3.3.75 \(\int \frac {\text {sech}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [275]

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

[Out]

Unintegrable(sech(d*x+c)/(f*x+e)/(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) (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)*(a + I*a*Sinh[c + d*x])),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 45.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {sech}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

-((d*f*x + d*e - f)*e^(d*x + c) - (a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2 - (a*d^2*f^2*x^2 + 2*a*d^2*f*x*e
+ a*d^2*e^2)*e^(2*d*x + 2*c) + 2*(I*a*d^2*f^2*x^2 + 2*I*a*d^2*f*x*e + I*a*d^2*e^2)*e^(d*x + c))*integral((-2*I
*f^2 + (d^2*f^2*x^2 + 2*d^2*f*x*e + d^2*e^2 - 2*f^2)*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)), x)
+ I*f)/(a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2 - (a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2)*e^(2*d*x + 2*c
) + 2*(I*a*d^2*f^2*x^2 + 2*I*a*d^2*f*x*e + I*a*d^2*e^2)*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 \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*Integral(sech(c + d*x)/(e*sinh(c + d*x) - I*e + f*x*sinh(c + d*x) - I*f*x), 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)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate(sech(d*x + c)/((f*x + e)*(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 )\,\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)*(a + a*sinh(c + d*x)*1i)),x)

[Out]

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

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