3.3.76 \(\int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\) [276]

Optimal. Leaf size=32 \[ \text {Int}\left (\frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, 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)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]
time = 180.01, 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]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]

[Out]

$Aborted

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-((d*f*x*e^c + d*e^(c + 1) - 2*f*e^c)*e^(d*x) + 2*I*f)/(a*d^2*f^3*x^3 + 3*a*d^2*f^2*x^2*e + 3*a*d^2*f*x*e^2 +
a*d^2*e^3 - (a*d^2*f^3*x^3*e^(2*c) + 3*a*d^2*f^2*x^2*e^(2*c + 1) + 3*a*d^2*f*x*e^(2*c + 2) + a*d^2*e^(2*c + 3)
)*e^(2*d*x) + 2*(I*a*d^2*f^3*x^3*e^c + 3*I*a*d^2*f^2*x^2*e^(c + 1) + 3*I*a*d^2*f*x*e^(c + 2) + I*a*d^2*e^(c +
3))*e^(d*x)) + 2*integrate((d^2*f^2*x^2 + 2*d^2*f*x*e + d^2*e^2 - 12*f^2)/(-4*I*a*d^2*f^4*x^4 - 16*I*a*d^2*f^3
*x^3*e - 24*I*a*d^2*f^2*x^2*e^2 - 16*I*a*d^2*f*x*e^3 - 4*I*a*d^2*e^4 + 4*(a*d^2*f^4*x^4*e^c + 4*a*d^2*f^3*x^3*
e^(c + 1) + 6*a*d^2*f^2*x^2*e^(c + 2) + 4*a*d^2*f*x*e^(c + 3) + a*d^2*e^(c + 4))*e^(d*x)), x) + 2*integrate(1/
(4*I*a*f^2*x^2 + 8*I*a*f*x*e + 4*I*a*e^2 + 4*(a*f^2*x^2*e^c + 2*a*f*x*e^(c + 1) + a*e^(c + 2))*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)/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-((d*f*x + d*e - 2*f)*e^(d*x + c) - (a*d^2*f^3*x^3 + 3*a*d^2*f^2*x^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3 - (a*d^2*
f^3*x^3 + 3*a*d^2*f^2*x^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3)*e^(2*d*x + 2*c) + 2*(I*a*d^2*f^3*x^3 + 3*I*a*d^2*f^
2*x^2*e + 3*I*a*d^2*f*x*e^2 + I*a*d^2*e^3)*e^(d*x + c))*integral((-6*I*f^2 + (d^2*f^2*x^2 + 2*d^2*f*x*e + d^2*
e^2 - 6*f^2)*e^(d*x + c))/(a*d^2*f^4*x^4 + 4*a*d^2*f^3*x^3*e + 6*a*d^2*f^2*x^2*e^2 + 4*a*d^2*f*x*e^3 + a*d^2*e
^4 + (a*d^2*f^4*x^4 + 4*a*d^2*f^3*x^3*e + 6*a*d^2*f^2*x^2*e^2 + 4*a*d^2*f*x*e^3 + a*d^2*e^4)*e^(2*d*x + 2*c)),
x) + 2*I*f)/(a*d^2*f^3*x^3 + 3*a*d^2*f^2*x^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3 - (a*d^2*f^3*x^3 + 3*a*d^2*f^2*x
^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3)*e^(2*d*x + 2*c) + 2*(I*a*d^2*f^3*x^3 + 3*I*a*d^2*f^2*x^2*e + 3*I*a*d^2*f*x
*e^2 + I*a*d^2*e^3)*e^(d*x + c))

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \int \frac {\operatorname {sech}{\left (c + d x \right )}}{e^{2} \sinh {\left (c + d x \right )} - i e^{2} + 2 e f x \sinh {\left (c + d x \right )} - 2 i e f x + f^{2} x^{2} \sinh {\left (c + d x \right )} - i f^{2} x^{2}}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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