3.566 \(\int \frac{1}{x^4 (c+a^2 c x^2)^3 \tan ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=153 \[ \frac{4 a \text{Unintegrable}\left (\frac{1}{x^3 \tan ^{-1}(a x)},x\right )}{c^3}-\frac{4 \text{Unintegrable}\left (\frac{1}{x^5 \tan ^{-1}(a x)},x\right )}{a c^3}-\frac{3 a^3 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a^3 \text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{2 a^3}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{a^3}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}+\frac{2 a}{c^3 x^2 \tan ^{-1}(a x)}-\frac{1}{a c^3 x^4 \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.896924, 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 = {} \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.51735, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ({a}^{2}c{x}^{2}+c \right ) ^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \,{\left (a^{5} c^{3} x^{8} + 2 \, a^{3} c^{3} x^{6} + a c^{3} x^{4}\right )} \arctan \left (a x\right ) \int \frac{2 \, a^{2} x^{2} + 1}{{\left (a^{7} c^{3} x^{11} + 3 \, a^{5} c^{3} x^{9} + 3 \, a^{3} c^{3} x^{7} + a c^{3} x^{5}\right )} \arctan \left (a x\right )}\,{d x} + 1}{{\left (a^{5} c^{3} x^{8} + 2 \, a^{3} c^{3} x^{6} + a c^{3} x^{4}\right )} \arctan \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-((a^5*c^3*x^8 + 2*a^3*c^3*x^6 + a*c^3*x^4)*arctan(a*x)*integrate(4*(2*a^2*x^2 + 1)/((a^7*c^3*x^11 + 3*a^5*c^3
*x^9 + 3*a^3*c^3*x^7 + a*c^3*x^5)*arctan(a*x)), x) + 1)/((a^5*c^3*x^8 + 2*a^3*c^3*x^6 + a*c^3*x^4)*arctan(a*x)
)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{6} c^{3} x^{10} + 3 \, a^{4} c^{3} x^{8} + 3 \, a^{2} c^{3} x^{6} + c^{3} x^{4}\right )} \arctan \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral(1/((a^6*c^3*x^10 + 3*a^4*c^3*x^8 + 3*a^2*c^3*x^6 + c^3*x^4)*arctan(a*x)^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{6} x^{10} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{4} x^{8} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{2} x^{6} \operatorname{atan}^{2}{\left (a x \right )} + x^{4} \operatorname{atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**6*x**10*atan(a*x)**2 + 3*a**4*x**8*atan(a*x)**2 + 3*a**2*x**6*atan(a*x)**2 + x**4*atan(a*x)**2)
, x)/c**3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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