3.11.39 \(\int \frac {1}{x^2 (c+a^2 c x^2)^{5/2} \text {ArcTan}(a x)^{3/2}} \, dx\) [1039]

Optimal. Leaf size=100 \[ -\frac {2}{a c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)}}-\frac {4 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {ArcTan}(a x)}},x\right )}{a}-10 a \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

-2/a/c/x^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2)-4*Unintegrable(1/x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)
/a-10*a*Unintegrable(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)

________________________________________________________________________________________

Rubi [A]
time = 0.23, 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}{x^2 \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]

[Out]

-2/(a*c*x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (4*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[
a*x]]), x])/a - 10*a*Defer[Int][1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {4 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(10 a) \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 6.87, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]

[Out]

Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]

________________________________________________________________________________________

Maple [A]
time = 0.86, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(3/2),x)

[Out]

int(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(3/2),x)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a**2*c*x**2+c)**(5/2)/atan(a*x)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3064 deep

________________________________________________________________________________________

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/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*atan(a*x)^(3/2)*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(1/(x^2*atan(a*x)^(3/2)*(c + a^2*c*x^2)^(5/2)), x)

________________________________________________________________________________________