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

Optimal. Leaf size=113 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tan ^{-1}(a x)},x\right )}{a c^3}+\frac {a x}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {a x}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}-\frac {3 \text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {1}{a c^3 x \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.66, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.78, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/(x*(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 \[ {\rm integral}\left (\frac {1}{{\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)*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/(a^2*c*x^2+c)^3/arctan(a*x)^2,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.65, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x/(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 \[ -\frac {{\left (a^{5} c^{3} x^{5} + 2 \, a^{3} c^{3} x^{3} + a c^{3} x\right )} \mathit {sage}_{0} x \arctan \left (a x\right ) + 1}{{\left (a^{5} c^{3} x^{5} + 2 \, a^{3} c^{3} x^{3} + a c^{3} x\right )} \arctan \left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x*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 \[ \frac {\int \frac {1}{a^{6} x^{7} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atan}^{2}{\left (a x \right )} + x \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**6*x**7*atan(a*x)**2 + 3*a**4*x**5*atan(a*x)**2 + 3*a**2*x**3*atan(a*x)**2 + x*atan(a*x)**2), x)
/c**3

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