\(\int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [410]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sinh(d*x+c)^2*tanh(d*x+c)/(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 = {} \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

Int[(Sinh[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][(Sinh[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 39.99 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

Integrate[(Sinh[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Integrate[(Sinh[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 1.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

\[\int \frac {\sinh \left (d x +c \right )^{2} \tanh \left (d x +c \right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}d x\]

[In]

int(sinh(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(sinh(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2} \tanh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sinh(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

integral(sinh(d*x + c)^2*tanh(d*x + c)/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)

Sympy [N/A]

Not integrable

Time = 2.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\sinh ^{2}{\left (c + d x \right )} \tanh {\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]

[In]

integrate(sinh(d*x+c)**2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

Integral(sinh(c + d*x)**2*tanh(c + d*x)/((a + b*sinh(c + d*x))*(e + f*x)), x)

Maxima [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 322, normalized size of antiderivative = 9.47 \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2} \tanh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sinh(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate(sinh(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 2.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\sinh ^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,\mathrm {tanh}\left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

int((sinh(c + d*x)^2*tanh(c + d*x))/((e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int((sinh(c + d*x)^2*tanh(c + d*x))/((e + f*x)*(a + b*sinh(c + d*x))), x)