3.111 \(\int \frac{1}{(c+d x) (a+i a \sinh (e+f x))} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a+i a \sinh (e+f x))},x\right ) \]

[Out]

Unintegrable[1/((c + d*x)*(a + I*a*Sinh[e + f*x])), x]

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Rubi [A]  time = 0.0641974, 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 = {} \[ \int \frac{1}{(c+d x) (a+i a \sinh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + d*x)*(a + I*a*Sinh[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)*(a + I*a*Sinh[e + f*x])), x]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x) (a+i a \sinh (e+f x))} \, dx &=\int \frac{1}{(c+d x) (a+i a \sinh (e+f x))} \, dx\\ \end{align*}

Mathematica [A]  time = 20.6237, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+i a \sinh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + d*x)*(a + I*a*Sinh[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)*(a + I*a*Sinh[e + f*x])), x]

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Maple [A]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+ia\sinh \left ( fx+e \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 i \, d \int \frac{1}{-i \, a d^{2} f x^{2} - 2 i \, a c d f x - i \, a c^{2} f +{\left (a d^{2} f x^{2} e^{e} + 2 \, a c d f x e^{e} + a c^{2} f e^{e}\right )} e^{\left (f x\right )}}\,{d x} + \frac{2 i}{-i \, a d f x - i \, a c f +{\left (a d f x e^{e} + a c f e^{e}\right )} e^{\left (f x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e)),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}{\left (i \, a \sinh \left (f x + e\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(I*a*sinh(f*x + e) + a)), x)