((3^(1/2)*e
xp(x))/12 - exp(2*x)/12 - 1/12))/12

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