3.1022 \(\int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac {2 \text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )}{a}-\frac {2 \sqrt {a^2 c x^2+c}}{a c x \sqrt {\tan ^{-1}(a x)}} \]

[Out]

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

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Rubi [A]  time = 0.21, 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 = {} \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.42, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), 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/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(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 [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 1.66, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(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/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(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 = -0.01 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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