3.4.90 \(\int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [390]

Optimal. Leaf size=37 \[ \text {Int}\left (\frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sech(d*x+c)*tanh(d*x+c)^2/(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 = {} \begin {gather*} \int \frac {\text {sech}(c+d x) \tanh ^2(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]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][(Sech[c + d*x]*Tanh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end {align*}

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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sech[c + d*x]*Tanh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

$Aborted

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {sech}\left (d x +c \right ) \left (\tanh ^{2}\left (d x +c \right )\right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)*tanh(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(sech(d*x+c)*tanh(d*x+c)^2/(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)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(b*f - (a*d*f*x*e^(3*c) - a*f*e^(3*c) + a*d*e^(3*c + 1))*e^(3*d*x) - (2*b*d*f*x*e^(2*c) - b*f*e^(2*c) + 2*b*d*
e^(2*c + 1))*e^(2*d*x) + (a*d*f*x*e^c + a*d*e^(c + 1) + a*f*e^c)*e^(d*x))/((a^2*d^2*f^2 + b^2*d^2*f^2)*x^2 + 2
*(a^2*d^2*f + b^2*d^2*f)*x*e + (a^2*d^2 + b^2*d^2)*e^2 + ((a^2*d^2*f^2*e^(4*c) + b^2*d^2*f^2*e^(4*c))*x^2 + 2*
(a^2*d^2*f*e^(4*c) + b^2*d^2*f*e^(4*c))*x*e + (a^2*d^2*e^(4*c) + b^2*d^2*e^(4*c))*e^2)*e^(4*d*x) + 2*((a^2*d^2
*f^2*e^(2*c) + b^2*d^2*f^2*e^(2*c))*x^2 + 2*(a^2*d^2*f*e^(2*c) + b^2*d^2*f*e^(2*c))*x*e + (a^2*d^2*e^(2*c) + b
^2*d^2*e^(2*c))*e^2)*e^(2*d*x)) + 2*integrate(1/2*(2*a^2*b*d^2*f^2*x^2 + 4*a^2*b*d^2*f*x*e + 2*a^2*b*d^2*e^2 +
2*a^2*b*f^2 + 2*b^3*f^2 + (2*a^3*f^2*e^c + 2*a*b^2*f^2*e^c + (a^3*d^2*f^2*e^c - a*b^2*d^2*f^2*e^c)*x^2 + 2*(a
^3*d^2*f*e^c - a*b^2*d^2*f*e^c)*x*e + (a^3*d^2*e^c - a*b^2*d^2*e^c)*e^2)*e^(d*x))/((a^4*d^2*f^3 + 2*a^2*b^2*d^
2*f^3 + b^4*d^2*f^3)*x^3 + 3*(a^4*d^2*f^2 + 2*a^2*b^2*d^2*f^2 + b^4*d^2*f^2)*x^2*e + 3*(a^4*d^2*f + 2*a^2*b^2*
d^2*f + b^4*d^2*f)*x*e^2 + (a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2)*e^3 + ((a^4*d^2*f^3*e^(2*c) + 2*a^2*b^2*d^2*f^3
*e^(2*c) + b^4*d^2*f^3*e^(2*c))*x^3 + 3*(a^4*d^2*f^2*e^(2*c) + 2*a^2*b^2*d^2*f^2*e^(2*c) + b^4*d^2*f^2*e^(2*c)
)*x^2*e + 3*(a^4*d^2*f*e^(2*c) + 2*a^2*b^2*d^2*f*e^(2*c) + b^4*d^2*f*e^(2*c))*x*e^2 + (a^4*d^2*e^(2*c) + 2*a^2
*b^2*d^2*e^(2*c) + b^4*d^2*e^(2*c))*e^3)*e^(2*d*x)), x) - 2*integrate(-(a^3*b*e^(d*x + c) - a^2*b^2)/((a^4*b*f
+ 2*a^2*b^3*f + b^5*f)*x + (a^4*b + 2*a^2*b^3 + b^5)*e - ((a^4*b*f*e^(2*c) + 2*a^2*b^3*f*e^(2*c) + b^5*f*e^(2
*c))*x + (a^4*b*e^(2*c) + 2*a^2*b^3*e^(2*c) + b^5*e^(2*c))*e)*e^(2*d*x) - 2*((a^5*f*e^c + 2*a^3*b^2*f*e^c + a*
b^4*f*e^c)*x + (a^5*e^c + 2*a^3*b^2*e^c + a*b^4*e^c)*e)*e^(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)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

integral(sech(d*x + c)*tanh(d*x + c)^2/(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 ^{2}{\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)**2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

Integral(tanh(c + d*x)**2*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)^2/(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 )}^2}{\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)^2/(cosh(c + d*x)*(e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(tanh(c + d*x)^2/(cosh(c + d*x)*(e + f*x)*(a + b*sinh(c + d*x))), x)

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