3.10.85 \(\int \frac {x}{(c+a^2 c x^2) \text {ArcTan}(a x)^{3/2}} \, dx\) [985]

Optimal. Leaf size=37 \[ -\frac {2 x}{a c \sqrt {\text {ArcTan}(a x)}}+\frac {2 \text {Int}\left (\frac {1}{\sqrt {\text {ArcTan}(a x)}},x\right )}{a c} \]

[Out]

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

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Rubi [A]
time = 0.03, 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 {x}{\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.78, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(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.

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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(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

sage0*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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