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

Optimal. Leaf size=83 \[ -\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}-\frac{2}{3} i b d^2 x \]

[Out]

((-2*I)/3)*b*d^2*x - (b*d^2*(1 + I*c*x)^2)/(6*c) - ((I/3)*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x]))/c - (4*b*d^2*
Log[1 - I*c*x])/(3*c)

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Rubi [A]  time = 0.0456393, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4862, 627, 43} \[ -\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}-\frac{2}{3} i b d^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)/3)*b*d^2*x - (b*d^2*(1 + I*c*x)^2)/(6*c) - ((I/3)*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x]))/c - (4*b*d^2*
Log[1 - I*c*x])/(3*c)

Rule 4862

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a + b*
ArcTan[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{
a, b, c, d, e, q}, x] && NeQ[q, -1]

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(i b) \int \frac{(d+i c d x)^3}{1+c^2 x^2} \, dx}{3 d}\\ &=-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(i b) \int \frac{(d+i c d x)^2}{\frac{1}{d}-\frac{i c x}{d}} \, dx}{3 d}\\ &=-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(i b) \int \left (-2 d^3+\frac{4 d^2}{\frac{1}{d}-\frac{i c x}{d}}-d^2 (d+i c d x)\right ) \, dx}{3 d}\\ &=-\frac{2}{3} i b d^2 x-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}\\ \end{align*}

Mathematica [A]  time = 0.0385873, size = 57, normalized size = 0.69 \[ \frac{1}{3} d^2 \left (-\frac{(c x-i)^3 \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac{1}{2} b x (c x-6 i)-\frac{4 b \log (c x+i)}{c}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*((b*x*(-6*I + c*x))/2 - ((-I + c*x)^3*(a + b*ArcTan[c*x]))/c - (4*b*Log[I + c*x])/c))/3

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Maple [A]  time = 0.028, size = 133, normalized size = 1.6 \begin{align*} -{\frac{{c}^{2}{x}^{3}a{d}^{2}}{3}}+ic{x}^{2}a{d}^{2}+ax{d}^{2}-{\frac{{\frac{i}{3}}{d}^{2}a}{c}}-{\frac{{c}^{2}{d}^{2}b\arctan \left ( cx \right ){x}^{3}}{3}}+ic{d}^{2}b\arctan \left ( cx \right ){x}^{2}+{d}^{2}bx\arctan \left ( cx \right ) +{\frac{i{d}^{2}b\arctan \left ( cx \right ) }{c}}-i{d}^{2}bx+{\frac{c{d}^{2}b{x}^{2}}{6}}-{\frac{2\,{d}^{2}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c^2*x^3*a*d^2+I*c*x^2*a*d^2+a*x*d^2-1/3*I/c*d^2*a-1/3*c^2*d^2*b*arctan(c*x)*x^3+I*c*d^2*b*arctan(c*x)*x^2
+d^2*b*x*arctan(c*x)+I/c*d^2*b*arctan(c*x)-I*d^2*b*x+1/6*c*d^2*b*x^2-2/3/c*d^2*b*ln(c^2*x^2+1)

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Maxima [B]  time = 1.49232, size = 186, normalized size = 2.24 \begin{align*} -\frac{1}{3} \, a c^{2} d^{2} x^{3} - \frac{1}{6} \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b c^{2} d^{2} + i \, a c d^{2} x^{2} + i \,{\left (x^{2} \arctan \left (c x\right ) - c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )}\right )} b c d^{2} + a d^{2} x + \frac{{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a*c^2*d^2*x^3 - 1/6*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*b*c^2*d^2 + I*a*c*d^2*x^2 +
I*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*b*c*d^2 + a*d^2*x + 1/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1
))*b*d^2/c

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Fricas [A]  time = 2.68781, size = 279, normalized size = 3.36 \begin{align*} -\frac{2 \, a c^{3} d^{2} x^{3} -{\left (6 i \, a + b\right )} c^{2} d^{2} x^{2} - 6 \,{\left (a - i \, b\right )} c d^{2} x + 7 \, b d^{2} \log \left (\frac{c x + i}{c}\right ) + b d^{2} \log \left (\frac{c x - i}{c}\right ) -{\left (-i \, b c^{3} d^{2} x^{3} - 3 \, b c^{2} d^{2} x^{2} + 3 i \, b c d^{2} x\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a*c^3*d^2*x^3 - (6*I*a + b)*c^2*d^2*x^2 - 6*(a - I*b)*c*d^2*x + 7*b*d^2*log((c*x + I)/c) + b*d^2*log((
c*x - I)/c) - (-I*b*c^3*d^2*x^3 - 3*b*c^2*d^2*x^2 + 3*I*b*c*d^2*x)*log(-(c*x + I)/(c*x - I)))/c

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Sympy [B]  time = 2.60729, size = 165, normalized size = 1.99 \begin{align*} - \frac{a c^{2} d^{2} x^{3}}{3} - \frac{b d^{2} \left (\frac{\log{\left (x - \frac{i}{c} \right )}}{6} + \frac{7 \log{\left (x + \frac{i}{c} \right )}}{6}\right )}{c} - x^{2} \left (- i a c d^{2} - \frac{b c d^{2}}{6}\right ) - x \left (- a d^{2} + i b d^{2}\right ) + \left (- \frac{i b c^{2} d^{2} x^{3}}{6} - \frac{b c d^{2} x^{2}}{2} + \frac{i b d^{2} x}{2}\right ) \log{\left (- i c x + 1 \right )} + \left (\frac{i b c^{2} d^{2} x^{3}}{6} + \frac{b c d^{2} x^{2}}{2} - \frac{i b d^{2} x}{2}\right ) \log{\left (i c x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*c**2*d**2*x**3/3 - b*d**2*(log(x - I/c)/6 + 7*log(x + I/c)/6)/c - x**2*(-I*a*c*d**2 - b*c*d**2/6) - x*(-a*d
**2 + I*b*d**2) + (-I*b*c**2*d**2*x**3/6 - b*c*d**2*x**2/2 + I*b*d**2*x/2)*log(-I*c*x + 1) + (I*b*c**2*d**2*x*
*3/6 + b*c*d**2*x**2/2 - I*b*d**2*x/2)*log(I*c*x + 1)

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Giac [A]  time = 1.15826, size = 176, normalized size = 2.12 \begin{align*} -\frac{2 \, b c^{3} d^{2} x^{3} \arctan \left (c x\right ) + 2 \, a c^{3} d^{2} x^{3} - 6 \, b c^{2} d^{2} i x^{2} \arctan \left (c x\right ) - 6 \, a c^{2} d^{2} i x^{2} - b c^{2} d^{2} x^{2} + 6 \, b c d^{2} i x - 6 \, b c d^{2} x \arctan \left (c x\right ) - 6 \, a c d^{2} x + 7 \, b d^{2} \log \left (c x + i\right ) + b d^{2} \log \left (c x - i\right )}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b*c^3*d^2*x^3*arctan(c*x) + 2*a*c^3*d^2*x^3 - 6*b*c^2*d^2*i*x^2*arctan(c*x) - 6*a*c^2*d^2*i*x^2 - b*c^
2*d^2*x^2 + 6*b*c*d^2*i*x - 6*b*c*d^2*x*arctan(c*x) - 6*a*c*d^2*x + 7*b*d^2*log(c*x + i) + b*d^2*log(c*x - i))
/c