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

Optimal. Leaf size=18 \[ \text{Unintegrable}\left ((d x)^m \left (a+b \tan ^{-1}(c x)\right )^2,x\right ) \]

[Out]

Unintegrable[(d*x)^m*(a + b*ArcTan[c*x])^2, x]

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Rubi [A]  time = 0.0212714, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (d x)^m \left (a+b \tan ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcTan[c*x])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.43325, size = 0, normalized size = 0. \[ \int (d x)^m \left (a+b \tan ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.633, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b\arctan \left ( cx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}\right )} \left (d x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arctan(c*x)^2 + 2*a*b*arctan(c*x) + a^2)*(d*x)^m, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \left (a + b \operatorname{atan}{\left (c x \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )}^{2} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctan(c*x) + a)^2*(d*x)^m, x)