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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{x \left (a^2 c x^2+c\right )^3 \sqrt {\tan ^{-1}(a x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx &=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.80, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), 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)^3/arctan(a*x)^(1/2),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)^3/arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 4.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{3} \sqrt {\arctan \left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),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)^3/arctan(a*x)^(1/2),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 = -0.04 \[ \int \frac {1}{x\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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