3.1099 \(\int \frac{1}{x^2 (c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=225 \[ \frac{8 \text{Unintegrable}\left (\frac{1}{x^4 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )}{a^2}+\frac{44}{3} \text{Unintegrable}\left (\frac{1}{x^2 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{8 \sqrt{2 \pi } a \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{4}{c x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}+\frac{8}{3 a^2 c x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}} \]

[Out]

-2/(3*a*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) +
4/(c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*a*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[Ar
cTan[a*x]]])/(c*Sqrt[c + a^2*c*x^2]) + (8*Unintegrable[1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a^
2 + (44*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3

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Rubi [A]  time = 0.833288, 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 = {} \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

-2/(3*a*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) +
4/(c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*a*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[Ar
cTan[a*x]]])/(c*Sqrt[c + a^2*c*x^2]) + (8*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a^2
+ (44*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{4 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-(2 a) \int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2}{3 a c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c x^3 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{c x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\left (8 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2}{3 a c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c x^3 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{c x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\frac{\left (8 a^2 \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{3 a c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c x^3 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{c x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\frac{\left (8 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{3 a c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c x^3 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{c x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\frac{\left (16 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{3 a c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c x^3 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{c x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 a \sqrt{2 \pi } \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}+4 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}\\ \end{align*}

Mathematica [A]  time = 13.0346, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]

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Maple [A]  time = 0.803, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{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^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)

[Out]

int(1/x^2/(a^2*c*x^2+c)^(3/2)/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^2/(a^2*c*x^2+c)^(3/2)/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^2/(a^2*c*x^2+c)^(3/2)/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**2/(a**2*c*x**2+c)**(3/2)/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 )}^{\frac{3}{2}} x^{2} \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^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((a^2*c*x^2 + c)^(3/2)*x^2*arctan(a*x)^(5/2)), x)