3.5.63 \(\int x^m \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3 \, dx\) [463]

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

[Out]

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

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Rubi [A]
time = 0.07, 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 x^m \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]

[Out]

Defer[Int][x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x]

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]

[Out]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x]

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Maple [A]
time = 2.19, size = 0, normalized size = 0.00 \[\int x^{m} \arctan \left (a x \right )^{3} \sqrt {a^{2} c \,x^{2}+c}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a^2*c*x^2 + c)*x^m*arctan(a*x)^3, x)

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int x^m\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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