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

Optimal. Leaf size=109 \[ \frac{a \text{Unintegrable}\left (\frac{1}{x^2 \tan ^{-1}(a x)},x\right )}{c^2}-\frac{3 \text{Unintegrable}\left (\frac{1}{x^4 \tan ^{-1}(a x)},x\right )}{a c^2}+\frac{a^2 \text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac{a^3 x}{c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+\frac{a}{c^2 x \tan ^{-1}(a x)} \]

[Out]

-(1/(a*c^2*x^3*ArcTan[a*x])) + a/(c^2*x*ArcTan[a*x]) - (a^3*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a^2*CosInteg
ral[2*ArcTan[a*x]])/c^2 - (3*Unintegrable[1/(x^4*ArcTan[a*x]), x])/(a*c^2) + (a*Unintegrable[1/(x^2*ArcTan[a*x
]), x])/c^2

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Rubi [A]  time = 0.499298, 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 = {} \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(a*c^2*x^3*ArcTan[a*x])) + a/(c^2*x*ArcTan[a*x]) - (a^3*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a^2*CosInteg
ral[2*ArcTan[a*x]])/c^2 - (3*Defer[Int][1/(x^4*ArcTan[a*x]), x])/(a*c^2) + (a*Defer[Int][1/(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 )^2 \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx\right )+\frac{\int \frac{1}{x^3 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+a^4 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^2}-\frac{a^2 \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+\frac{a}{c^2 x \tan ^{-1}(a x)}-\frac{a^3 x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+a^3 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx-a^5 \int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^2}\\ &=-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+\frac{a}{c^2 x \tan ^{-1}(a x)}-\frac{a^3 x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+\frac{a}{c^2 x \tan ^{-1}(a x)}-\frac{a^3 x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^2}-\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2}+\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+\frac{a}{c^2 x \tan ^{-1}(a x)}-\frac{a^3 x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^2}+2 \frac{a^2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^2}\\ &=-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+\frac{a}{c^2 x \tan ^{-1}(a x)}-\frac{a^3 x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{a^2 \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^2}+\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^2}\\ \end{align*}

Mathematica [A]  time = 3.0934, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.986, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ({a}^{2}c{x}^{2}+c \right ) ^{2} \left ( \arctan \left ( ax \right ) \right ) ^{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)^2/arctan(a*x)^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\frac{{\left (a^{3} c^{2} x^{5} + a c^{2} x^{3}\right )}{\left (5 \, a^{2} \int \frac{x^{2}}{a^{4} x^{8} \arctan \left (a x\right ) + 2 \, a^{2} x^{6} \arctan \left (a x\right ) + x^{4} \arctan \left (a x\right )}\,{d x} + 3 \, \int \frac{1}{a^{4} x^{8} \arctan \left (a x\right ) + 2 \, a^{2} x^{6} \arctan \left (a x\right ) + x^{4} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )}{a c^{2}} + 1}{{\left (a^{3} c^{2} x^{5} + a c^{2} x^{3}\right )} \arctan \left (a 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)^2/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-((a^3*c^2*x^5 + a*c^2*x^3)*arctan(a*x)*integrate((5*a^2*x^2 + 3)/((a^5*c^2*x^8 + 2*a^3*c^2*x^6 + a*c^2*x^4)*a
rctan(a*x)), x) + 1)/((a^3*c^2*x^5 + a*c^2*x^3)*arctan(a*x))

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} 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)^2/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral(1/((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*arctan(a*x)^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{4} x^{7} \operatorname{atan}^{2}{\left (a x \right )} + 2 a^{2} x^{5} \operatorname{atan}^{2}{\left (a x \right )} + x^{3} \operatorname{atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**4*x**7*atan(a*x)**2 + 2*a**2*x**5*atan(a*x)**2 + x**3*atan(a*x)**2), x)/c**2

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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^{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)^2/arctan(a*x)^2,x, algorithm="giac")

[Out]

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