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

Optimal. Leaf size=167 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right )}{a c^3}-\frac{a^2 x}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{2 a^2 x}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}+\frac{a}{2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}+\frac{2 a \text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac{a \text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.732928, 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 = {} \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx\right )+\frac{\int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac{a}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\left (2 a^3\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx+\frac{\int \frac{1}{x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c^2}-\frac{a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{2 a^2 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\left (2 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\left (6 a^4\right ) \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac{\int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{a c^3}+\frac{a^3 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{2 a^2 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{a^2 x}{c^3 \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^3}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{(6 a) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}-\frac{a^4 \int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{2 a^2 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{a^2 x}{c^3 \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^3}+\frac{a \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{a \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{(2 a) \operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cos (2 x)}{2 x}+\frac{\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{(6 a) \operatorname{Subst}\left (\int \left (\frac{1}{8 x}-\frac{\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{2 a^2 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{a^2 x}{c^3 \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^3}+\frac{a \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}+\frac{a \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\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^3}+\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^3}\\ &=-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{2 a^2 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{a^2 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{a \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac{a \text{Ci}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{\int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{a c^3}+2 \frac{a \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}\\ &=-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{2 a^2 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{a^2 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 a \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac{a \text{Ci}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{\int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{a c^3}\\ \end{align*}

Mathematica [A]  time = 3.64928, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.391, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) ^{3} \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-a x + 2 \,{\left (3 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) + \frac{2 \,{\left (a^{6} c^{3} x^{7} + 2 \, a^{4} c^{3} x^{5} + a^{2} c^{3} x^{3}\right )}{\left (15 \, a^{4} \int \frac{x^{4}}{a^{6} x^{10} \arctan \left (a x\right ) + 3 \, a^{4} x^{8} \arctan \left (a x\right ) + 3 \, a^{2} x^{6} \arctan \left (a x\right ) + x^{4} \arctan \left (a x\right )}\,{d x} + 10 \, a^{2} \int \frac{x^{2}}{a^{6} x^{10} \arctan \left (a x\right ) + 3 \, a^{4} x^{8} \arctan \left (a x\right ) + 3 \, a^{2} x^{6} \arctan \left (a x\right ) + x^{4} \arctan \left (a x\right )}\,{d x} + 3 \, \int \frac{1}{a^{6} x^{10} \arctan \left (a x\right ) + 3 \, a^{4} x^{8} \arctan \left (a x\right ) + 3 \, a^{2} x^{6} \arctan \left (a x\right ) + x^{4} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2}}{a^{2} c^{3}}}{2 \,{\left (a^{6} c^{3} x^{7} + 2 \, a^{4} c^{3} x^{5} + a^{2} c^{3} x^{3}\right )} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(a^6*c^3*x^7 + 2*a^4*c^3*x^5 + a^2*c^3*x^3)*arctan(a*x)^2*integrate((15*a^4*x^4 + 10*a^2*x^2 + 3)/((a^8
*c^3*x^10 + 3*a^6*c^3*x^8 + 3*a^4*c^3*x^6 + a^2*c^3*x^4)*arctan(a*x)), x) - a*x + 2*(3*a^2*x^2 + 1)*arctan(a*x
))/((a^6*c^3*x^7 + 2*a^4*c^3*x^5 + a^2*c^3*x^3)*arctan(a*x)^2)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^3), x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{6} x^{8} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname{atan}^{3}{\left (a x \right )} + x^{2} \operatorname{atan}^{3}{\left (a x \right )}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**6*x**8*atan(a*x)**3 + 3*a**4*x**6*atan(a*x)**3 + 3*a**2*x**4*atan(a*x)**3 + x**2*atan(a*x)**3),
x)/c**3

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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