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

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

[Out]

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

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Rubi [A]  time = 0.662014, 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 = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

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*}

Mathematica [A]  time = 1.601, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]

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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Maple [A]  time = 0.396, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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/(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., size = 0, normalized size = 0. \begin{align*} -\frac{\frac{{\left (a^{5} c^{3} x^{5} + 2 \, a^{3} c^{3} x^{3} + a c^{3} x\right )}{\left (5 \, a^{2} \int \frac{x^{2}}{a^{6} x^{8} \arctan \left (a x\right ) + 3 \, a^{4} x^{6} \arctan \left (a x\right ) + 3 \, a^{2} x^{4} \arctan \left (a x\right ) + x^{2} \arctan \left (a x\right )}\,{d x} + \int \frac{1}{{\left (a^{2} x^{2} + 1\right )}^{3} x^{2} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )}{a c^{3}} + 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 )} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

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

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]

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