3.10.94 \(\int \frac {1}{x (c+a^2 c x^2)^2 \text {ArcTan}(a x)^{3/2}} \, dx\) [994]

Optimal. Leaf size=102 \[ -\frac {2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\text {ArcTan}(a x)}}-\frac {6 \sqrt {\text {ArcTan}(a x)}}{c^2}-\frac {3 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{c^2}-\frac {2 \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\text {ArcTan}(a x)}},x\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.15, 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 \left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 3.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(1/x/(a^2*c*x^2+c)^2/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 {1}{a^{4} x^{5} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 2 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^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**4*x**5*atan(a*x)**(3/2) + 2*a**2*x**3*atan(a*x)**(3/2) + x*atan(a*x)**(3/2)), x)/c**2

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \end {gather*}

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)^2),x)

[Out]

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

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