3.246 \(\int \cosh (x) \text {csch}(3 x) \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{3} \log (\sinh (x))-\frac {1}{6} \log \left (4 \sinh ^2(x)+3\right ) \]

[Out]

1/3*ln(sinh(x))-1/6*ln(3+4*sinh(x)^2)

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4356, 266, 36, 29, 31} \[ \frac {1}{3} \log (\sinh (x))-\frac {1}{6} \log \left (4 \sinh ^2(x)+3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Sinh[x]]/3 - Log[3 + 4*Sinh[x]^2]/6

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ \frac {1}{3} \log (\sinh (x))-\frac {1}{6} \log \left (4 \sinh ^2(x)+3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Sinh[x]]/3 - Log[3 + 4*Sinh[x]^2]/6

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fricas [B]  time = 0.40, size = 52, normalized size = 2.48 \[ -\frac {1}{6} \, \log \left (\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} + 1}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) + \frac {1}{3} \, \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*log((2*cosh(x)^2 + 2*sinh(x)^2 + 1)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1/3*log(2*sinh(x)/(cos
h(x) - sinh(x)))

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giac [B]  time = 0.11, size = 40, normalized size = 1.90 \[ -\frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) - \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) + \frac {1}{3} \, \log \left (e^{x} + 1\right ) + \frac {1}{3} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*log(e^(2*x) + e^x + 1) - 1/6*log(e^(2*x) - e^x + 1) + 1/3*log(e^x + 1) + 1/3*log(abs(e^x - 1))

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maple [A]  time = 0.22, size = 24, normalized size = 1.14 \[ \frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{3}-\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{4 x}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*ln(exp(2*x)-1)-1/6*ln(1+exp(2*x)+exp(4*x))

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maxima [B]  time = 0.44, size = 47, normalized size = 2.24 \[ -\frac {1}{6} \, \log \left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{3} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{3} \, \log \left (e^{\left (-x\right )} - 1\right ) - \frac {1}{6} \, \log \left (-e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*log(e^(-x) + e^(-2*x) + 1) + 1/3*log(e^(-x) + 1) + 1/3*log(e^(-x) - 1) - 1/6*log(-e^(-x) + e^(-2*x) + 1)

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mupad [B]  time = 1.49, size = 29, normalized size = 1.38 \[ \frac {\ln \left (3\,{\mathrm {e}}^{2\,x}-3\right )}{3}-\frac {\ln \left (-{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{4\,x}-1\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*exp(2*x) - 3)/3 - log(- exp(2*x) - exp(4*x) - 1)/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \operatorname {csch}{\left (3 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*csch(3*x), x)

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