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

Optimal. Leaf size=72 \[ \frac {\text {Int}\left (\frac {1}{\tan ^{-1}(a x)},x\right )}{a^3 c^2}-\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{a^4 c^2}-\frac {x}{a^3 c^2 \tan ^{-1}(a x)}+\frac {x}{a^3 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)} \]

[Out]

-x/a^3/c^2/arctan(a*x)+x/a^3/c^2/(a^2*x^2+1)/arctan(a*x)-Ci(2*arctan(a*x))/a^4/c^2+Unintegrable(1/arctan(a*x),
x)/a^3/c^2

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Rubi [A]  time = 0.33, 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 = {} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 7.09, size = 0, normalized size = 0.00 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.91, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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