3.2.17 \(\int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))^2} \, dx\) [117]
Optimal. Leaf size=26 \[ \text {Int}\left (\frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))^2},x\right ) \]
[Out]
Unintegrable(1/(d*x+c)^2/(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)^2 (a+i a \sinh (e+f x))^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])^2),x]
[Out]
Defer[Int][1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])^2), x]
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))^2} \, dx &=\int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 26.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])^2),x]
[Out]
Integrate[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])^2), x]
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right )^{2} \left (a +i a \sinh \left (f x +e \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d*x+c)^2/(a+I*a*sinh(f*x+e))^2,x)
[Out]
int(1/(d*x+c)^2/(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)^2/(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 - 6*I*d^2 + 2*(-I*d^2*f*x*e^(2*e) + (-I*c*d*f + 3*I*d^2)*e^(2*e)
)*e^(2*f*x) - (3*d^2*f^2*x^2*e^e + 2*(3*c*d*f^2 + d^2*f)*x*e^e + (3*c^2*f^2 + 2*c*d*f - 12*d^2)*e^e)*e^(f*x))/
(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*e^(3*e) + 4*a^2*c*d^3*f^3*x^3*e^(3*e) + 6*a^2*c^2*d^2*f^3*x^2*e^(3*e) + 4*a^2*c^3*d*f
^3*x*e^(3*e) + a^2*c^4*f^3*e^(3*e))*e^(3*f*x) - 9*(I*a^2*d^4*f^3*x^4*e^(2*e) + 4*I*a^2*c*d^3*f^3*x^3*e^(2*e) +
6*I*a^2*c^2*d^2*f^3*x^2*e^(2*e) + 4*I*a^2*c^3*d*f^3*x*e^(2*e) + I*a^2*c^4*f^3*e^(2*e))*e^(2*f*x) - 9*(a^2*d^4
*f^3*x^4*e^e + 4*a^2*c*d^3*f^3*x^3*e^e + 6*a^2*c^2*d^2*f^3*x^2*e^e + 4*a^2*c^3*d*f^3*x*e^e + a^2*c^4*f^3*e^e)*
e^(f*x)) - integrate(4/3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 12*d^3)/(a^2*d^5*f^3*x^5 + 5*a^2*c*d^4*f^3
*x^4 + 10*a^2*c^2*d^3*f^3*x^3 + 10*a^2*c^3*d^2*f^3*x^2 + 5*a^2*c^4*d*f^3*x + a^2*c^5*f^3 - (-I*a^2*d^5*f^3*x^5
*e^e - 5*I*a^2*c*d^4*f^3*x^4*e^e - 10*I*a^2*c^2*d^3*f^3*x^3*e^e - 10*I*a^2*c^3*d^2*f^3*x^2*e^e - 5*I*a^2*c^4*d
*f^3*x*e^e - I*a^2*c^5*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)^2/(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 + 12*I*d^2 - 4*(-I*d^2*f*x - I*c*d*f + 3*I*d^2)*e^(2*f*x + 2*e
) + 2*(3*d^2*f^2*x^2 + 3*c^2*f^2 + 2*c*d*f - 12*d^2 + 2*(3*c*d*f^2 + d^2*f)*x)*e^(f*x + e) - 3*(-I*a^2*d^4*f^3
*x^4 - 4*I*a^2*c*d^3*f^3*x^3 - 6*I*a^2*c^2*d^2*f^3*x^2 - 4*I*a^2*c^3*d*f^3*x - I*a^2*c^4*f^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^(3*f*x + 3*e) + 3*(I*a^2*
d^4*f^3*x^4 + 4*I*a^2*c*d^3*f^3*x^3 + 6*I*a^2*c^2*d^2*f^3*x^2 + 4*I*a^2*c^3*d*f^3*x + I*a^2*c^4*f^3)*e^(2*f*x
+ 2*e) + 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))*integral(-4*(-I*d^3*f^2*x^2 - 2*I*c*d^2*f^2*x - I*c^2*d*f^2 + 12*I*d^3)/(-3*I*a^2*d^5*f^3*x^5 - 15
*I*a^2*c*d^4*f^3*x^4 - 30*I*a^2*c^2*d^3*f^3*x^3 - 30*I*a^2*c^3*d^2*f^3*x^2 - 15*I*a^2*c^4*d*f^3*x - 3*I*a^2*c^
5*f^3 + 3*(a^2*d^5*f^3*x^5 + 5*a^2*c*d^4*f^3*x^4 + 10*a^2*c^2*d^3*f^3*x^3 + 10*a^2*c^3*d^2*f^3*x^2 + 5*a^2*c^4
*d*f^3*x + a^2*c^5*f^3)*e^(f*x + e)), x))/(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^(3*f*x + 3*e) - 9*(I*a^2*d^4*f^3*x^4 + 4*I*a^2*c*d^3*f^3*x^3 + 6*I*a^2
*c^2*d^2*f^3*x^2 + 4*I*a^2*c^3*d*f^3*x + I*a^2*c^4*f^3)*e^(2*f*x + 2*e) - 9*(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))
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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)**2/(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)^2/(a+I*a*sinh(f*x+e))^2,x, algorithm="giac")
[Out]
integrate(1/((d*x + c)^2*(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 )}^2} \,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)^2),x)
[Out]
int(1/((a + a*sinh(e + f*x)*1i)^2*(c + d*x)^2), x)
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