3.232 \(\int \frac {\sinh ^2(c+d x)}{x (a+b \cosh (c+d x))} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\sinh ^2(c+d x)}{x (a+b \cosh (c+d x))},x\right ) \]

[Out]

Unintegrable(sinh(d*x+c)^2/x/(a+b*cosh(d*x+c)),x)

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Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh ^2(c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sinh[c + d*x]^2/(x*(a + b*Cosh[c + d*x])),x]

[Out]

Defer[Int][Sinh[c + d*x]^2/(x*(a + b*Cosh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\sinh ^2(c+d x)}{x (a+b \cosh (c+d x))} \, dx &=\int \frac {\sinh ^2(c+d x)}{x (a+b \cosh (c+d x))} \, dx\\ \end {align*}

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Mathematica [A]  time = 116.73, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^2(c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sinh[c + d*x]^2/(x*(a + b*Cosh[c + d*x])),x]

[Out]

Integrate[Sinh[c + d*x]^2/(x*(a + b*Cosh[c + d*x])), x]

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fricas [A]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sinh \left (d x + c\right )^{2}}{b x \cosh \left (d x + c\right ) + a x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/x/(a+b*cosh(d*x+c)),x, algorithm="fricas")

[Out]

integral(sinh(d*x + c)^2/(b*x*cosh(d*x + c) + a*x), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \cosh \left (d x + c\right ) + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/x/(a+b*cosh(d*x+c)),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)^2/((b*cosh(d*x + c) + a)*x), x)

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maple [A]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}\left (d x +c \right )}{x \left (a +b \cosh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2/x/(a+b*cosh(d*x+c)),x)

[Out]

int(sinh(d*x+c)^2/x/(a+b*cosh(d*x+c)),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, {\left (a^{2} e^{c} - b^{2} e^{c}\right )} \int \frac {e^{\left (d x\right )}}{b^{3} x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} x e^{\left (d x + c\right )} + b^{3} x}\,{d x} + \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )}}{2 \, b} + \frac {{\rm Ei}\left (d x\right ) e^{c}}{2 \, b} - \frac {a \log \relax (x)}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/x/(a+b*cosh(d*x+c)),x, algorithm="maxima")

[Out]

2*(a^2*e^c - b^2*e^c)*integrate(e^(d*x)/(b^3*x*e^(2*d*x + 2*c) + 2*a*b^2*x*e^(d*x + c) + b^3*x), x) + 1/2*Ei(-
d*x)*e^(-c)/b + 1/2*Ei(d*x)*e^c/b - a*log(x)/b^2

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{x\,\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^2/(x*(a + b*cosh(c + d*x))),x)

[Out]

int(sinh(c + d*x)^2/(x*(a + b*cosh(c + d*x))), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\left (c + d x \right )}}{x \left (a + b \cosh {\left (c + d x \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**2/x/(a+b*cosh(d*x+c)),x)

[Out]

Integral(sinh(c + d*x)**2/(x*(a + b*cosh(c + d*x))), x)

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