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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{1}{x (a+i a \sinh (c+d x))^{5/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0900751, 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 = {} \[ \int \frac{1}{x (a+i a \sinh (c+d x))^{5/2}} \, dx \]

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*}

Mathematica [A]  time = 35.33, size = 0, normalized size = 0. \[ \int \frac{1}{x (a+i a \sinh (c+d x))^{5/2}} \, dx \]

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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Maple [A]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+ia\sinh \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}} x}\,{d x} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{\frac{1}{2}}{\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{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} +{\left (24 \, a^{3} d^{4} x^{4} e^{\left (5 \, d x + 5 \, c\right )} - 120 i \, a^{3} d^{4} x^{4} e^{\left (4 \, d x + 4 \, c\right )} - 240 \, a^{3} d^{4} x^{4} e^{\left (3 \, d x + 3 \, c\right )} + 240 i \, a^{3} d^{4} x^{4} e^{\left (2 \, d x + 2 \, c\right )} + 120 \, a^{3} d^{4} x^{4} e^{\left (d x + c\right )} - 24 i \, a^{3} d^{4} x^{4}\right )}{\rm integral}\left (\frac{\sqrt{\frac{1}{2}}{\left (-9 i \, d^{4} x^{4} + 80 i \, d^{2} x^{2} - 384 i\right )} \sqrt{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{48 \, a^{3} d^{4} x^{5} e^{\left (2 \, d x + 2 \, c\right )} - 96 i \, a^{3} d^{4} x^{5} e^{\left (d x + c\right )} - 48 \, a^{3} d^{4} x^{5}}, x\right )}{24 \, a^{3} d^{4} x^{4} e^{\left (5 \, d x + 5 \, c\right )} - 120 i \, a^{3} d^{4} x^{4} e^{\left (4 \, d x + 4 \, c\right )} - 240 \, a^{3} d^{4} x^{4} e^{\left (3 \, d x + 3 \, c\right )} + 240 i \, a^{3} d^{4} x^{4} e^{\left (2 \, d x + 2 \, c\right )} + 120 \, a^{3} d^{4} x^{4} e^{\left (d x + c\right )} - 24 i \, a^{3} d^{4} x^{4}} \end{align*}

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]

(sqrt(1/2)*((-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(I*a*e^(2*d*x + 2*c) + 2*a*e^(d*x + c) - I*a)*e^(-1/2*d*x - 1/2*c) + (24*
a^3*d^4*x^4*e^(5*d*x + 5*c) - 120*I*a^3*d^4*x^4*e^(4*d*x + 4*c) - 240*a^3*d^4*x^4*e^(3*d*x + 3*c) + 240*I*a^3*
d^4*x^4*e^(2*d*x + 2*c) + 120*a^3*d^4*x^4*e^(d*x + c) - 24*I*a^3*d^4*x^4)*integral(sqrt(1/2)*(-9*I*d^4*x^4 + 8
0*I*d^2*x^2 - 384*I)*sqrt(I*a*e^(2*d*x + 2*c) + 2*a*e^(d*x + c) - I*a)*e^(1/2*d*x + 1/2*c)/(48*a^3*d^4*x^5*e^(
2*d*x + 2*c) - 96*I*a^3*d^4*x^5*e^(d*x + c) - 48*a^3*d^4*x^5), x))/(24*a^3*d^4*x^4*e^(5*d*x + 5*c) - 120*I*a^3
*d^4*x^4*e^(4*d*x + 4*c) - 240*a^3*d^4*x^4*e^(3*d*x + 3*c) + 240*I*a^3*d^4*x^4*e^(2*d*x + 2*c) + 120*a^3*d^4*x
^4*e^(d*x + c) - 24*I*a^3*d^4*x^4)

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}} x}\,{d x} \end{align*}

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)