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

Optimal. Leaf size=21 \[ \text {Int}\left ((d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2,x\right ) \]

[Out]

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

________________________________________________________________________________________

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 = {} \[ \int (d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.28, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arctan \left (c x^{2}\right )^{2} + 2 \, a b \arctan \left (c x^{2}\right ) + a^{2}\right )} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} + \frac {7 \, b^{2} d^{m} x x^{m} \arctan \left (c x^{2}\right )^{2} - \frac {3}{4} \, b^{2} d^{m} x x^{m} \log \left (c^{2} x^{4} + 1\right )^{2} + {\left (m + 1\right )} \int \frac {24 \, b^{2} c^{2} d^{m} x^{4} x^{m} \log \left (c^{2} x^{4} + 1\right ) + 36 \, {\left ({\left (b^{2} c^{2} d^{m} m + b^{2} c^{2} d^{m}\right )} x^{4} + b^{2} d^{m} m + b^{2} d^{m}\right )} x^{m} \arctan \left (c x^{2}\right )^{2} + 3 \, {\left ({\left (b^{2} c^{2} d^{m} m + b^{2} c^{2} d^{m}\right )} x^{4} + b^{2} d^{m} m + b^{2} d^{m}\right )} x^{m} \log \left (c^{2} x^{4} + 1\right )^{2} - 16 \, {\left (7 \, b^{2} c d^{m} x^{2} - 8 \, {\left (a b c^{2} d^{m} m + a b c^{2} d^{m}\right )} x^{4} - 8 \, a b d^{m} m - 8 \, a b d^{m}\right )} x^{m} \arctan \left (c x^{2}\right )}{4 \, {\left ({\left (c^{2} m + c^{2}\right )} x^{4} + m + 1\right )}}\,{d x}}{16 \, {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*atan(c*x^2))^2,x)

[Out]

int((d*x)^m*(a + b*atan(c*x^2))^2, x)

________________________________________________________________________________________

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**2))**2,x)

[Out]

Timed out

________________________________________________________________________________________