3.1063 \(\int \frac{1}{x (c+a^2 c x^2)^2 \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=178 \[ \frac{8 \text{Unintegrable}\left (\frac{1}{x^3 \left (a^2 c x^2+c\right )^2 \sqrt{\tan ^{-1}(a x)}},x\right )}{3 a^2}+\frac{16}{3} \text{Unintegrable}\left (\frac{1}{x \left (a^2 c x^2+c\right )^2 \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{4}{c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}+\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{c^2} \]

[Out]

-2/(3*a*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 4/(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^2*x^2*(1
+ a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 + (8*Unintegrable[1
/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (16*Unintegrable[1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[
a*x]]), x])/3

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Rubi [A]  time = 0.36048, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]

[Out]

-2/(3*a*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 4/(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^2*x^2*(1
+ a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 + (8*Defer[Int][1/(
x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (16*Defer[Int][1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]
]), x])/3

Rubi steps

\begin{align*} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac{2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-(2 a) \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{4}{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{16}{3} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{3 a^2}+\left (8 a^2\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{4}{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{16}{3} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac{8 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{4}{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{16}{3} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac{8 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{4}{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{16}{3} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac{4 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{4}{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{16}{3} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac{8 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{c^2}\\ &=-\frac{2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{4}{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{c^2}+\frac{16}{3} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 4.25049, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]

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Maple [A]  time = 0.535, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{2}} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(5/2),x)

[Out]

int(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(5/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a**2*c*x**2+c)**2/atan(a*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((a^2*c*x^2 + c)^2*x*arctan(a*x)^(5/2)), x)