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

Optimal. Leaf size=64 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)},x\right )}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.22, 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 = {} \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{{\left (a^{2} c x^{3} + c x\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)/((a^2*c*x^3 + c*x)*arctan(a*x)^2), x)

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

[Out]

sage0*x

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maple [A]  time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \arctan \left (a x \right )^{2} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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