\(\int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [395]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cosh (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 = 35.62 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.88 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 326, normalized size of antiderivative = 9.59 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

Giac [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.77 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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