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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{x^m \sinh (c+d x)}{a+b \cosh (c+d x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0381827, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 5.09777, size = 0, normalized size = 0. \[ \int \frac{x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}\sinh \left ( dx+c \right ) }{a+b\cosh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x e^{\left (2 \, d x + m \log \left (x\right ) + 2 \, c\right )}}{b{\left (m + 1\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a{\left (m + 1\right )} e^{\left (d x + c\right )} + b{\left (m + 1\right )}} - \frac{1}{2} \, \int \frac{2 \,{\left (2 \, a d x e^{\left (3 \, d x + 3 \, c\right )} + 2 \, a{\left (m + 1\right )} e^{\left (d x + c\right )} + b{\left (m + 1\right )} +{\left (2 \, b d x e^{\left (2 \, c\right )} + b{\left (m + 1\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )} x^{m}}{b^{2}{\left (m + 1\right )} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a b{\left (m + 1\right )} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, a b{\left (m + 1\right )} e^{\left (d x + c\right )} + b^{2}{\left (m + 1\right )} + 2 \,{\left (2 \, a^{2}{\left (m + 1\right )} e^{\left (2 \, c\right )} + b^{2}{\left (m + 1\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \sinh{\left (c + d x \right )}}{a + b \cosh{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)