\(\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [317]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sech(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 65.56 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.87 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 3.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]

[In]

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

[Out]

Integral(sech(c + d*x)**3/((a + b*sinh(c + d*x))*(e + f*x)), x)

Maxima [N/A]

Not integrable

Time = 1.84 (sec) , antiderivative size = 1100, normalized size of antiderivative = 39.29 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-(b*f - (a*d*f*x*e^(3*c) + (d*e - f)*a*e^(3*c))*e^(3*d*x) - (2*b*d*f*x*e^(2*c) + (2*d*e - f)*b*e^(2*c))*e^(2*d
*x) + (a*d*f*x*e^c + (d*e + f)*a*e^c)*e^(d*x))/(a^2*d^2*e^2 + b^2*d^2*e^2 + (a^2*d^2*f^2 + b^2*d^2*f^2)*x^2 +
2*(a^2*d^2*e*f + b^2*d^2*e*f)*x + (a^2*d^2*e^2*e^(4*c) + b^2*d^2*e^2*e^(4*c) + (a^2*d^2*f^2*e^(4*c) + b^2*d^2*
f^2*e^(4*c))*x^2 + 2*(a^2*d^2*e*f*e^(4*c) + b^2*d^2*e*f*e^(4*c))*x)*e^(4*d*x) + 2*(a^2*d^2*e^2*e^(2*c) + b^2*d
^2*e^2*e^(2*c) + (a^2*d^2*f^2*e^(2*c) + b^2*d^2*f^2*e^(2*c))*x^2 + 2*(a^2*d^2*e*f*e^(2*c) + b^2*d^2*e*f*e^(2*c
))*x)*e^(2*d*x)) + 8*integrate(1/8*(2*b^3*d^2*f^2*x^2 + 4*b^3*d^2*e*f*x - 2*a^2*b*f^2 + 2*(d^2*e^2 - f^2)*b^3
+ ((d^2*e^2 - 2*f^2)*a^3*e^c + (3*d^2*e^2 - 2*f^2)*a*b^2*e^c + (a^3*d^2*f^2*e^c + 3*a*b^2*d^2*f^2*e^c)*x^2 + 2
*(a^3*d^2*e*f*e^c + 3*a*b^2*d^2*e*f*e^c)*x)*e^(d*x))/(a^4*d^2*e^3 + 2*a^2*b^2*d^2*e^3 + b^4*d^2*e^3 + (a^4*d^2
*f^3 + 2*a^2*b^2*d^2*f^3 + b^4*d^2*f^3)*x^3 + 3*(a^4*d^2*e*f^2 + 2*a^2*b^2*d^2*e*f^2 + b^4*d^2*e*f^2)*x^2 + 3*
(a^4*d^2*e^2*f + 2*a^2*b^2*d^2*e^2*f + b^4*d^2*e^2*f)*x + (a^4*d^2*e^3*e^(2*c) + 2*a^2*b^2*d^2*e^3*e^(2*c) + b
^4*d^2*e^3*e^(2*c) + (a^4*d^2*f^3*e^(2*c) + 2*a^2*b^2*d^2*f^3*e^(2*c) + b^4*d^2*f^3*e^(2*c))*x^3 + 3*(a^4*d^2*
e*f^2*e^(2*c) + 2*a^2*b^2*d^2*e*f^2*e^(2*c) + b^4*d^2*e*f^2*e^(2*c))*x^2 + 3*(a^4*d^2*e^2*f*e^(2*c) + 2*a^2*b^
2*d^2*e^2*f*e^(2*c) + b^4*d^2*e^2*f*e^(2*c))*x)*e^(2*d*x)), x) - 8*integrate(-1/4*(a*b^3*e^(d*x + c) - b^4)/(a
^4*b*e + 2*a^2*b^3*e + b^5*e + (a^4*b*f + 2*a^2*b^3*f + b^5*f)*x - (a^4*b*e*e^(2*c) + 2*a^2*b^3*e*e^(2*c) + b^
5*e*e^(2*c) + (a^4*b*f*e^(2*c) + 2*a^2*b^3*f*e^(2*c) + b^5*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^5*e*e^c + 2*a^3*b^2*
e*e^c + a*b^4*e*e^c + (a^5*f*e^c + 2*a^3*b^2*f*e^c + a*b^4*f*e^c)*x)*e^(d*x)), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 3.75 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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