3.595 \(\int \frac {1}{x^3 (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=238 \[ \frac {2 a \text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)},x\right )}{c^2}+\frac {\text {Int}\left (\frac {1}{x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{c^2}+\frac {2 a \sqrt {a^2 c x^2+c}}{c^3 x \tan ^{-1}(a x)}+\frac {9 a^2 \sqrt {a^2 x^2+1} \text {Ci}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 a^3 x}{c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}-\frac {a^3 x}{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

-a^3*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-2*a^3*x/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)+9/4*a^2*Ci(arctan(a*x))*(
a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+3/4*a^2*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+2
*a*(a^2*c*x^2+c)^(1/2)/c^3/x/arctan(a*x)+Unintegrable(1/x^3/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/c^2+2*a*Unint
egrable(1/x^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2),x)/c^2

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Rubi [A]  time = 2.05, 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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

-((a^3*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (2*a*Sqrt
[c + a^2*c*x^2])/(c^3*x*ArcTan[a*x]) + (9*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*
c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^3
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 + (2*a*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])/c^2

Rubi steps

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

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Mathematica [A]  time = 7.95, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

Integrate[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{{\left (a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 1.97, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

[Out]

int(1/x^3/(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 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*atan(a*x)^2*(c + a^2*c*x^2)^(5/2)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a**2*c*x**2+c)**(5/2)/atan(a*x)**2,x)

[Out]

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

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