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

Optimal. Leaf size=114 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tan ^{-1}(a x)^2},x\right )}{2 a c^2}+\frac {a x}{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {1}{2 a c^2 x \tan ^{-1}(a x)^2} \]

[Out]

-1/2/a/c^2/x/arctan(a*x)^2+1/2*a*x/c^2/(a^2*x^2+1)/arctan(a*x)^2+1/2*(-a^2*x^2+1)/c^2/(a^2*x^2+1)/arctan(a*x)+
Si(2*arctan(a*x))/c^2-1/2*Unintegrable(1/x^2/arctan(a*x)^2,x)/a/c^2

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Rubi [A]  time = 0.26, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.94, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*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/(a^2*c*x^2+c)^2/arctan(a*x)^3,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x/(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 \[ \frac {2 \, {\left (a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )} \mathit {sage}_{0} x \arctan \left (a x\right )^{2} - a x + {\left (3 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{2 \, {\left (a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )} \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x*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 \[ \frac {\int \frac {1}{a^{4} x^{5} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )} + x \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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