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

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

[Out]

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

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Rubi [A]  time = 0.0496738, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{sech}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

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*}

Mathematica [A]  time = 65.4834, size = 0, normalized size = 0. \[ \int \frac{\text{sech}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

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 = 0.714, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm sech} \left (dx+c\right )}{ \left ( fx+e \right ) \left ( a+ia\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left ({\left (d f x e^{c} +{\left (d e - f\right )} e^{c}\right )} e^{\left (d x\right )} + i \, f\right )}}{2 \, a d^{2} f^{2} x^{2} + 4 \, a d^{2} e f x + 2 \, a d^{2} e^{2} - 2 \,{\left (a d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \, a d^{2} e f x e^{\left (2 \, c\right )} + a d^{2} e^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} -{\left (-4 i \, a d^{2} f^{2} x^{2} e^{c} - 8 i \, a d^{2} e f x e^{c} - 4 i \, a d^{2} e^{2} e^{c}\right )} e^{\left (d x\right )}} + 2 \, \int \frac{d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + d^{2} e^{2} - 4 \, f^{2}}{-4 i \, a d^{2} f^{3} x^{3} - 12 i \, a d^{2} e f^{2} x^{2} - 12 i \, a d^{2} e^{2} f x - 4 i \, a d^{2} e^{3} + 4 \,{\left (a d^{2} f^{3} x^{3} e^{c} + 3 \, a d^{2} e f^{2} x^{2} e^{c} + 3 \, a d^{2} e^{2} f x e^{c} + a d^{2} e^{3} e^{c}\right )} e^{\left (d x\right )}}\,{d x} + 2 \, \int \frac{1}{4 i \, a f x + 4 i \, a e + 4 \,{\left (a f x e^{c} + a e e^{c}\right )} e^{\left (d x\right )}}\,{d x} \end{align*}

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]

-2*((d*f*x*e^c + (d*e - f)*e^c)*e^(d*x) + I*f)/(2*a*d^2*f^2*x^2 + 4*a*d^2*e*f*x + 2*a*d^2*e^2 - 2*(a*d^2*f^2*x
^2*e^(2*c) + 2*a*d^2*e*f*x*e^(2*c) + a*d^2*e^2*e^(2*c))*e^(2*d*x) - (-4*I*a*d^2*f^2*x^2*e^c - 8*I*a*d^2*e*f*x*
e^c - 4*I*a*d^2*e^2*e^c)*e^(d*x)) + 2*integrate((d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2 - 4*f^2)/(-4*I*a*d^2*f^3*
x^3 - 12*I*a*d^2*e*f^2*x^2 - 12*I*a*d^2*e^2*f*x - 4*I*a*d^2*e^3 + 4*(a*d^2*f^3*x^3*e^c + 3*a*d^2*e*f^2*x^2*e^c
+ 3*a*d^2*e^2*f*x*e^c + a*d^2*e^3*e^c)*e^(d*x)), x) + 2*integrate(1/(4*I*a*f*x + 4*I*a*e + 4*(a*f*x*e^c + a*e
*e^c)*e^(d*x)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (d f x + d e - f\right )} e^{\left (d x + c\right )} -{\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2} -{\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} -{\left (-2 i \, a d^{2} f^{2} x^{2} - 4 i \, a d^{2} e f x - 2 i \, a d^{2} e^{2}\right )} e^{\left (d x + c\right )}\right )}{\rm integral}\left (\frac{-2 i \, f^{2} +{\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + d^{2} e^{2} - 2 \, f^{2}\right )} e^{\left (d x + c\right )}}{a d^{2} f^{3} x^{3} + 3 \, a d^{2} e f^{2} x^{2} + 3 \, a d^{2} e^{2} f x + a d^{2} e^{3} +{\left (a d^{2} f^{3} x^{3} + 3 \, a d^{2} e f^{2} x^{2} + 3 \, a d^{2} e^{2} f x + a d^{2} e^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}, x\right ) + i \, f}{a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2} -{\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} -{\left (-2 i \, a d^{2} f^{2} x^{2} - 4 i \, a d^{2} e f x - 2 i \, a d^{2} e^{2}\right )} e^{\left (d x + c\right )}} \end{align*}

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

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

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]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (d x + c\right )}{{\left (f x + e\right )}{\left (i \, a \sinh \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}

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)