3.313 \(\int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 66.07, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (d x + c\right )^{2}}{a f x + a e + {\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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maple [A]  time = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (d x +c \right )^{2}}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sech(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 \[ 4 \, b^{2} \int -\frac {e^{\left (d x + c\right )}}{2 \, {\left (a^{2} b e + b^{3} e + {\left (a^{2} b f + b^{3} f\right )} x - {\left (a^{2} b e e^{\left (2 \, c\right )} + b^{3} e e^{\left (2 \, c\right )} + {\left (a^{2} b f e^{\left (2 \, c\right )} + b^{3} f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} e e^{c} + a b^{2} e e^{c} + {\left (a^{3} f e^{c} + a b^{2} f e^{c}\right )} x\right )} e^{\left (d x\right )}\right )}}\,{d x} + \frac {2 \, {\left (b e^{\left (d x + c\right )} - a\right )}}{a^{2} d e + b^{2} d e + {\left (a^{2} d f + b^{2} d f\right )} x + {\left (a^{2} d e e^{\left (2 \, c\right )} + b^{2} d e e^{\left (2 \, c\right )} + {\left (a^{2} d f e^{\left (2 \, c\right )} + b^{2} d f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}} + 4 \, \int \frac {b f e^{\left (d x + c\right )} - a f}{2 \, {\left (a^{2} d e^{2} + b^{2} d e^{2} + {\left (a^{2} d f^{2} + b^{2} d f^{2}\right )} x^{2} + 2 \, {\left (a^{2} d e f + b^{2} d e f\right )} x + {\left (a^{2} d e^{2} e^{\left (2 \, c\right )} + b^{2} d e^{2} e^{\left (2 \, c\right )} + {\left (a^{2} d f^{2} e^{\left (2 \, c\right )} + b^{2} d f^{2} e^{\left (2 \, c\right )}\right )} x^{2} + 2 \, {\left (a^{2} d e f e^{\left (2 \, c\right )} + b^{2} d e f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b^2*integrate(-1/2*e^(d*x + c)/(a^2*b*e + b^3*e + (a^2*b*f + b^3*f)*x - (a^2*b*e*e^(2*c) + b^3*e*e^(2*c) + (
a^2*b*f*e^(2*c) + b^3*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^3*e*e^c + a*b^2*e*e^c + (a^3*f*e^c + a*b^2*f*e^c)*x)*e^(d
*x)), x) + 2*(b*e^(d*x + c) - a)/(a^2*d*e + b^2*d*e + (a^2*d*f + b^2*d*f)*x + (a^2*d*e*e^(2*c) + b^2*d*e*e^(2*
c) + (a^2*d*f*e^(2*c) + b^2*d*f*e^(2*c))*x)*e^(2*d*x)) + 4*integrate(1/2*(b*f*e^(d*x + c) - a*f)/(a^2*d*e^2 +
b^2*d*e^2 + (a^2*d*f^2 + b^2*d*f^2)*x^2 + 2*(a^2*d*e*f + b^2*d*e*f)*x + (a^2*d*e^2*e^(2*c) + b^2*d*e^2*e^(2*c)
+ (a^2*d*f^2*e^(2*c) + b^2*d*f^2*e^(2*c))*x^2 + 2*(a^2*d*e*f*e^(2*c) + b^2*d*e*f*e^(2*c))*x)*e^(2*d*x)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(1/(cosh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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