3.10.14 \(\int \frac {x^m (c+a^2 c x^2)}{\sqrt {\text {ArcTan}(a x)}} \, dx\) [914]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, 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 = {} \begin {gather*} \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\text {ArcTan}(a x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\text {ArcTan}(a x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.74, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )}{\sqrt {\arctan \left (a x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*(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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c \left (\int \frac {x^{m}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx + \int \frac {a^{2} x^{2} x^{m}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^m\,\left (c\,a^2\,x^2+c\right )}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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