3.227 \(\int \frac {\sinh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\)

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\sinh (c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sinh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sinh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][Sinh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\sinh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac {\sinh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end {align*}

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Mathematica [A]  time = 9.09, size = 0, normalized size = 0.00 \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sinh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Integrate[Sinh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sinh \left (d x + c\right )}{a f x + a e + {\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

integral(sinh(d*x + c)/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)/((f*x + e)*(b*sinh(d*x + c) + a)), x)

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x +c \right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(sinh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, a \int -\frac {e^{\left (d x + c\right )}}{b^{2} f x + b^{2} e - {\left (b^{2} f x e^{\left (2 \, c\right )} + b^{2} e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a b f x e^{c} + a b e e^{c}\right )} e^{\left (d x\right )}}\,{d x} + \frac {\log \left (f x + e\right )}{b f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-2*a*integrate(-e^(d*x + c)/(b^2*f*x + b^2*e - (b^2*f*x*e^(2*c) + b^2*e*e^(2*c))*e^(2*d*x) - 2*(a*b*f*x*e^c +
a*b*e*e^c)*e^(d*x)), x) + log(f*x + e)/(b*f)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {sinh}\left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)/((e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(sinh(c + d*x)/((e + f*x)*(a + b*sinh(c + d*x))), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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