### 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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0305192, antiderivative size = 21, normalized size of antiderivative = 1., 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 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])

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 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 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]

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*}

Mathematica [A]  time = 0.0092672, size = 21, normalized size = 1. $\frac{1}{3} \log (\sinh (x))-\frac{1}{6} \log \left (4 \sinh ^2(x)+3\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 24, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({{\rm e}^{2\,x}}-1 \right ) }{3}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+{{\rm e}^{2\,x}}+1 \right ) }{6}} \end{align*}

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(exp(4*x)+exp(2*x)+1)

________________________________________________________________________________________

Maxima [B]  time = 1.63466, size = 63, normalized size = 3. \begin{align*} -\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 ) \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 2.0435, size = 173, normalized size = 8.24 \begin{align*} -\frac{1}{6} \, \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac{1}{3} \, \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end{align*}

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)))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [B]  time = 1.15222, size = 54, normalized size = 2.57 \begin{align*} -\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 ) \end{align*}

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))