3.127 \(\int (d x)^m (a+b \tan ^{-1}(c x^3))^3 \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left ((d x)^m \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3,x\right ) \]

[Out]

Unintegrable((d*x)^m*(a+b*arctan(c*x^3))^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 = {} \[ \int (d x)^m \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.98, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \arctan \left (c x^{3}\right )^{3} + 3 \, a b^{2} \arctan \left (c x^{3}\right )^{2} + 3 \, a^{2} b \arctan \left (c x^{3}\right ) + a^{3}\right )} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctan \left (c \,x^{3}\right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} + \frac {\frac {15}{2} \, b^{3} d^{m} x x^{m} \arctan \left (c x^{3}\right )^{3} - \frac {21}{8} \, b^{3} d^{m} x x^{m} \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right )^{2} + {\left (m + 1\right )} \int \frac {252 \, b^{3} c^{2} d^{m} x^{6} x^{m} \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right ) + 196 \, {\left ({\left (b^{3} c^{2} d^{m} m + b^{3} c^{2} d^{m}\right )} x^{6} + b^{3} d^{m} m + b^{3} d^{m}\right )} x^{m} \arctan \left (c x^{3}\right )^{3} - 12 \, {\left (45 \, b^{3} c d^{m} x^{3} - 64 \, {\left (a b^{2} c^{2} d^{m} m + a b^{2} c^{2} d^{m}\right )} x^{6} - 64 \, a b^{2} d^{m} m - 64 \, a b^{2} d^{m}\right )} x^{m} \arctan \left (c x^{3}\right )^{2} + 768 \, {\left ({\left (a^{2} b c^{2} d^{m} m + a^{2} b c^{2} d^{m}\right )} x^{6} + a^{2} b d^{m} m + a^{2} b d^{m}\right )} x^{m} \arctan \left (c x^{3}\right ) + 21 \, {\left (3 \, b^{3} c d^{m} x^{3} x^{m} + {\left ({\left (b^{3} c^{2} d^{m} m + b^{3} c^{2} d^{m}\right )} x^{6} + b^{3} d^{m} m + b^{3} d^{m}\right )} x^{m} \arctan \left (c x^{3}\right )\right )} \log \left (c^{2} x^{6} + 1\right )^{2}}{8 \, {\left ({\left (c^{2} m + c^{2}\right )} x^{6} + m + 1\right )}}\,{d x}}{32 \, {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctan(c*x^3))^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 - 3*b^3*d^m*x*x^m*arctan(c*x^3)*log(c^2*
x^6 + 1)^2 + 32*(m + 1)*integrate(1/32*(36*b^3*c^2*d^m*x^6*x^m*arctan(c*x^3)*log(c^2*x^6 + 1) + 28*((b^3*c^2*d
^m*m + b^3*c^2*d^m)*x^6 + b^3*d^m*m + b^3*d^m)*x^m*arctan(c*x^3)^3 - 12*(3*b^3*c*d^m*x^3 - 8*(a*b^2*c^2*d^m*m
+ a*b^2*c^2*d^m)*x^6 - 8*a*b^2*d^m*m - 8*a*b^2*d^m)*x^m*arctan(c*x^3)^2 + 96*((a^2*b*c^2*d^m*m + a^2*b*c^2*d^m
)*x^6 + a^2*b*d^m*m + a^2*b*d^m)*x^m*arctan(c*x^3) + 3*(3*b^3*c*d^m*x^3*x^m + ((b^3*c^2*d^m*m + b^3*c^2*d^m)*x
^6 + b^3*d^m*m + b^3*d^m)*x^m*arctan(c*x^3))*log(c^2*x^6 + 1)^2)/((c^2*m + c^2)*x^6 + m + 1), x))/(m + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*atan(c*x^3))^3,x)

[Out]

int((d*x)^m*(a + b*atan(c*x^3))^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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