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

Optimal. Leaf size=112 \[ -\frac {4 \text {Int}\left (\frac {1}{x^5 \tan ^{-1}(a x)},x\right )}{a c^2}+\frac {2 a \text {Int}\left (\frac {1}{x^3 \tan ^{-1}(a x)},x\right )}{c^2}-\frac {a^3 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {a^3}{c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac {1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac {a}{c^2 x^2 \tan ^{-1}(a x)} \]

[Out]

-1/a/c^2/x^4/arctan(a*x)+a/c^2/x^2/arctan(a*x)-a^3/c^2/(a^2*x^2+1)/arctan(a*x)-a^3*Si(2*arctan(a*x))/c^2-4*Uni
ntegrable(1/x^5/arctan(a*x),x)/a/c^2+2*a*Unintegrable(1/x^3/arctan(a*x),x)/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.40, 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 = {} \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 3.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{8} + 2 \, a^{2} c^{2} x^{6} + c^{2} x^{4}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 2.49, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (a^{3} c^{2} x^{6} + a c^{2} x^{4}\right )} \arctan \left (a x\right ) \int \frac {3 \, a^{2} x^{2} + 2}{{\left (a^{5} c^{2} x^{9} + 2 \, a^{3} c^{2} x^{7} + a c^{2} x^{5}\right )} \arctan \left (a x\right )}\,{d x} + 1}{{\left (a^{3} c^{2} x^{6} + a c^{2} x^{4}\right )} \arctan \left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} x^{8} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + x^{4} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________