3.987 \(\int \frac {1}{x (c+a^2 c x^2) \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=43 \[ -\frac {2 \text {Int}\left (\frac {1}{x^2 \sqrt {\tan ^{-1}(a x)}},x\right )}{a c}-\frac {2}{a c x \sqrt {\tan ^{-1}(a x)}} \]

[Out]

-2/a/c/x/arctan(a*x)^(1/2)-2*Unintegrable(1/x^2/arctan(a*x)^(1/2),x)/a/c

________________________________________________________________________________________

Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-2/(a*c*x*Sqrt[ArcTan[a*x]]) - (2*Defer[Int][1/(x^2*Sqrt[ArcTan[a*x]]), x])/(a*c)

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________