\(\int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx\) [221]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Int}\left (\frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.94 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{m} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^{m} \sinh {\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 202, normalized size of antiderivative = 9.18 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{m} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

x*e^(2*d*x + m*log(x) + 2*c)/(b*(m + 1)*e^(2*d*x + 2*c) + 2*a*(m + 1)*e^(d*x + c) + b*(m + 1)) - 1/2*integrate
(2*(2*a*d*x*e^(3*d*x + 3*c) + 2*a*(m + 1)*e^(d*x + c) + b*(m + 1) + (2*b*d*x*e^(2*c) + b*(m + 1)*e^(2*c))*e^(2
*d*x))*x^m/(b^2*(m + 1)*e^(4*d*x + 4*c) + 4*a*b*(m + 1)*e^(3*d*x + 3*c) + 4*a*b*(m + 1)*e^(d*x + c) + b^2*(m +
1) + 2*(2*a^2*(m + 1)*e^(2*c) + b^2*(m + 1)*e^(2*c))*e^(2*d*x)), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{m} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^m\,\mathrm {sinh}\left (c+d\,x\right )}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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