3.2.16 \(\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx\) [116]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {1}{(c+d x) (a+i a \sinh (e+f x))^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, 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 {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 24.78, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right ) \left (a +i a \sinh \left (f x +e \right )\right )^{2}}\, dx\]

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 = 0.00, 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(1/(d*x+c)/(a+I*a*sinh(f*x+e))^2,x, algorithm="maxima")

[Out]

-2*(I*d^2*f^2*x^2 + 2*I*c*d*f^2*x + I*c^2*f^2 - 2*I*d^2 + (-I*d^2*f*x*e^(2*e) + (-I*c*d*f + 2*I*d^2)*e^(2*e))*
e^(2*f*x) - (3*d^2*f^2*x^2*e^e + (6*c*d*f^2 + d^2*f)*x*e^e + (3*c^2*f^2 + c*d*f - 4*d^2)*e^e)*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) + 3*I*a^2*c*d^2*f^3*x^2*e^(2*e) + 3*I*a^2*c^2*d*f^3*x*e^(2*e) + 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
(2/3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 6*d^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 - (-I*a^2*d^4*f^3*x^4*e^e - 4*I*a^2*c*d^3*f^3*x^3*e^e - 6*I*a^2*c^2*
d^2*f^3*x^2*e^e - 4*I*a^2*c^3*d*f^3*x*e^e - I*a^2*c^4*f^3*e^e)*e^(f*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(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 - I*c*d*f + 2*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 -
3*I*a^2*c*d^2*f^3*x^2 - 3*I*a^2*c^2*d*f^3*x - I*a^2*c^3*f^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) + 3*(I*a^2*d^3*f^3*x^3 + 3*I*a^2*c*d^2*f^3*x^2 + 3*I*a^2*c^2*d*f^3*
x + I*a^2*c^3*f^3)*e^(2*f*x + 2*e) + 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^(f*x + e))*integral(-2*(-I*d^3*f^2*x^2 - 2*I*c*d^2*f^2*x - I*c^2*d*f^2 + 6*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 + 3*I*a^2*c*d^2*f^3*x^2 +
3*I*a^2*c^2*d*f^3*x + 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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, 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(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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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