3.238 \(\int \frac{\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))},x\right ) \]

[Out]

Unintegrable[Sinh[c + d*x]^3/(x*(a + b*Cosh[c + d*x])), x]

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Rubi [A]  time = 0.0550612, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sinh[c + d*x]^3/(x*(a + b*Cosh[c + d*x])),x]

[Out]

Defer[Int][Sinh[c + d*x]^3/(x*(a + b*Cosh[c + d*x])), x]

Rubi steps

\begin{align*} \int \frac{\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx &=\int \frac{\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx\\ \end{align*}

Mathematica [F]  time = 180., size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[Sinh[c + d*x]^3/(x*(a + b*Cosh[c + d*x])),x]

[Out]

$Aborted

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Maple [A]  time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{x \left ( a+b\cosh \left ( dx+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3/x/(a+b*cosh(d*x+c)),x)

[Out]

int(sinh(d*x+c)^3/x/(a+b*cosh(d*x+c)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\rm Ei}\left (2 \, d x\right ) e^{\left (2 \, c\right )}}{4 \, b} + \frac{a{\rm Ei}\left (-d x\right ) e^{\left (-c\right )}}{2 \, b^{2}} - \frac{{\rm Ei}\left (-2 \, d x\right ) e^{\left (-2 \, c\right )}}{4 \, b} - \frac{a{\rm Ei}\left (d x\right ) e^{c}}{2 \, b^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (x\right )}{b^{3}} - \frac{1}{8} \, \int \frac{16 \,{\left (a^{2} b - b^{3} +{\left (a^{3} e^{c} - a b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{4} x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} x e^{\left (d x + c\right )} + b^{4} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/x/(a+b*cosh(d*x+c)),x, algorithm="maxima")

[Out]

1/4*Ei(2*d*x)*e^(2*c)/b + 1/2*a*Ei(-d*x)*e^(-c)/b^2 - 1/4*Ei(-2*d*x)*e^(-2*c)/b - 1/2*a*Ei(d*x)*e^c/b^2 + (a^2
- b^2)*log(x)/b^3 - 1/8*integrate(16*(a^2*b - b^3 + (a^3*e^c - a*b^2*e^c)*e^(d*x))/(b^4*x*e^(2*d*x + 2*c) + 2
*a*b^3*x*e^(d*x + c) + b^4*x), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sinh \left (d x + c\right )^{3}}{b x \cosh \left (d x + c\right ) + a x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/x/(a+b*cosh(d*x+c)),x, algorithm="fricas")

[Out]

integral(sinh(d*x + c)^3/(b*x*cosh(d*x + c) + a*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**3/x/(a+b*cosh(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (d x + c\right )^{3}}{{\left (b \cosh \left (d x + c\right ) + a\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/x/(a+b*cosh(d*x+c)),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)^3/((b*cosh(d*x + c) + a)*x), x)