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

Optimal. Leaf size=22 \[ \text {Int}\left (\left (a^2 c x^2+c\right ) \sqrt {\tan ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, 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 = {} \[ \int \left (c+a^2 c x^2\right ) \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.43, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right ) \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 1.62, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right ) \sqrt {\arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*c*x^2+c)*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((a^2*c*x^2+c)*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 = -0.05 \[ \int \sqrt {\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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