\(\int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [405]

Optimal result

Integrand size = 36, antiderivative size = 36 \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 28.68 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.80 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00

\[\int \frac {\cosh \left (d x +c \right )^{3} \sinh \left (d x +c \right )^{3}}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Timed out} \]

[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

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 551, normalized size of antiderivative = 15.31 \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[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)