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

Optimal. Leaf size=104 \[ -\frac {\text {Int}\left (\frac {1}{x^3 \tan ^{-1}(a x)^2},x\right )}{a c^2}-\frac {a^2 x}{c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {a}{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac {a \text {Ci}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {1}{2 a c^2 x^2 \tan ^{-1}(a x)^2} \]

[Out]

-1/2/a/c^2/x^2/arctan(a*x)^2+1/2*a/c^2/(a^2*x^2+1)/arctan(a*x)^2-a^2*x/c^2/(a^2*x^2+1)/arctan(a*x)+a*Ci(2*arct
an(a*x))/c^2-Unintegrable(1/x^3/arctan(a*x)^2,x)/a/c^2

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Rubi [A]  time = 0.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.82, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*arctan(a*x)^3), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (a^{4} c^{2} x^{5} + a^{2} c^{2} x^{3}\right )} \mathit {sage}_{0} x \arctan \left (a x\right )^{2} - a x + 2 \, {\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{2 \, {\left (a^{4} c^{2} x^{5} + a^{2} c^{2} x^{3}\right )} \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(a^4*c^2*x^5 + a^2*c^2*x^3)*arctan(a*x)^2*integrate((6*a^4*x^4 + 7*a^2*x^2 + 3)/((a^6*c^2*x^8 + 2*a^4*c
^2*x^6 + a^2*c^2*x^4)*arctan(a*x)), x) - a*x + 2*(2*a^2*x^2 + 1)*arctan(a*x))/((a^4*c^2*x^5 + a^2*c^2*x^3)*arc
tan(a*x)^2)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + x^{2} \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**4*x**6*atan(a*x)**3 + 2*a**2*x**4*atan(a*x)**3 + x**2*atan(a*x)**3), x)/c**2

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