\(\int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx\) [177]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\text {Int}\left (\frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)^2/(a+b*sinh(f*x+e))^2,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx \]

[In]

Int[1/((c + d*x)^2*(a + b*Sinh[e + f*x])^2),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + b*Sinh[e + f*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 26.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx \]

[In]

Integrate[1/((c + d*x)^2*(a + b*Sinh[e + f*x])^2),x]

[Out]

Integrate[1/((c + d*x)^2*(a + b*Sinh[e + f*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.50 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d x +c \right )^{2} \left (a +b \sinh \left (f x +e \right )\right )^{2}}d x\]

[In]

int(1/(d*x+c)^2/(a+b*sinh(f*x+e))^2,x)

[Out]

int(1/(d*x+c)^2/(a+b*sinh(f*x+e))^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.80 \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+b*sinh(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sinh(f*x + e)^2 + 2*(a
*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*sinh(f*x + e)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(d*x+c)**2/(a+b*sinh(f*x+e))**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.86 (sec) , antiderivative size = 606, normalized size of antiderivative = 30.30 \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+b*sinh(f*x+e))^2,x, algorithm="maxima")

[Out]

-2*(a*e^(f*x + e) - b)/(a^2*b*c^2*f + b^3*c^2*f + (a^2*b*d^2*f + b^3*d^2*f)*x^2 + 2*(a^2*b*c*d*f + b^3*c*d*f)*
x - (a^2*b*c^2*f*e^(2*e) + b^3*c^2*f*e^(2*e) + (a^2*b*d^2*f*e^(2*e) + b^3*d^2*f*e^(2*e))*x^2 + 2*(a^2*b*c*d*f*
e^(2*e) + b^3*c*d*f*e^(2*e))*x)*e^(2*f*x) - 2*(a^3*c^2*f*e^e + a*b^2*c^2*f*e^e + (a^3*d^2*f*e^e + a*b^2*d^2*f*
e^e)*x^2 + 2*(a^3*c*d*f*e^e + a*b^2*c*d*f*e^e)*x)*e^(f*x)) + integrate(2*(2*b*d - (a*d*f*x*e^e + (c*f*e^e + 2*
d*e^e)*a)*e^(f*x))/(a^2*b*c^3*f + b^3*c^3*f + (a^2*b*d^3*f + b^3*d^3*f)*x^3 + 3*(a^2*b*c*d^2*f + b^3*c*d^2*f)*
x^2 + 3*(a^2*b*c^2*d*f + b^3*c^2*d*f)*x - (a^2*b*c^3*f*e^(2*e) + b^3*c^3*f*e^(2*e) + (a^2*b*d^3*f*e^(2*e) + b^
3*d^3*f*e^(2*e))*x^3 + 3*(a^2*b*c*d^2*f*e^(2*e) + b^3*c*d^2*f*e^(2*e))*x^2 + 3*(a^2*b*c^2*d*f*e^(2*e) + b^3*c^
2*d*f*e^(2*e))*x)*e^(2*f*x) - 2*(a^3*c^3*f*e^e + a*b^2*c^3*f*e^e + (a^3*d^3*f*e^e + a*b^2*d^3*f*e^e)*x^3 + 3*(
a^3*c*d^2*f*e^e + a*b^2*c*d^2*f*e^e)*x^2 + 3*(a^3*c^2*d*f*e^e + a*b^2*c^2*d*f*e^e)*x)*e^(f*x)), x)

Giac [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+b*sinh(f*x+e))^2,x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^2*(b*sinh(f*x + e) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \sinh (e+f x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(1/((a + b*sinh(e + f*x))^2*(c + d*x)^2),x)

[Out]

int(1/((a + b*sinh(e + f*x))^2*(c + d*x)^2), x)