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

Optimal. Leaf size=39 \[ \text {Int}\left (\frac {\sinh ^3(c+d x) \cosh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.13, 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 = {} \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{a f x + a e + {\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cosh ^{3}\left (d x +c \right )\right ) \left (\sinh ^{3}\left (d x +c \right )\right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {e^{\left (-5 \, c + \frac {5 \, d e}{f}\right )} E_{1}\left (\frac {5 \, {\left (f x + e\right )} d}{f}\right )}{32 \, b f} - \frac {a e^{\left (-4 \, c + \frac {4 \, d e}{f}\right )} E_{1}\left (\frac {4 \, {\left (f x + e\right )} d}{f}\right )}{16 \, b^{2} f} + \frac {a e^{\left (4 \, c - \frac {4 \, d e}{f}\right )} E_{1}\left (-\frac {4 \, {\left (f x + e\right )} d}{f}\right )}{16 \, b^{2} f} - \frac {e^{\left (5 \, c - \frac {5 \, d e}{f}\right )} E_{1}\left (-\frac {5 \, {\left (f x + e\right )} d}{f}\right )}{32 \, b f} - \frac {{\left (4 \, a^{2} + b^{2}\right )} e^{\left (-3 \, c + \frac {3 \, d e}{f}\right )} E_{1}\left (\frac {3 \, {\left (f x + e\right )} d}{f}\right )}{32 \, b^{3} f} - \frac {{\left (4 \, a^{2} e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (-\frac {3 \, d e}{f}\right )} E_{1}\left (-\frac {3 \, {\left (f x + e\right )} d}{f}\right )}{32 \, b^{3} f} - \frac {{\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-2 \, c + \frac {2 \, d e}{f}\right )} E_{1}\left (\frac {2 \, {\left (f x + e\right )} d}{f}\right )}{8 \, b^{4} f} + \frac {{\left (2 \, a^{3} e^{\left (2 \, c\right )} + a b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (-\frac {2 \, d e}{f}\right )} E_{1}\left (-\frac {2 \, {\left (f x + e\right )} d}{f}\right )}{8 \, b^{4} f} - \frac {{\left (8 \, a^{4} + 6 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-c + \frac {d e}{f}\right )} E_{1}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{16 \, b^{5} f} - \frac {{\left (8 \, a^{4} e^{c} + 6 \, a^{2} b^{2} e^{c} - b^{4} e^{c}\right )} e^{\left (-\frac {d e}{f}\right )} E_{1}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{16 \, b^{5} f} - \frac {{\left (a^{5} + a^{3} b^{2}\right )} \log \left (f x + e\right )}{b^{6} f} + \frac {1}{64} \, \int \frac {128 \, {\left (a^{5} b + a^{3} b^{3} - {\left (a^{6} e^{c} + a^{4} b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{7} f x + b^{7} e - {\left (b^{7} f x e^{\left (2 \, c\right )} + b^{7} e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a b^{6} f x e^{c} + a b^{6} e e^{c}\right )} e^{\left (d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*e^(-5*c + 5*d*e/f)*exp_integral_e(1, 5*(f*x + e)*d/f)/(b*f) - 1/16*a*e^(-4*c + 4*d*e/f)*exp_integral_e(1
, 4*(f*x + e)*d/f)/(b^2*f) + 1/16*a*e^(4*c - 4*d*e/f)*exp_integral_e(1, -4*(f*x + e)*d/f)/(b^2*f) - 1/32*e^(5*
c - 5*d*e/f)*exp_integral_e(1, -5*(f*x + e)*d/f)/(b*f) - 1/32*(4*a^2 + b^2)*e^(-3*c + 3*d*e/f)*exp_integral_e(
1, 3*(f*x + e)*d/f)/(b^3*f) - 1/32*(4*a^2*e^(3*c) + b^2*e^(3*c))*e^(-3*d*e/f)*exp_integral_e(1, -3*(f*x + e)*d
/f)/(b^3*f) - 1/8*(2*a^3 + a*b^2)*e^(-2*c + 2*d*e/f)*exp_integral_e(1, 2*(f*x + e)*d/f)/(b^4*f) + 1/8*(2*a^3*e
^(2*c) + a*b^2*e^(2*c))*e^(-2*d*e/f)*exp_integral_e(1, -2*(f*x + e)*d/f)/(b^4*f) - 1/16*(8*a^4 + 6*a^2*b^2 - b
^4)*e^(-c + d*e/f)*exp_integral_e(1, (f*x + e)*d/f)/(b^5*f) - 1/16*(8*a^4*e^c + 6*a^2*b^2*e^c - b^4*e^c)*e^(-d
*e/f)*exp_integral_e(1, -(f*x + e)*d/f)/(b^5*f) - (a^5 + a^3*b^2)*log(f*x + e)/(b^6*f) + 1/64*integrate(128*(a
^5*b + a^3*b^3 - (a^6*e^c + a^4*b^2*e^c)*e^(d*x))/(b^7*f*x + b^7*e - (b^7*f*x*e^(2*c) + b^7*e*e^(2*c))*e^(2*d*
x) - 2*(a*b^6*f*x*e^c + a*b^6*e*e^c)*e^(d*x)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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