3.11.4 \(\int \frac {1}{x^2 (c+a^2 c x^2)^3 \text {ArcTan}(a x)^{3/2}} \, dx\) [1004]

Optimal. Leaf size=93 \[ -\frac {2}{a c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)}}-\frac {4 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right )}{a}-12 a \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

-2/a/c^3/x^2/(a^2*x^2+1)^2/arctan(a*x)^(1/2)-4*Unintegrable(1/x^3/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)/a-12*a*
Unintegrable(1/x/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

________________________________________________________________________________________

Rubi [A]
time = 0.14, 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 )^3 \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)^3*ArcTan[a*x]^(3/2)),x]

[Out]

-2/(a*c^3*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (4*Defer[Int][1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),
x])/a - 12*a*Defer[Int][1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {4 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(12 a) \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 4.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \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)^3*ArcTan[a*x]^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.04, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^3/arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{6} x^{8} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**6*x**8*atan(a*x)**(3/2) + 3*a**4*x**6*atan(a*x)**(3/2) + 3*a**2*x**4*atan(a*x)**(3/2) + x**2*at
an(a*x)**(3/2)), x)/c**3

________________________________________________________________________________________

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)^3/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 )}^3} \,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)^3),x)

[Out]

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

________________________________________________________________________________________