3.1031 \(\int \frac {1}{x^3 (c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=100 \[ -8 a \text {Int}\left (\frac {1}{x^2 \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}},x\right )-\frac {6 \text {Int}\left (\frac {1}{x^4 \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}},x\right )}{a}-\frac {2}{a c x^3 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}} \]

[Out]

-2/a/c/x^3/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2)-6*Unintegrable(1/x^4/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)
/a-8*a*Unintegrable(1/x^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)

________________________________________________________________________________________

Rubi [A]  time = 0.37, 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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-2/(a*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (6*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*
x]]), x])/a - 8*a*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c x^3 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}-\frac {6 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(8 a) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 17.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 3.95, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^3/(a^2*c*x^2+c)^(3/2)/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^3/(a^2*c*x^2+c)^(3/2)/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 = -0.01 \[ \int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [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**3/(a**2*c*x**2+c)**(3/2)/atan(a*x)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________