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

Optimal. Leaf size=236 \[ \frac{2 a \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right )}{c^2}+\frac{\text{Unintegrable}\left (\frac{1}{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{c^2}+\frac{9 a^2 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 a^3 x}{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{2 a \sqrt{a^2 c x^2+c}}{c^3 x \tan ^{-1}(a x)}-\frac{a^3 x}{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

-((a^3*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (2*a*Sqrt
[c + a^2*c*x^2])/(c^3*x*ArcTan[a*x]) + (9*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*
c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x
^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 + (2*a*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])
/c^2

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Rubi [A]  time = 2.04828, 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 = {} \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((a^3*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (2*a*Sqrt
[c + a^2*c*x^2])/(c^3*x*ArcTan[a*x]) + (9*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*
c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^3
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 + (2*a*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])/c^2

Rubi steps

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

Mathematica [A]  time = 6.79928, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.762, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c}}{{\left (a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )} \arctan \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)/((a^6*c^3*x^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^3)*arctan(a*x)^2), x)

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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**3/(a**2*c*x**2+c)**(5/2)/atan(a*x)**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{5}{2}} x^{3} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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