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

Optimal. Leaf size=206 \[ \frac{2 a \text{Unintegrable}\left (\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right )}{c^3}-\frac{2 \text{Unintegrable}\left (\frac{1}{x^5 \tan ^{-1}(a x)^2},x\right )}{a c^3}-\frac{3 a^3 \text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a^3 \text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{c^3}+\frac{2 a^4 x}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{2 a^4 x}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}-\frac{a^3}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}+\frac{a}{c^3 x^2 \tan ^{-1}(a x)^2}-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 1.40316, 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 = {} \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-1/(2*a*c^3*x^4*ArcTan[a*x]^2) + a/(c^3*x^2*ArcTan[a*x]^2) - a^3/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - a^3/(
c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + (2*a^4*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (2*a^4*x)/(c^3*(1 + a^2*x^2)*
ArcTan[a*x]) - (3*a^3*CosIntegral[2*ArcTan[a*x]])/c^3 - (a^3*CosIntegral[4*ArcTan[a*x]])/c^3 - (2*Defer[Int][1
/(x^5*ArcTan[a*x]^2), x])/(a*c^3) + (2*a*Defer[Int][1/(x^3*ArcTan[a*x]^2), 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)^3} \, dx &=-\left (a^2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx\right )+\frac{\int \frac{1}{x^4 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx+\frac{\int \frac{1}{x^4 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, 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)^3} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\left (2 a^5\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx-\frac{2 \int \frac{1}{x^5 \tan ^{-1}(a x)^2} \, 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)^3} \, dx}{c^2}-\frac{a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\right )\\ &=-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{2 a^4 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\left (2 a^4\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx+\left (6 a^6\right ) \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac{2 \int \frac{1}{x^5 \tan ^{-1}(a x)^2} \, dx}{a c^3}-2 \left (-\frac{a}{2 c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a^3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{a \int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{c^3}+\frac{a^5 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\right )\\ &=-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{2 a^4 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{2 \int \frac{1}{x^5 \tan ^{-1}(a x)^2} \, dx}{a c^3}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{\left (6 a^3\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-2 \left (-\frac{a}{2 c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a^3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{a^4 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{c^3}+\frac{a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}-\frac{a^6 \int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\right )\\ &=-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{2 a^4 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{2 \int \frac{1}{x^5 \tan ^{-1}(a x)^2} \, dx}{a c^3}-2 \left (-\frac{a}{2 c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a^3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{a^4 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{c^3}+\frac{a^3 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{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{\left (2 a^3\right ) \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{\left (6 a^3\right ) \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{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{2 a^4 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{2 \int \frac{1}{x^5 \tan ^{-1}(a x)^2} \, dx}{a c^3}-\frac{a^3 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}-\frac{a^3 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-2 \left (-\frac{a}{2 c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a^3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{a^4 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{c^3}-\frac{a^3 \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{a^3 \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}\right )\\ &=-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{2 a^4 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{a^3 \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a^3 \text{Ci}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{2 \int \frac{1}{x^5 \tan ^{-1}(a x)^2} \, dx}{a c^3}-2 \left (-\frac{a}{2 c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a^3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{a^4 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{c^3}+2 \frac{a^3 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}\right )\\ &=-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{2 a^4 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{a^3 \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a^3 \text{Ci}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{2 \int \frac{1}{x^5 \tan ^{-1}(a x)^2} \, dx}{a c^3}-2 \left (-\frac{a}{2 c^3 x^2 \tan ^{-1}(a x)^2}+\frac{a^3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}-\frac{a^4 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{a^3 \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a \int \frac{1}{x^3 \tan ^{-1}(a x)^2} \, dx}{c^3}\right )\\ \end{align*}

Mathematica [A]  time = 9.02906, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.156, 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 ) ^{3}}}\, 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)^3,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \,{\left (a^{6} c^{3} x^{9} + 2 \, a^{4} c^{3} x^{7} + a^{2} c^{3} x^{5}\right )} \arctan \left (a x\right )^{2} \int \frac{14 \, a^{4} x^{4} + 15 \, a^{2} x^{2} + 5}{{\left (a^{8} c^{3} x^{12} + 3 \, a^{6} c^{3} x^{10} + 3 \, a^{4} c^{3} x^{8} + a^{2} c^{3} x^{6}\right )} \arctan \left (a x\right )}\,{d x} - a x + 4 \,{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{2 \,{\left (a^{6} c^{3} x^{9} + 2 \, a^{4} c^{3} x^{7} + a^{2} c^{3} x^{5}\right )} \arctan \left (a x\right )^{2}} \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)^3,x, algorithm="maxima")

[Out]

1/2*(2*(a^6*c^3*x^9 + 2*a^4*c^3*x^7 + a^2*c^3*x^5)*arctan(a*x)^2*integrate(2*(14*a^4*x^4 + 15*a^2*x^2 + 5)/((a
^8*c^3*x^12 + 3*a^6*c^3*x^10 + 3*a^4*c^3*x^8 + a^2*c^3*x^6)*arctan(a*x)), x) - a*x + 4*(2*a^2*x^2 + 1)*arctan(
a*x))/((a^6*c^3*x^9 + 2*a^4*c^3*x^7 + a^2*c^3*x^5)*arctan(a*x)^2)

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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 )^{3}}, 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)^3,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)^3), 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}^{3}{\left (a x \right )} + 3 a^{4} x^{8} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{2} x^{6} \operatorname{atan}^{3}{\left (a x \right )} + x^{4} \operatorname{atan}^{3}{\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)**3,x)

[Out]

Integral(1/(a**6*x**10*atan(a*x)**3 + 3*a**4*x**8*atan(a*x)**3 + 3*a**2*x**6*atan(a*x)**3 + x**4*atan(a*x)**3)
, 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 )^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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