3.1.54 \(\int (d x)^m (a+b \text {ArcTan}(c x))^3 \, dx\) [54]

Optimal. Leaf size=19 \[ \text {Int}\left ((d x)^m (a+b \text {ArcTan}(c x))^3,x\right ) \]

[Out]

Unintegrable((d*x)^m*(a+b*arctan(c*x))^3,x)

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Rubi [A]
time = 0.02, 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 (d x)^m (a+b \text {ArcTan}(c x))^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcTan[c*x])^3,x]

[Out]

Defer[Int][(d*x)^m*(a + b*ArcTan[c*x])^3, x]

Rubi steps

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

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Mathematica [A]
time = 2.71, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m (a+b \text {ArcTan}(c x))^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(d*x)^m*(a + b*ArcTan[c*x])^3,x]

[Out]

Integrate[(d*x)^m*(a + b*ArcTan[c*x])^3, x]

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Maple [A]
time = 1.53, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \arctan \left (c x \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*arctan(c*x))^3,x)

[Out]

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

[Out]

(d*x)^(m + 1)*a^3/(d*(m + 1)) + 1/32*(4*b^3*d^m*x*x^m*arctan(c*x)^3 - 3*b^3*d^m*x*x^m*arctan(c*x)*log(c^2*x^2
+ 1)^2 + 32*(m + 1)*integrate(1/32*(12*b^3*c^2*d^m*x^2*x^m*arctan(c*x)*log(c^2*x^2 + 1) + 28*(b^3*d^m*m + b^3*
d^m + (b^3*c^2*d^m*m + b^3*c^2*d^m)*x^2)*x^m*arctan(c*x)^3 - 12*(b^3*c*d^m*x - 8*a*b^2*d^m*m - 8*a*b^2*d^m - 8
*(a*b^2*c^2*d^m*m + a*b^2*c^2*d^m)*x^2)*x^m*arctan(c*x)^2 + 96*(a^2*b*d^m*m + a^2*b*d^m + (a^2*b*c^2*d^m*m + a
^2*b*c^2*d^m)*x^2)*x^m*arctan(c*x) + 3*(b^3*c*d^m*x*x^m + (b^3*d^m*m + b^3*d^m + (b^3*c^2*d^m*m + b^3*c^2*d^m)
*x^2)*x^m*arctan(c*x))*log(c^2*x^2 + 1)^2)/((c^2*m + c^2)*x^2 + m + 1), x))/(m + 1)

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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((d*x)^m*(a+b*arctan(c*x))^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*atan(c*x))**3,x)

[Out]

Integral((d*x)**m*(a + b*atan(c*x))**3, 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((d*x)^m*(a+b*arctan(c*x))^3,x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3\,{\left (d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))^3*(d*x)^m,x)

[Out]

int((a + b*atan(c*x))^3*(d*x)^m, x)

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