3.1063 \(\int \frac {1}{x (c+a^2 c x^2)^2 \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=180 \[ \frac {16}{3} \text {Int}\left (\frac {1}{x \left (a^2 c x^2+c\right )^2 \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {8 \text {Int}\left (\frac {1}{x^3 \left (a^2 c x^2+c\right )^2 \sqrt {\tan ^{-1}(a x)}},x\right )}{3 a^2}+\frac {4}{c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{c^2} \]

[Out]

-2/3/a/c^2/x/(a^2*x^2+1)/arctan(a*x)^(3/2)+4*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/c^2+4/c^2/(a^2*x^
2+1)/arctan(a*x)^(1/2)+4/3/a^2/c^2/x^2/(a^2*x^2+1)/arctan(a*x)^(1/2)+8/3*Unintegrable(1/x^3/(a^2*c*x^2+c)^2/ar
ctan(a*x)^(1/2),x)/a^2+16/3*Unintegrable(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.36, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-2/(3*a*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 4/(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^2*x^2*(1
+ a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 + (8*Defer[Int][1/(
x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (16*Defer[Int][1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]
]), x])/3

Rubi steps

\begin {align*} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-(2 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4}{c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\left (8 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4}{c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {8 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4}{c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {8 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4}{c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {4 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4}{c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {8 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {4}{c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{c^2}+\frac {16}{3} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}\\ \end {align*}

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Mathematica [A]  time = 4.54, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

[Out]

Timed out

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maple [A]  time = 2.67, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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