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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx \]

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*}

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Mathematica [A]  time = 5.53, size = 0, normalized size = 0.00 \[ \int \frac {x^m \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx \]

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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fricas [A]  time = 2.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sinh \left (d x + c\right )}{b \cosh \left (d x + c\right ) + a}\,{d x} \]

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)

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maple [A]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sinh \left (d x +c \right )}{a +b \cosh \left (d x +c \right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {x e^{\left (2 \, d x + m \log \relax (x) + 2 \, c\right )}}{b {\left (m + 1\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a {\left (m + 1\right )} e^{\left (d x + c\right )} + b {\left (m + 1\right )}} - \frac {1}{2} \, \int \frac {2 \, {\left (2 \, a d x e^{\left (3 \, d x + 3 \, c\right )} + 2 \, a {\left (m + 1\right )} e^{\left (d x + c\right )} + b {\left (m + 1\right )} + {\left (2 \, b d x e^{\left (2 \, c\right )} + b {\left (m + 1\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )} x^{m}}{b^{2} {\left (m + 1\right )} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a b {\left (m + 1\right )} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, a b {\left (m + 1\right )} e^{\left (d x + c\right )} + b^{2} {\left (m + 1\right )} + 2 \, {\left (2 \, a^{2} {\left (m + 1\right )} e^{\left (2 \, c\right )} + b^{2} {\left (m + 1\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^m\,\mathrm {sinh}\left (c+d\,x\right )}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sinh {\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]

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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