3.11.23 \(\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}} \, dx\) [1023]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, 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 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 7.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.92, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} c \,x^{2}+c}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^2/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),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/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),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/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),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*} \int \frac {1}{x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c}} \,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)^(1/2)),x)

[Out]

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

________________________________________________________________________________________