3.7.77 \(\int \frac {x^m}{(c+a^2 c x^2) \text {ArcTan}(a x)^3} \, dx\) [677]

Optimal. Leaf size=46 \[ -\frac {x^m}{2 a c \text {ArcTan}(a x)^2}+\frac {m \text {Int}\left (\frac {x^{-1+m}}{\text {ArcTan}(a x)^2},x\right )}{2 a c} \]

[Out]

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

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Rubi [A]
time = 0.06, 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 ) \text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m}{\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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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)^3,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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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)^3,x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )} \,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)),x)

[Out]

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

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