\(\int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx\) [176]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 27.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 173.54 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx=\int \frac {1}{\left (a + b \cosh {\left (e + f x \right )}\right )^{2} \left (c + d x\right )}\, dx \]

[In]

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

[Out]

Integral(1/((a + b*cosh(e + f*x))**2*(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 416, normalized size of antiderivative = 20.80 \[ \int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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