\(\int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx\) [180]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\text {Int}\left (\frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3},x\right ) \]

[Out]

Unintegrable(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,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 = {} \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx \]

[In]

Int[1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3),x]

[Out]

Defer[Int][1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 46.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx \]

[In]

Integrate[1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3),x]

[Out]

Integrate[1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (f x +e \right )^{2} \left (a +b \sinh \left (d x +c \right )\right )^{3}}d x\]

[In]

int(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x)

[Out]

int(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 7.05 \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\int { \frac {1}{{\left (f x + e\right )}^{2} {\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(1/(a^3*f^2*x^2 + 2*a^3*e*f*x + a^3*e^2 + (b^3*f^2*x^2 + 2*b^3*e*f*x + b^3*e^2)*sinh(d*x + c)^3 + 3*(a
*b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e^2)*sinh(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x + a^2*b*e^2)*sinh
(d*x + c)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(f*x+e)**2/(a+b*sinh(d*x+c))**3,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 3.62 (sec) , antiderivative size = 2122, normalized size of antiderivative = 106.10 \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\int { \frac {1}{{\left (f x + e\right )}^{2} {\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x, algorithm="maxima")

[Out]

(3*a*b^2*d*f*x + 3*a*b^2*d*e + (2*(d*e + f)*a^2*b*e^(3*c) - (d*e - 2*f)*b^3*e^(3*c) + (2*a^2*b*d*f*e^(3*c) - b
^3*d*f*e^(3*c))*x)*e^(3*d*x) + (2*(3*d*e + 2*f)*a^3*e^(2*c) - (3*d*e - 4*f)*a*b^2*e^(2*c) + 3*(2*a^3*d*f*e^(2*
c) - a*b^2*d*f*e^(2*c))*x)*e^(2*d*x) - (2*(5*d*e + f)*a^2*b*e^c + (d*e + 2*f)*b^3*e^c + (10*a^2*b*d*f*e^c + b^
3*d*f*e^c)*x)*e^(d*x))/(a^4*b^2*d^2*e^3 + 2*a^2*b^4*d^2*e^3 + b^6*d^2*e^3 + (a^4*b^2*d^2*f^3 + 2*a^2*b^4*d^2*f
^3 + b^6*d^2*f^3)*x^3 + 3*(a^4*b^2*d^2*e*f^2 + 2*a^2*b^4*d^2*e*f^2 + b^6*d^2*e*f^2)*x^2 + 3*(a^4*b^2*d^2*e^2*f
+ 2*a^2*b^4*d^2*e^2*f + b^6*d^2*e^2*f)*x + (a^4*b^2*d^2*e^3*e^(4*c) + 2*a^2*b^4*d^2*e^3*e^(4*c) + b^6*d^2*e^3
*e^(4*c) + (a^4*b^2*d^2*f^3*e^(4*c) + 2*a^2*b^4*d^2*f^3*e^(4*c) + b^6*d^2*f^3*e^(4*c))*x^3 + 3*(a^4*b^2*d^2*e*
f^2*e^(4*c) + 2*a^2*b^4*d^2*e*f^2*e^(4*c) + b^6*d^2*e*f^2*e^(4*c))*x^2 + 3*(a^4*b^2*d^2*e^2*f*e^(4*c) + 2*a^2*
b^4*d^2*e^2*f*e^(4*c) + b^6*d^2*e^2*f*e^(4*c))*x)*e^(4*d*x) + 4*(a^5*b*d^2*e^3*e^(3*c) + 2*a^3*b^3*d^2*e^3*e^(
3*c) + a*b^5*d^2*e^3*e^(3*c) + (a^5*b*d^2*f^3*e^(3*c) + 2*a^3*b^3*d^2*f^3*e^(3*c) + a*b^5*d^2*f^3*e^(3*c))*x^3
+ 3*(a^5*b*d^2*e*f^2*e^(3*c) + 2*a^3*b^3*d^2*e*f^2*e^(3*c) + a*b^5*d^2*e*f^2*e^(3*c))*x^2 + 3*(a^5*b*d^2*e^2*
f*e^(3*c) + 2*a^3*b^3*d^2*e^2*f*e^(3*c) + a*b^5*d^2*e^2*f*e^(3*c))*x)*e^(3*d*x) + 2*(2*a^6*d^2*e^3*e^(2*c) + 3
*a^4*b^2*d^2*e^3*e^(2*c) - b^6*d^2*e^3*e^(2*c) + (2*a^6*d^2*f^3*e^(2*c) + 3*a^4*b^2*d^2*f^3*e^(2*c) - b^6*d^2*
f^3*e^(2*c))*x^3 + 3*(2*a^6*d^2*e*f^2*e^(2*c) + 3*a^4*b^2*d^2*e*f^2*e^(2*c) - b^6*d^2*e*f^2*e^(2*c))*x^2 + 3*(
2*a^6*d^2*e^2*f*e^(2*c) + 3*a^4*b^2*d^2*e^2*f*e^(2*c) - b^6*d^2*e^2*f*e^(2*c))*x)*e^(2*d*x) - 4*(a^5*b*d^2*e^3
*e^c + 2*a^3*b^3*d^2*e^3*e^c + a*b^5*d^2*e^3*e^c + (a^5*b*d^2*f^3*e^c + 2*a^3*b^3*d^2*f^3*e^c + a*b^5*d^2*f^3*
e^c)*x^3 + 3*(a^5*b*d^2*e*f^2*e^c + 2*a^3*b^3*d^2*e*f^2*e^c + a*b^5*d^2*e*f^2*e^c)*x^2 + 3*(a^5*b*d^2*e^2*f*e^
c + 2*a^3*b^3*d^2*e^2*f*e^c + a*b^5*d^2*e^2*f*e^c)*x)*e^(d*x)) + integrate((6*a*b*d*f^2*x + 6*a*b*d*e*f - (2*(
d^2*e^2 + 3*d*e*f + 3*f^2)*a^2*e^c - (d^2*e^2 - 6*f^2)*b^2*e^c + (2*a^2*d^2*f^2*e^c - b^2*d^2*f^2*e^c)*x^2 - 2
*(b^2*d^2*e*f*e^c - (2*d^2*e*f + 3*d*f^2)*a^2*e^c)*x)*e^(d*x))/(a^4*b*d^2*e^4 + 2*a^2*b^3*d^2*e^4 + b^5*d^2*e^
4 + (a^4*b*d^2*f^4 + 2*a^2*b^3*d^2*f^4 + b^5*d^2*f^4)*x^4 + 4*(a^4*b*d^2*e*f^3 + 2*a^2*b^3*d^2*e*f^3 + b^5*d^2
*e*f^3)*x^3 + 6*(a^4*b*d^2*e^2*f^2 + 2*a^2*b^3*d^2*e^2*f^2 + b^5*d^2*e^2*f^2)*x^2 + 4*(a^4*b*d^2*e^3*f + 2*a^2
*b^3*d^2*e^3*f + b^5*d^2*e^3*f)*x - (a^4*b*d^2*e^4*e^(2*c) + 2*a^2*b^3*d^2*e^4*e^(2*c) + b^5*d^2*e^4*e^(2*c) +
(a^4*b*d^2*f^4*e^(2*c) + 2*a^2*b^3*d^2*f^4*e^(2*c) + b^5*d^2*f^4*e^(2*c))*x^4 + 4*(a^4*b*d^2*e*f^3*e^(2*c) +
2*a^2*b^3*d^2*e*f^3*e^(2*c) + b^5*d^2*e*f^3*e^(2*c))*x^3 + 6*(a^4*b*d^2*e^2*f^2*e^(2*c) + 2*a^2*b^3*d^2*e^2*f^
2*e^(2*c) + b^5*d^2*e^2*f^2*e^(2*c))*x^2 + 4*(a^4*b*d^2*e^3*f*e^(2*c) + 2*a^2*b^3*d^2*e^3*f*e^(2*c) + b^5*d^2*
e^3*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^5*d^2*e^4*e^c + 2*a^3*b^2*d^2*e^4*e^c + a*b^4*d^2*e^4*e^c + (a^5*d^2*f^4*e^
c + 2*a^3*b^2*d^2*f^4*e^c + a*b^4*d^2*f^4*e^c)*x^4 + 4*(a^5*d^2*e*f^3*e^c + 2*a^3*b^2*d^2*e*f^3*e^c + a*b^4*d^
2*e*f^3*e^c)*x^3 + 6*(a^5*d^2*e^2*f^2*e^c + 2*a^3*b^2*d^2*e^2*f^2*e^c + a*b^4*d^2*e^2*f^2*e^c)*x^2 + 4*(a^5*d^
2*e^3*f*e^c + 2*a^3*b^2*d^2*e^3*f*e^c + a*b^4*d^2*e^3*f*e^c)*x)*e^(d*x)), x)

Giac [N/A]

Not integrable

Time = 60.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\int { \frac {1}{{\left (f x + e\right )}^{2} {\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((f*x + e)^2*(b*sinh(d*x + c) + a)^3), x)

Mupad [N/A]

Not integrable

Time = 1.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx=\int \frac {1}{{\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int(1/((e + f*x)^2*(a + b*sinh(c + d*x))^3),x)

[Out]

int(1/((e + f*x)^2*(a + b*sinh(c + d*x))^3), x)