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

Optimal. Leaf size=23 \[ \text {Int}\left (x^m \left (a^2 c x^2+c\right ) \tan ^{-1}(a x)^2,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int x^m \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.96, size = 0, normalized size = 0.00 \[ \int x^m \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} x^{m} \arctan \left (a x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 1.54, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {7 \, {\left ({\left (a^{2} c m + a^{2} c\right )} x^{3} + {\left (c m + 3 \, c\right )} x\right )} x^{m} \arctan \left (a x\right )^{2} - \frac {3}{4} \, {\left ({\left (a^{2} c m + a^{2} c\right )} x^{3} + {\left (c m + 3 \, c\right )} x\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )^{2} + {\left (m^{2} + 4 \, m + 3\right )} \int \frac {36 \, {\left ({\left (a^{4} c m^{2} + 4 \, a^{4} c m + 3 \, a^{4} c\right )} x^{4} + c m^{2} + 2 \, {\left (a^{2} c m^{2} + 4 \, a^{2} c m + 3 \, a^{2} c\right )} x^{2} + 4 \, c m + 3 \, c\right )} x^{m} \arctan \left (a x\right )^{2} + 3 \, {\left ({\left (a^{4} c m^{2} + 4 \, a^{4} c m + 3 \, a^{4} c\right )} x^{4} + c m^{2} + 2 \, {\left (a^{2} c m^{2} + 4 \, a^{2} c m + 3 \, a^{2} c\right )} x^{2} + 4 \, c m + 3 \, c\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )^{2} - 56 \, {\left ({\left (a^{3} c m + a^{3} c\right )} x^{3} + {\left (a c m + 3 \, a c\right )} x\right )} x^{m} \arctan \left (a x\right ) + 12 \, {\left ({\left (a^{4} c m + a^{4} c\right )} x^{4} + {\left (a^{2} c m + 3 \, a^{2} c\right )} x^{2}\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )}{4 \, {\left ({\left (a^{2} m^{2} + 4 \, a^{2} m + 3 \, a^{2}\right )} x^{2} + m^{2} + 4 \, m + 3\right )}}\,{d x}}{16 \, {\left (m^{2} + 4 \, m + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*((a^2*c*m + a^2*c)*x^3 + (c*m + 3*c)*x)*x^m*arctan(a*x)^2 - ((a^2*c*m + a^2*c)*x^3 + (c*m + 3*c)*x)*x^
m*log(a^2*x^2 + 1)^2 + 16*(m^2 + 4*m + 3)*integrate(1/16*(12*((a^4*c*m^2 + 4*a^4*c*m + 3*a^4*c)*x^4 + c*m^2 +
2*(a^2*c*m^2 + 4*a^2*c*m + 3*a^2*c)*x^2 + 4*c*m + 3*c)*x^m*arctan(a*x)^2 + ((a^4*c*m^2 + 4*a^4*c*m + 3*a^4*c)*
x^4 + c*m^2 + 2*(a^2*c*m^2 + 4*a^2*c*m + 3*a^2*c)*x^2 + 4*c*m + 3*c)*x^m*log(a^2*x^2 + 1)^2 - 8*((a^3*c*m + a^
3*c)*x^3 + (a*c*m + 3*a*c)*x)*x^m*arctan(a*x) + 4*((a^4*c*m + a^4*c)*x^4 + (a^2*c*m + 3*a^2*c)*x^2)*x^m*log(a^
2*x^2 + 1))/((a^2*m^2 + 4*a^2*m + 3*a^2)*x^2 + m^2 + 4*m + 3), x))/(m^2 + 4*m + 3)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int x^m\,{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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