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

Optimal. Leaf size=117 \[ \frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d x^2}{10 c^2}-\frac{i b d \log \left (c^2 x^2+1\right )}{10 c^4}+\frac{b d x}{4 c^3}-\frac{b d \tan ^{-1}(c x)}{4 c^4}-\frac{b d x^3}{12 c}-\frac{1}{20} i b d x^4 \]

[Out]

(b*d*x)/(4*c^3) + ((I/10)*b*d*x^2)/c^2 - (b*d*x^3)/(12*c) - (I/20)*b*d*x^4 - (b*d*ArcTan[c*x])/(4*c^4) + (d*x^
4*(a + b*ArcTan[c*x]))/4 + (I/5)*c*d*x^5*(a + b*ArcTan[c*x]) - ((I/10)*b*d*Log[1 + c^2*x^2])/c^4

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Rubi [A]  time = 0.103317, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {43, 4872, 12, 801, 635, 203, 260} \[ \frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d x^2}{10 c^2}-\frac{i b d \log \left (c^2 x^2+1\right )}{10 c^4}+\frac{b d x}{4 c^3}-\frac{b d \tan ^{-1}(c x)}{4 c^4}-\frac{b d x^3}{12 c}-\frac{1}{20} i b d x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*x)/(4*c^3) + ((I/10)*b*d*x^2)/c^2 - (b*d*x^3)/(12*c) - (I/20)*b*d*x^4 - (b*d*ArcTan[c*x])/(4*c^4) + (d*x^
4*(a + b*ArcTan[c*x]))/4 + (I/5)*c*d*x^5*(a + b*ArcTan[c*x]) - ((I/10)*b*d*Log[1 + c^2*x^2])/c^4

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])

Rule 4872

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^
2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0
]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int x^3 (d+i c d x) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{d x^4 (5+4 i c x)}{20 \left (1+c^2 x^2\right )} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{20} (b c d) \int \frac{x^4 (5+4 i c x)}{1+c^2 x^2} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{20} (b c d) \int \left (-\frac{5}{c^4}-\frac{4 i x}{c^3}+\frac{5 x^2}{c^2}+\frac{4 i x^3}{c}+\frac{5+4 i c x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{b d x}{4 c^3}+\frac{i b d x^2}{10 c^2}-\frac{b d x^3}{12 c}-\frac{1}{20} i b d x^4+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{(b d) \int \frac{5+4 i c x}{1+c^2 x^2} \, dx}{20 c^3}\\ &=\frac{b d x}{4 c^3}+\frac{i b d x^2}{10 c^2}-\frac{b d x^3}{12 c}-\frac{1}{20} i b d x^4+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{(b d) \int \frac{1}{1+c^2 x^2} \, dx}{4 c^3}-\frac{(i b d) \int \frac{x}{1+c^2 x^2} \, dx}{5 c^2}\\ &=\frac{b d x}{4 c^3}+\frac{i b d x^2}{10 c^2}-\frac{b d x^3}{12 c}-\frac{1}{20} i b d x^4-\frac{b d \tan ^{-1}(c x)}{4 c^4}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} i c d x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{i b d \log \left (1+c^2 x^2\right )}{10 c^4}\\ \end{align*}

Mathematica [A]  time = 0.074168, size = 98, normalized size = 0.84 \[ \frac{d \left (3 a c^4 x^4 (5+4 i c x)+b c x \left (-3 i c^3 x^3-5 c^2 x^2+6 i c x+15\right )-6 i b \log \left (c^2 x^2+1\right )+3 b \left (4 i c^5 x^5+5 c^4 x^4-5\right ) \tan ^{-1}(c x)\right )}{60 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(3*a*c^4*x^4*(5 + (4*I)*c*x) + b*c*x*(15 + (6*I)*c*x - 5*c^2*x^2 - (3*I)*c^3*x^3) + 3*b*(-5 + 5*c^4*x^4 + (
4*I)*c^5*x^5)*ArcTan[c*x] - (6*I)*b*Log[1 + c^2*x^2]))/(60*c^4)

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Maple [A]  time = 0.027, size = 108, normalized size = 0.9 \begin{align*}{\frac{i}{5}}cda{x}^{5}+{\frac{da{x}^{4}}{4}}+{\frac{i}{5}}cdb\arctan \left ( cx \right ){x}^{5}+{\frac{db\arctan \left ( cx \right ){x}^{4}}{4}}+{\frac{dbx}{4\,{c}^{3}}}-{\frac{i}{20}}bd{x}^{4}-{\frac{db{x}^{3}}{12\,c}}+{\frac{{\frac{i}{10}}bd{x}^{2}}{{c}^{2}}}-{\frac{{\frac{i}{10}}bd\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{4}}}-{\frac{db\arctan \left ( cx \right ) }{4\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*I*c*d*a*x^5+1/4*d*a*x^4+1/5*I*c*d*b*arctan(c*x)*x^5+1/4*d*b*arctan(c*x)*x^4+1/4*b*d*x/c^3-1/20*I*b*d*x^4-1
/12*b*d*x^3/c+1/10*I*b*d*x^2/c^2-1/10*I*b*d*ln(c^2*x^2+1)/c^4-1/4*b*d*arctan(c*x)/c^4

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Maxima [A]  time = 1.48333, size = 147, normalized size = 1.26 \begin{align*} \frac{1}{5} i \, a c d x^{5} + \frac{1}{4} \, a d x^{4} + \frac{1}{20} i \,{\left (4 \, x^{5} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d + \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*I*a*c*d*x^5 + 1/4*a*d*x^4 + 1/20*I*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6)
)*b*c*d + 1/12*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*b*d

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Fricas [A]  time = 2.86457, size = 302, normalized size = 2.58 \begin{align*} \frac{24 i \, a c^{5} d x^{5} + 6 \,{\left (5 \, a - i \, b\right )} c^{4} d x^{4} - 10 \, b c^{3} d x^{3} + 12 i \, b c^{2} d x^{2} + 30 \, b c d x - 27 i \, b d \log \left (\frac{c x + i}{c}\right ) + 3 i \, b d \log \left (\frac{c x - i}{c}\right ) -{\left (12 \, b c^{5} d x^{5} - 15 i \, b c^{4} d x^{4}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{120 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*I*a*c^5*d*x^5 + 6*(5*a - I*b)*c^4*d*x^4 - 10*b*c^3*d*x^3 + 12*I*b*c^2*d*x^2 + 30*b*c*d*x - 27*I*b*d*
log((c*x + I)/c) + 3*I*b*d*log((c*x - I)/c) - (12*b*c^5*d*x^5 - 15*I*b*c^4*d*x^4)*log(-(c*x + I)/(c*x - I)))/c
^4

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Sympy [A]  time = 1.84808, size = 153, normalized size = 1.31 \begin{align*} \frac{i a c d x^{5}}{5} - \frac{b d x^{3}}{12 c} + \frac{i b d x^{2}}{10 c^{2}} + \frac{b d x}{4 c^{3}} + \frac{i b d \log{\left (x - \frac{i}{c} \right )}}{40 c^{4}} - \frac{9 i b d \log{\left (x + \frac{i}{c} \right )}}{40 c^{4}} + x^{4} \left (\frac{a d}{4} - \frac{i b d}{20}\right ) + \left (- \frac{b c d x^{5}}{10} + \frac{i b d x^{4}}{8}\right ) \log{\left (- i c x + 1 \right )} + \left (\frac{b c d x^{5}}{10} - \frac{i b d x^{4}}{8}\right ) \log{\left (i c x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*a*c*d*x**5/5 - b*d*x**3/(12*c) + I*b*d*x**2/(10*c**2) + b*d*x/(4*c**3) + I*b*d*log(x - I/c)/(40*c**4) - 9*I*
b*d*log(x + I/c)/(40*c**4) + x**4*(a*d/4 - I*b*d/20) + (-b*c*d*x**5/10 + I*b*d*x**4/8)*log(-I*c*x + 1) + (b*c*
d*x**5/10 - I*b*d*x**4/8)*log(I*c*x + 1)

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Giac [A]  time = 1.30961, size = 162, normalized size = 1.38 \begin{align*} -\frac{24 \, b c^{5} d x^{5} \arctan \left (c x\right ) + 24 \, a c^{5} d x^{5} - 30 \, b c^{4} d i x^{4} \arctan \left (c x\right ) - 30 \, a c^{4} d i x^{4} - 6 \, b c^{4} d x^{4} + 10 \, b c^{3} d i x^{3} + 12 \, b c^{2} d x^{2} - 30 \, b c d i x + 3 \, b d \log \left (c i x + 1\right ) - 27 \, b d \log \left (-c i x + 1\right )}{120 \, c^{4} i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(24*b*c^5*d*x^5*arctan(c*x) + 24*a*c^5*d*x^5 - 30*b*c^4*d*i*x^4*arctan(c*x) - 30*a*c^4*d*i*x^4 - 6*b*c^
4*d*x^4 + 10*b*c^3*d*i*x^3 + 12*b*c^2*d*x^2 - 30*b*c*d*i*x + 3*b*d*log(c*i*x + 1) - 27*b*d*log(-c*i*x + 1))/(c
^4*i)