3.1073 \(\int \frac{1}{x^2 (c+a^2 c x^2)^3 \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=229 \[ \frac{80}{3} \text{Unintegrable}\left (\frac{1}{x^2 \left (a^2 c x^2+c\right )^3 \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{8 \text{Unintegrable}\left (\frac{1}{x^4 \left (a^2 c x^2+c\right )^3 \sqrt{\tan ^{-1}(a x)}},x\right )}{a^2}+\frac{8}{c^3 x \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^3 x^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{5 \sqrt{\frac{\pi }{2}} a \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{c^3}+\frac{20 \sqrt{\pi } a \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{c^3}+\frac{30 a \sqrt{\tan ^{-1}(a x)}}{c^3} \]

[Out]

-2/(3*a*c^3*x^2*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 8/(
c^3*x*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (30*a*Sqrt[ArcTan[a*x]])/c^3 + (5*a*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi
]*Sqrt[ArcTan[a*x]]])/c^3 + (20*a*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^3 + (8*Unintegrable[1/(
x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a^2 + (80*Unintegrable[1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]
]), x])/3

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Rubi [A]  time = 0.508072, 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 = {} \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-2/(3*a*c^3*x^2*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 8/(
c^3*x*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (30*a*Sqrt[ArcTan[a*x]])/c^3 + (5*a*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi
]*Sqrt[ArcTan[a*x]]])/c^3 + (20*a*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^3 + (8*Defer[Int][1/(x^
4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a^2 + (80*Defer[Int][1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),
x])/3

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c^3 x^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-(4 a) \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2}{3 a c^3 x^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{8}{c^3 x \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+8 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{56}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\left (40 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2}{3 a c^3 x^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{8}{c^3 x \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+8 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{56}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\frac{(40 a) \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{2}{3 a c^3 x^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{8}{c^3 x \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+8 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{56}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\frac{(40 a) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{2}{3 a c^3 x^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{8}{c^3 x \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{30 a \sqrt{\tan ^{-1}(a x)}}{c^3}+8 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{56}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\frac{(5 a) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{(20 a) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{2}{3 a c^3 x^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{8}{c^3 x \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{30 a \sqrt{\tan ^{-1}(a x)}}{c^3}+8 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{56}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\frac{(10 a) \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{c^3}+\frac{(40 a) \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{c^3}\\ &=-\frac{2}{3 a c^3 x^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{8}{c^3 x \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{30 a \sqrt{\tan ^{-1}(a x)}}{c^3}+\frac{5 a \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{c^3}+\frac{20 a \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{c^3}+8 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{56}{3} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}\\ \end{align*}

Mathematica [A]  time = 7.28295, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.584, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) ^{3}} \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/arctan(a*x)^(5/2),x)

[Out]

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

[Out]

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