3.596 \(\int \frac {1}{x^4 (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=209 \[ -\frac {2 a^2 \text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{c^2}+\frac {\text {Int}\left (\frac {1}{x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{c^2}-\frac {11 a^3 \sqrt {a^2 x^2+1} \text {Si}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {a^2 c x^2+c}}-\frac {3 a^3 \sqrt {a^2 x^2+1} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 a^3}{c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}-\frac {a^3}{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

-a^3/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-2*a^3/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)-11/4*a^3*Si(arctan(a*x))*(a^2
*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-3/4*a^3*Si(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+Unin
tegrable(1/x^4/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/c^2-2*a^2*Unintegrable(1/x^2/arctan(a*x)^2/(a^2*c*x^2+c)^(
1/2),x)/c^2

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Rubi [A]  time = 1.61, 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 = {} \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

-(a^3/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (11*a^3*Sqrt[1
+ a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*a^3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcT
an[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 - (2*a^2*
Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/c^2

Rubi steps

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

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Mathematica [A]  time = 7.48, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

Integrate[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{{\left (a^{6} c^{3} x^{10} + 3 \, a^{4} c^{3} x^{8} + 3 \, a^{2} c^{3} x^{6} + c^{3} x^{4}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)/((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)^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^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 3.88, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

[Out]

int(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

integrate(1/((a^2*c*x^2 + c)^(5/2)*x^4*arctan(a*x)^2), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*atan(a*x)^2*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(1/(x^4*atan(a*x)^2*(c + a^2*c*x^2)^(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a**2*c*x**2+c)**(5/2)/atan(a*x)**2,x)

[Out]

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

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