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

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

[Out]

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

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Rubi [A]  time = 0.0604981, 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 = {} \[ \int \frac{1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 36.1759, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.823, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+ia\sinh \left ( fx+e \right ) \right ) ^{2}}}\, 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))^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-2 i \, d^{2} f^{2} x^{2} - 4 i \, c d f^{2} x - 2 i \, c^{2} f^{2} + 4 i \, d^{2} +{\left (2 i \, d^{2} f x e^{\left (2 \, e\right )} + 2 i \, c d f e^{\left (2 \, e\right )} - 4 i \, d^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 2 \,{\left (3 \, d^{2} f^{2} x^{2} e^{e} + 3 \, c^{2} f^{2} e^{e} + c d f e^{e} - 4 \, d^{2} e^{e} +{\left (6 \, c d f^{2} e^{e} + d^{2} f e^{e}\right )} x\right )} e^{\left (f x\right )}}{3 i \, a^{2} d^{3} f^{3} x^{3} + 9 i \, a^{2} c d^{2} f^{3} x^{2} + 9 i \, a^{2} c^{2} d f^{3} x + 3 i \, a^{2} c^{3} f^{3} + 3 \,{\left (a^{2} d^{3} f^{3} x^{3} e^{\left (3 \, e\right )} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{\left (3 \, e\right )} + 3 \, a^{2} c^{2} d f^{3} x e^{\left (3 \, e\right )} + a^{2} c^{3} f^{3} e^{\left (3 \, e\right )}\right )} e^{\left (3 \, f x\right )} +{\left (-9 i \, a^{2} d^{3} f^{3} x^{3} e^{\left (2 \, e\right )} - 27 i \, a^{2} c d^{2} f^{3} x^{2} e^{\left (2 \, e\right )} - 27 i \, a^{2} c^{2} d f^{3} x e^{\left (2 \, e\right )} - 9 i \, a^{2} c^{3} f^{3} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 9 \,{\left (a^{2} d^{3} f^{3} x^{3} e^{e} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{e} + 3 \, a^{2} c^{2} d f^{3} x e^{e} + a^{2} c^{3} f^{3} e^{e}\right )} e^{\left (f x\right )}} - \int \frac{2 \,{\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} - 6 \, d^{3}\right )}}{3 \, a^{2} d^{4} f^{3} x^{4} + 12 \, a^{2} c d^{3} f^{3} x^{3} + 18 \, a^{2} c^{2} d^{2} f^{3} x^{2} + 12 \, a^{2} c^{3} d f^{3} x + 3 \, a^{2} c^{4} f^{3} +{\left (3 i \, a^{2} d^{4} f^{3} x^{4} e^{e} + 12 i \, a^{2} c d^{3} f^{3} x^{3} e^{e} + 18 i \, a^{2} c^{2} d^{2} f^{3} x^{2} e^{e} + 12 i \, a^{2} c^{3} d f^{3} x e^{e} + 3 i \, a^{2} c^{4} f^{3} e^{e}\right )} e^{\left (f x\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))^2,x, algorithm="maxima")

[Out]

(-2*I*d^2*f^2*x^2 - 4*I*c*d*f^2*x - 2*I*c^2*f^2 + 4*I*d^2 + (2*I*d^2*f*x*e^(2*e) + 2*I*c*d*f*e^(2*e) - 4*I*d^2
*e^(2*e))*e^(2*f*x) + 2*(3*d^2*f^2*x^2*e^e + 3*c^2*f^2*e^e + c*d*f*e^e - 4*d^2*e^e + (6*c*d*f^2*e^e + d^2*f*e^
e)*x)*e^(f*x))/(3*I*a^2*d^3*f^3*x^3 + 9*I*a^2*c*d^2*f^3*x^2 + 9*I*a^2*c^2*d*f^3*x + 3*I*a^2*c^3*f^3 + 3*(a^2*d
^3*f^3*x^3*e^(3*e) + 3*a^2*c*d^2*f^3*x^2*e^(3*e) + 3*a^2*c^2*d*f^3*x*e^(3*e) + a^2*c^3*f^3*e^(3*e))*e^(3*f*x)
+ (-9*I*a^2*d^3*f^3*x^3*e^(2*e) - 27*I*a^2*c*d^2*f^3*x^2*e^(2*e) - 27*I*a^2*c^2*d*f^3*x*e^(2*e) - 9*I*a^2*c^3*
f^3*e^(2*e))*e^(2*f*x) - 9*(a^2*d^3*f^3*x^3*e^e + 3*a^2*c*d^2*f^3*x^2*e^e + 3*a^2*c^2*d*f^3*x*e^e + a^2*c^3*f^
3*e^e)*e^(f*x)) - integrate(2*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 6*d^3)/(3*a^2*d^4*f^3*x^4 + 12*a^2*c*
d^3*f^3*x^3 + 18*a^2*c^2*d^2*f^3*x^2 + 12*a^2*c^3*d*f^3*x + 3*a^2*c^4*f^3 + (3*I*a^2*d^4*f^3*x^4*e^e + 12*I*a^
2*c*d^3*f^3*x^3*e^e + 18*I*a^2*c^2*d^2*f^3*x^2*e^e + 12*I*a^2*c^3*d*f^3*x*e^e + 3*I*a^2*c^4*f^3*e^e)*e^(f*x)),
x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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))^2,x, algorithm="fricas")

[Out]

(-2*I*d^2*f^2*x^2 - 4*I*c*d*f^2*x - 2*I*c^2*f^2 + 4*I*d^2 + (2*I*d^2*f*x + 2*I*c*d*f - 4*I*d^2)*e^(2*f*x + 2*e
) + 2*(3*d^2*f^2*x^2 + 3*c^2*f^2 + c*d*f - 4*d^2 + (6*c*d*f^2 + d^2*f)*x)*e^(f*x + e) + (3*I*a^2*d^3*f^3*x^3 +
9*I*a^2*c*d^2*f^3*x^2 + 9*I*a^2*c^2*d*f^3*x + 3*I*a^2*c^3*f^3 + 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*
a^2*c^2*d*f^3*x + a^2*c^3*f^3)*e^(3*f*x + 3*e) + (-9*I*a^2*d^3*f^3*x^3 - 27*I*a^2*c*d^2*f^3*x^2 - 27*I*a^2*c^2
*d*f^3*x - 9*I*a^2*c^3*f^3)*e^(2*f*x + 2*e) - 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a
^2*c^3*f^3)*e^(f*x + e))*integral((2*I*d^3*f^2*x^2 + 4*I*c*d^2*f^2*x + 2*I*c^2*d*f^2 - 12*I*d^3)/(-3*I*a^2*d^4
*f^3*x^4 - 12*I*a^2*c*d^3*f^3*x^3 - 18*I*a^2*c^2*d^2*f^3*x^2 - 12*I*a^2*c^3*d*f^3*x - 3*I*a^2*c^4*f^3 + 3*(a^2
*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*e^(f*x + e)), x)
)/(3*I*a^2*d^3*f^3*x^3 + 9*I*a^2*c*d^2*f^3*x^2 + 9*I*a^2*c^2*d*f^3*x + 3*I*a^2*c^3*f^3 + 3*(a^2*d^3*f^3*x^3 +
3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*e^(3*f*x + 3*e) + (-9*I*a^2*d^3*f^3*x^3 - 27*I*a^2*c*d^
2*f^3*x^2 - 27*I*a^2*c^2*d*f^3*x - 9*I*a^2*c^3*f^3)*e^(2*f*x + 2*e) - 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2
+ 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*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))**2,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 )}^{2}}\,{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))^2,x, algorithm="giac")

[Out]

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