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

Optimal. Leaf size=111 \[ -\frac {2 \text {Int}\left (\frac {1}{x^3 \tan ^{-1}(a x)},x\right )}{a c^3}+\frac {a}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {a}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}+\frac {2 a \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac {a \text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.43, 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 = {} \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(a*c^3*x^2*ArcTan[a*x])) + a/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + a/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (2*a*
SinIntegral[2*ArcTan[a*x]])/c^3 + (a*SinIntegral[4*ArcTan[a*x]])/(2*c^3) - (2*Defer[Int][1/(x^3*ArcTan[a*x]),
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)^2} \, dx &=-\left (a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx\right )+\frac {\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\left (4 a^3\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx+\frac {\int \frac {1}{x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac {a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac {\left (2 a^3\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac {(4 a) \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^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {a \operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 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 \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {a \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac {a \text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {a \operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {2 a \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac {a \text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}\\ \end {align*}

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Mathematica [A]  time = 2.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

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)^2,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)^2), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)^2,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{2}}\, dx \]

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)^2,x)

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{6} x^{8} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + x^{2} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]

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)**2,x)

[Out]

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

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