3.1.42 \(\int (e+f x)^m (a+b \text {ArcTan}(c+d x))^2 \, dx\) [42]

Optimal. Leaf size=23 \[ \text {Int}\left ((e+f x)^m (a+b \text {ArcTan}(c+d x))^2,x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, 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 (e+f x)^m (a+b \text {ArcTan}(c+d x))^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcTan[x])^2, x], x, c + d*x]/d

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 3.35, size = 0, normalized size = 0.00 \begin {gather*} \int (e+f x)^m (a+b \text {ArcTan}(c+d x))^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (f x +e \right )^{m} \left (a +b \arctan \left (d x +c \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^m*(a + b*atan(c + d*x))^2,x)

[Out]

int((e + f*x)^m*(a + b*atan(c + d*x))^2, x)

________________________________________________________________________________________