\(\int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx\) [227]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Int}\left (\frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)},x\right ) \]

[Out]

Unintegrable(x^m*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx \]

[In]

Int[(x^m*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]

[Out]

Defer[Int][(x^m*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]), x]

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

Mathematica [N/A]

Not integrable

Time = 6.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx \]

[In]

Integrate[(x^m*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]

[Out]

Integrate[(x^m*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

int(x^m*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)

[Out]

int(x^m*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{m} \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(x^m*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="fricas")

[Out]

integral(x^m*sinh(d*x + c)^2/(b*cosh(d*x + c) + a), x)

Sympy [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^{m} \sinh ^{2}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]

[In]

integrate(x**m*sinh(d*x+c)**2/(a+b*cosh(d*x+c)),x)

[Out]

Integral(x**m*sinh(c + d*x)**2/(a + b*cosh(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{m} \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(x^m*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="maxima")

[Out]

integrate(x^m*sinh(d*x + c)^2/(b*cosh(d*x + c) + a), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{m} \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(x^m*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="giac")

[Out]

integrate(x^m*sinh(d*x + c)^2/(b*cosh(d*x + c) + a), x)

Mupad [N/A]

Not integrable

Time = 1.78 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^m\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]

[In]

int((x^m*sinh(c + d*x)^2)/(a + b*cosh(c + d*x)),x)

[Out]

int((x^m*sinh(c + d*x)^2)/(a + b*cosh(c + d*x)), x)