\(\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [361]
Optimal result
Integrand size = 34, antiderivative size = 34 \[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right )
\]
[Out]
Unintegrable(sech(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)
Rubi [N/A]
Not integrable
Time = 0.06 (sec) , antiderivative size = 34, 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 = {} \[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx
\]
[In]
Int[(Sech[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])),x]
[Out]
Defer[Int][(Sech[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 75.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06
\[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx
\]
[In]
Integrate[(Sech[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])),x]
[Out]
Integrate[(Sech[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]
Maple [N/A] (verified)
Not integrable
Time = 0.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {sech}\left (d x +c \right )^{2} \tanh \left (d x +c \right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}d x\]
[In]
int(sech(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)
[Out]
int(sech(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)
Fricas [N/A]
Not integrable
Time = 4.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24
\[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{2} \tanh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x }
\]
[In]
integrate(sech(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")
[Out]
integral(sech(d*x + c)^2*tanh(d*x + c)/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)
Sympy [N/A]
Not integrable
Time = 3.70 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91
\[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\tanh {\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx
\]
[In]
integrate(sech(d*x+c)**2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)
[Out]
Integral(tanh(c + d*x)*sech(c + d*x)**2/((a + b*sinh(c + d*x))*(e + f*x)), x)
Maxima [N/A]
Not integrable
Time = 1.96 (sec) , antiderivative size = 1104, normalized size of antiderivative = 32.47
\[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{2} \tanh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x }
\]
[In]
integrate(sech(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")
[Out]
(a*f + (b*d*f*x*e^(3*c) + (d*e - f)*b*e^(3*c))*e^(3*d*x) - (2*a*d*f*x*e^(2*c) + (2*d*e - f)*a*e^(2*c))*e^(2*d*
x) - (b*d*f*x*e^c + (d*e + f)*b*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)) - 4*integrate(1/4*(2*a*b^2*d^2*f^2*x^2 + 4*a*b^2*d^2*e*f*x - 2*a^3*f^2 + 2*(d^2*e^2 - f^2)*a*b
^2 + ((d^2*e^2 + 2*f^2)*a^2*b*e^c - (d^2*e^2 - 2*f^2)*b^3*e^c + (a^2*b*d^2*f^2*e^c - b^3*d^2*f^2*e^c)*x^2 + 2*
(a^2*b*d^2*e*f*e^c - b^3*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) + 4*integrate(-1/2*(a^2*b^2*e^(d*x + c) - a*b^3)/(
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. \[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Timed out}
\]
[In]
integrate(sech(d*x+c)^2*tanh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")
[Out]
Timed out
Mupad [N/A]
Not integrable
Time = 3.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06
\[
\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\mathrm {tanh}\left (c+d\,x\right )}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x
\]
[In]
int(tanh(c + d*x)/(cosh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))),x)
[Out]
int(tanh(c + d*x)/(cosh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))), x)