3.112 \(\int \frac{1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{1}{(c+d x)^2 (a+i a \sinh (e+f x))},x\right ) \]

[Out]

Unintegrable[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])), x]

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Rubi [A]  time = 0.0597691, 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 = {} \[ \int \frac{1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])), x]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx &=\int \frac{1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx\\ \end{align*}

Mathematica [A]  time = 21.0866, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])), x]

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Maple [A]  time = 0.164, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2} \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)^2/(a+I*a*sinh(f*x+e)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+I*a*sinh(f*x+e)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**2/(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 )}^{2}{\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)^2/(a+I*a*sinh(f*x+e)),x, algorithm="giac")

[Out]

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