3.4.57 \(\int \frac {\text {sech}(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [357]

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {\text {sech}(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sech(d*x+c)*tanh(d*x+c)/(f*x+e)/(a+b*sinh(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 = {} \begin {gather*} \int \frac {\text {sech}(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 46.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {sech}(c+d x) \tanh (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sech(d*x+c)*tanh(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*b*integrate(-e^(d*x + c)/((a^2*b*f + b^3*f)*x + (a^2*b + b^3)*e - ((a^2*b*f*e^(2*c) + b^3*f*e^(2*c))*x +
(a^2*b*e^(2*c) + b^3*e^(2*c))*e)*e^(2*d*x) - 2*((a^3*f*e^c + a*b^2*f*e^c)*x + (a^3*e^c + a*b^2*e^c)*e)*e^(d*x)
), x) - 2*(a*e^(d*x + c) + b)/((a^2*d*f + b^2*d*f)*x + (a^2*d + b^2*d)*e + ((a^2*d*f*e^(2*c) + b^2*d*f*e^(2*c)
)*x + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e)*e^(2*d*x)) - 2*integrate((a*f*e^(d*x + c) + b*f)/((a^2*d*f^2 + b^2*d*
f^2)*x^2 + 2*(a^2*d*f + b^2*d*f)*x*e + (a^2*d + b^2*d)*e^2 + ((a^2*d*f^2*e^(2*c) + b^2*d*f^2*e^(2*c))*x^2 + 2*
(a^2*d*f*e^(2*c) + b^2*d*f*e^(2*c))*x*e + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e^2)*e^(2*d*x)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (c + d x \right )} \operatorname {sech}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(c + d*x)*sech(c + d*x)/((a + b*sinh(c + d*x))*(e + f*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {tanh}\left (c+d\,x\right )}{\mathrm {cosh}\left (c+d\,x\right )\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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