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

Optimal. Leaf size=246 \[ \frac{a \text{Unintegrable}\left (\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right )}{c^3}-\frac{3 \text{Unintegrable}\left (\frac{1}{x^4 \tan ^{-1}(a x)^2},x\right )}{2 a c^3}-\frac{5 a^2 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{a^2 \text{Si}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{a^3 x}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}-\frac{a^2 \left (1-a^2 x^2\right )}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{3 a^2}{2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{2 a^2}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}+\frac{a}{c^3 x \tan ^{-1}(a x)^2} \]

[Out]

-1/(2*a*c^3*x^3*ArcTan[a*x]^2) + a/(c^3*x*ArcTan[a*x]^2) - (a^3*x)/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - (a^
3*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*a^2)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (3*a^2)/(2*c^3*(1 + a^2*x
^2)*ArcTan[a*x]) - (a^2*(1 - a^2*x^2))/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (5*a^2*SinIntegral[2*ArcTan[a*x]])/(2
*c^3) - (a^2*SinIntegral[4*ArcTan[a*x]])/c^3 - (3*Unintegrable[1/(x^4*ArcTan[a*x]^2), x])/(2*a*c^3) + (a*Unint
egrable[1/(x^2*ArcTan[a*x]^2), x])/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.28108, 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 = {} \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-1/(2*a*c^3*x^3*ArcTan[a*x]^2) + a/(c^3*x*ArcTan[a*x]^2) - (a^3*x)/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - (a^
3*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*a^2)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (3*a^2)/(2*c^3*(1 + a^2*x
^2)*ArcTan[a*x]) - (a^2*(1 - a^2*x^2))/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (5*a^2*SinIntegral[2*ArcTan[a*x]])/(2
*c^3) - (a^2*SinIntegral[4*ArcTan[a*x]])/c^3 - (3*Defer[Int][1/(x^4*ArcTan[a*x]^2), x])/(2*a*c^3) + (a*Defer[I
nt][1/(x^2*ArcTan[a*x]^2), x])/c^3

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx\right )+\frac{\int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=a^4 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx+\frac{\int \frac{1}{x^3 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c^2}-2 \frac{a^2 \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{1}{2} a^3 \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx-\frac{1}{2} \left (3 a^5\right ) \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (\frac{a^2 \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c^2}-\frac{a^4 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\right )\\ &=-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{a^2}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{1}{2} \left (3 a^3\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx-\left (2 a^4\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-\frac{\left (3 a^3\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{2 c}-2 \left (-\frac{a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac{\left (2 a^4\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\right )\\ &=-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\left (6 a^4\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac{a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{\left (3 a^4\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac{a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 x}+\frac{\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}-2 \left (-\frac{a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 x}+\frac{\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a^2 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{a^2 \text{Si}\left (4 \tan ^{-1}(a x)\right )}{4 c^3}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac{a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{a^2 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}\right )-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}\\ &=-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a^2 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{a^2 \text{Si}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac{a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac{a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{a^2 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}\right )\\ \end{align*}

Mathematica [A]  time = 4.81972, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.508, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \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^3/(a^2*c*x^2+c)^3/arctan(a*x)^3,x)

[Out]

int(1/x^3/(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 +{\left (7 \, a^{2} x^{2} + 3\right )} \arctan \left (a x\right ) + \frac{2 \,{\left (a^{6} c^{3} x^{8} + 2 \, a^{4} c^{3} x^{6} + a^{2} c^{3} x^{4}\right )}{\left (21 \, a^{4} \int \frac{x^{4}}{a^{6} x^{11} \arctan \left (a x\right ) + 3 \, a^{4} x^{9} \arctan \left (a x\right ) + 3 \, a^{2} x^{7} \arctan \left (a x\right ) + x^{5} \arctan \left (a x\right )}\,{d x} + 19 \, a^{2} \int \frac{x^{2}}{a^{6} x^{11} \arctan \left (a x\right ) + 3 \, a^{4} x^{9} \arctan \left (a x\right ) + 3 \, a^{2} x^{7} \arctan \left (a x\right ) + x^{5} \arctan \left (a x\right )}\,{d x} + 6 \, \int \frac{1}{a^{6} x^{11} \arctan \left (a x\right ) + 3 \, a^{4} x^{9} \arctan \left (a x\right ) + 3 \, a^{2} x^{7} \arctan \left (a x\right ) + x^{5} \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^{8} + 2 \, a^{4} c^{3} x^{6} + a^{2} c^{3} x^{4}\right )} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(a^6*c^3*x^8 + 2*a^4*c^3*x^6 + a^2*c^3*x^4)*arctan(a*x)^2*integrate((21*a^4*x^4 + 19*a^2*x^2 + 6)/((a^8
*c^3*x^11 + 3*a^6*c^3*x^9 + 3*a^4*c^3*x^7 + a^2*c^3*x^5)*arctan(a*x)), x) - a*x + (7*a^2*x^2 + 3)*arctan(a*x))
/((a^6*c^3*x^8 + 2*a^4*c^3*x^6 + a^2*c^3*x^4)*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^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\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^3/(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral(1/((a^6*c^3*x^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^3)*arctan(a*x)^3), x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{6} x^{9} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{4} x^{7} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{2} x^{5} \operatorname{atan}^{3}{\left (a x \right )} + x^{3} \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**3/(a**2*c*x**2+c)**3/atan(a*x)**3,x)

[Out]

Integral(1/(a**6*x**9*atan(a*x)**3 + 3*a**4*x**7*atan(a*x)**3 + 3*a**2*x**5*atan(a*x)**3 + x**3*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^{3} \arctan \left (a x\right )^{3}}\,{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)^3/arctan(a*x)^3,x, algorithm="giac")

[Out]

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