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

Optimal. Leaf size=248 \[ -\frac {3 \text {Int}\left (\frac {1}{x^4 \tan ^{-1}(a x)^2},x\right )}{2 a c^3}+\frac {a \text {Int}\left (\frac {1}{x^2 \tan ^{-1}(a x)^2},x\right )}{c^3}-\frac {5 a^2 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {a^2 \text {Si}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac {a^2 \left (1-a^2 x^2\right )}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {3 a^2}{2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac {2 a^2}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}-\frac {a^3 x}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}-\frac {1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}+\frac {a}{c^3 x \tan ^{-1}(a x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.28, 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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx\right )+\frac {\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx+\frac {\int \frac {1}{x^3 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c^2}-2 \frac {a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=-\frac {1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac {1}{2} a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx-\frac {1}{2} \left (3 a^5\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx-\frac {3 \int \frac {1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (\frac {a^2 \int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c^2}-\frac {a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\right )\\ &=-\frac {1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {a^2}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {1}{2} \left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx-\left (2 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac {3 \int \frac {1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-\frac {\left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{2 c}-2 \left (-\frac {a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {a \int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac {\left (2 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\right )\\ &=-\frac {1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\left (6 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac {3 \int \frac {1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {a \int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac {\left (3 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac {1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {3 \int \frac {1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {a \int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )-\frac {\left (2 a^2\right ) \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 {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {3 \int \frac {1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}-2 \left (-\frac {a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {a \int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac {\left (6 a^2\right ) \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}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {a^2 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {a^2 \text {Si}\left (4 \tan ^{-1}(a x)\right )}{4 c^3}-\frac {3 \int \frac {1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {a^2 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac {a \int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}\right )-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}\\ &=-\frac {1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {a^2 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {a^2 \text {Si}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac {3 \int \frac {1}{x^4 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \tan ^{-1}(a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {a^2 \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac {a \int \frac {1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 c^3}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 5.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 4.45, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (a^{6} c^{3} x^{8} + 2 \, a^{4} c^{3} x^{6} + a^{2} c^{3} x^{4}\right )} \mathit {sage}_{0} x \arctan \left (a x\right )^{2} - a x + {\left (7 \, a^{2} x^{2} + 3\right )} \arctan \left (a x\right )}{2 \, {\left (a^{6} c^{3} x^{8} + 2 \, a^{4} c^{3} x^{6} + a^{2} c^{3} x^{4}\right )} \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(a^6*c^3*x^8 + 2*a^4*c^3*x^6 + a^2*c^3*x^4)*arctan(a*x)^2*integrate((21*a^4*x^4 + 19*a^2*x^2 + 6)/((a^8
*c^3*x^11 + 3*a^6*c^3*x^9 + 3*a^4*c^3*x^7 + a^2*c^3*x^5)*arctan(a*x)), x) - a*x + (7*a^2*x^2 + 3)*arctan(a*x))
/((a^6*c^3*x^8 + 2*a^4*c^3*x^6 + a^2*c^3*x^4)*arctan(a*x)^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**6*x**9*atan(a*x)**3 + 3*a**4*x**7*atan(a*x)**3 + 3*a**2*x**5*atan(a*x)**3 + x**3*atan(a*x)**3),
x)/c**3

________________________________________________________________________________________