3.149 \(\int \frac {1}{x (a+i a \sinh (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {1}{x (a+i a \sinh (c+d x))^{5/2}},x\right ) \]

[Out]

Unintegrable(1/x/(a+I*a*sinh(d*x+c))^(5/2),x)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {1}{x (a+i a \sinh (c+d x))^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + I*a*Sinh[c + d*x])^(5/2)),x]

[Out]

Defer[Int][1/(x*(a + I*a*Sinh[c + d*x])^(5/2)), x]

Rubi steps

\begin {align*} \int \frac {1}{x (a+i a \sinh (c+d x))^{5/2}} \, dx &=\int \frac {1}{x (a+i a \sinh (c+d x))^{5/2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 34.80, size = 0, normalized size = 0.00 \[ \int \frac {1}{x (a+i a \sinh (c+d x))^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(a + I*a*Sinh[c + d*x])^(5/2)),x]

[Out]

Integrate[1/(x*(a + I*a*Sinh[c + d*x])^(5/2)), x]

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fricas [A]  time = 0.91, size = 0, normalized size = 0.00 \[ \frac {{\left (24 \, a^{3} d^{4} x^{4} e^{\left (4 \, d x + 4 \, c\right )} - 96 i \, a^{3} d^{4} x^{4} e^{\left (3 \, d x + 3 \, c\right )} - 144 \, a^{3} d^{4} x^{4} e^{\left (2 \, d x + 2 \, c\right )} + 96 i \, a^{3} d^{4} x^{4} e^{\left (d x + c\right )} + 24 \, a^{3} d^{4} x^{4}\right )} {\rm integral}\left (\frac {{\left (-9 i \, d^{4} x^{4} + 80 i \, d^{2} x^{2} - 384 i\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} e^{\left (d x + c\right )}}{48 \, a^{3} d^{4} x^{5} e^{\left (d x + c\right )} - 48 i \, a^{3} d^{4} x^{5}}, x\right ) + {\left ({\left (-9 i \, d^{3} x^{3} + 18 i \, d^{2} x^{2} + 8 i \, d x - 48 i\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (33 \, d^{3} x^{3} - 70 \, d^{2} x^{2} - 8 \, d x + 144\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-33 i \, d^{3} x^{3} - 70 i \, d^{2} x^{2} + 8 i \, d x + 144 i\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (9 \, d^{3} x^{3} + 18 \, d^{2} x^{2} - 8 \, d x - 48\right )} e^{\left (d x + c\right )}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}}{24 \, a^{3} d^{4} x^{4} e^{\left (4 \, d x + 4 \, c\right )} - 96 i \, a^{3} d^{4} x^{4} e^{\left (3 \, d x + 3 \, c\right )} - 144 \, a^{3} d^{4} x^{4} e^{\left (2 \, d x + 2 \, c\right )} + 96 i \, a^{3} d^{4} x^{4} e^{\left (d x + c\right )} + 24 \, a^{3} d^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

((24*a^3*d^4*x^4*e^(4*d*x + 4*c) - 96*I*a^3*d^4*x^4*e^(3*d*x + 3*c) - 144*a^3*d^4*x^4*e^(2*d*x + 2*c) + 96*I*a
^3*d^4*x^4*e^(d*x + c) + 24*a^3*d^4*x^4)*integral((-9*I*d^4*x^4 + 80*I*d^2*x^2 - 384*I)*sqrt(1/2*I*a*e^(-d*x -
c))*e^(d*x + c)/(48*a^3*d^4*x^5*e^(d*x + c) - 48*I*a^3*d^4*x^5), x) + ((-9*I*d^3*x^3 + 18*I*d^2*x^2 + 8*I*d*x
- 48*I)*e^(4*d*x + 4*c) - (33*d^3*x^3 - 70*d^2*x^2 - 8*d*x + 144)*e^(3*d*x + 3*c) + (-33*I*d^3*x^3 - 70*I*d^2
*x^2 + 8*I*d*x + 144*I)*e^(2*d*x + 2*c) - (9*d^3*x^3 + 18*d^2*x^2 - 8*d*x - 48)*e^(d*x + c))*sqrt(1/2*I*a*e^(-
d*x - c)))/(24*a^3*d^4*x^4*e^(4*d*x + 4*c) - 96*I*a^3*d^4*x^4*e^(3*d*x + 3*c) - 144*a^3*d^4*x^4*e^(2*d*x + 2*c
) + 96*I*a^3*d^4*x^4*e^(d*x + c) + 24*a^3*d^4*x^4)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((I*a*sinh(d*x + c) + a)^(5/2)*x), x)

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maple [A]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a+I*a*sinh(d*x+c))^(5/2),x)

[Out]

int(1/x/(a+I*a*sinh(d*x+c))^(5/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((I*a*sinh(d*x + c) + a)^(5/2)*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x\,{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + a*sinh(c + d*x)*1i)^(5/2)),x)

[Out]

int(1/(x*(a + a*sinh(c + d*x)*1i)^(5/2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (i a \left (\sinh {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(d*x+c))**(5/2),x)

[Out]

Integral(1/(x*(I*a*(sinh(c + d*x) - I))**(5/2)), x)

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