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

Optimal. Leaf size=36 \[ \text{Unintegrable}\left (\frac{\sinh (c+d x) \cosh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0842915, 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 = {} \[ \int \frac{\cosh ^3(c+d x) \sinh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [A]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}\sinh \left ( dx+c \right ) }{ \left ( fx+e \right ) \left ( a+b\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e^{\left (-3 \, c + \frac{3 \, d e}{f}\right )} E_{1}\left (\frac{3 \,{\left (f x + e\right )} d}{f}\right )}{8 \, b f} - \frac{a e^{\left (-2 \, c + \frac{2 \, d e}{f}\right )} E_{1}\left (\frac{2 \,{\left (f x + e\right )} d}{f}\right )}{4 \, b^{2} f} + \frac{a e^{\left (2 \, c - \frac{2 \, d e}{f}\right )} E_{1}\left (-\frac{2 \,{\left (f x + e\right )} d}{f}\right )}{4 \, b^{2} f} - \frac{e^{\left (3 \, c - \frac{3 \, d e}{f}\right )} E_{1}\left (-\frac{3 \,{\left (f x + e\right )} d}{f}\right )}{8 \, b f} - \frac{{\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-c + \frac{d e}{f}\right )} E_{1}\left (\frac{{\left (f x + e\right )} d}{f}\right )}{8 \, b^{3} f} - \frac{{\left (4 \, a^{2} e^{c} + 3 \, b^{2} e^{c}\right )} e^{\left (-\frac{d e}{f}\right )} E_{1}\left (-\frac{{\left (f x + e\right )} d}{f}\right )}{8 \, b^{3} f} - \frac{{\left (a^{3} + a b^{2}\right )} \log \left (f x + e\right )}{b^{4} f} + \frac{1}{16} \, \int \frac{32 \,{\left (a^{3} b + a b^{3} -{\left (a^{4} e^{c} + a^{2} b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{5} f x + b^{5} e -{\left (b^{5} f x e^{\left (2 \, c\right )} + b^{5} e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a b^{4} f x e^{c} + a b^{4} e e^{c}\right )} e^{\left (d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e^(-3*c + 3*d*e/f)*exp_integral_e(1, 3*(f*x + e)*d/f)/(b*f) - 1/4*a*e^(-2*c + 2*d*e/f)*exp_integral_e(1,
2*(f*x + e)*d/f)/(b^2*f) + 1/4*a*e^(2*c - 2*d*e/f)*exp_integral_e(1, -2*(f*x + e)*d/f)/(b^2*f) - 1/8*e^(3*c -
3*d*e/f)*exp_integral_e(1, -3*(f*x + e)*d/f)/(b*f) - 1/8*(4*a^2 + 3*b^2)*e^(-c + d*e/f)*exp_integral_e(1, (f*x
+ e)*d/f)/(b^3*f) - 1/8*(4*a^2*e^c + 3*b^2*e^c)*e^(-d*e/f)*exp_integral_e(1, -(f*x + e)*d/f)/(b^3*f) - (a^3 +
a*b^2)*log(f*x + e)/(b^4*f) + 1/16*integrate(32*(a^3*b + a*b^3 - (a^4*e^c + a^2*b^2*e^c)*e^(d*x))/(b^5*f*x +
b^5*e - (b^5*f*x*e^(2*c) + b^5*e*e^(2*c))*e^(2*d*x) - 2*(a*b^4*f*x*e^c + a*b^4*e*e^c)*e^(d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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