\(\int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx\) [119]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 21.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.85 \[ \int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx=\frac {\int \frac {1}{c \cosh ^{2}{\left (e + f x \right )} + 2 c \cosh {\left (e + f x \right )} + c + d x \cosh ^{2}{\left (e + f x \right )} + 2 d x \cosh {\left (e + f x \right )} + d x}\, dx}{a^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 593, normalized size of antiderivative = 29.65 \[ \int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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