\(\int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx\) [172]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \cosh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 37.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx=\int \frac {1}{\left (a + b \cosh {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \cosh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \cosh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \cosh (e+f x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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