3.32 \(\int x^2 (d+i c d x)^4 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=193 \[ \frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{5 i b d^4 x}{3 c^2}+\frac{176 b d^4 \log (c x+i)}{105 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3 \]

[Out]

(((5*I)/3)*b*d^4*x)/c^2 - (88*b*d^4*x^2)/(105*c) - ((5*I)/9)*b*d^4*x^3 + (47*b*c*d^4*x^4)/140 + ((2*I)/15)*b*c
^2*d^4*x^5 - (b*c^3*d^4*x^6)/42 + ((I/5)*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/c^3 - ((I/3)*d^4*(1 + I*c*x)^6
*(a + b*ArcTan[c*x]))/c^3 + ((I/7)*d^4*(1 + I*c*x)^7*(a + b*ArcTan[c*x]))/c^3 + (176*b*d^4*Log[I + c*x])/(105*
c^3)

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Rubi [A]  time = 0.166718, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {43, 4872, 12, 893} \[ \frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{5 i b d^4 x}{3 c^2}+\frac{176 b d^4 \log (c x+i)}{105 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]

[Out]

(((5*I)/3)*b*d^4*x)/c^2 - (88*b*d^4*x^2)/(105*c) - ((5*I)/9)*b*d^4*x^3 + (47*b*c*d^4*x^4)/140 + ((2*I)/15)*b*c
^2*d^4*x^5 - (b*c^3*d^4*x^6)/42 + ((I/5)*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/c^3 - ((I/3)*d^4*(1 + I*c*x)^6
*(a + b*ArcTan[c*x]))/c^3 + ((I/7)*d^4*(1 + I*c*x)^7*(a + b*ArcTan[c*x]))/c^3 + (176*b*d^4*Log[I + c*x])/(105*
c^3)

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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int x^2 (d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-(b c) \int \frac{d^4 (i-c x)^4 \left (-1+5 i c x+15 c^2 x^2\right )}{105 c^3 (i+c x)} \, dx\\ &=\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{\left (b d^4\right ) \int \frac{(i-c x)^4 \left (-1+5 i c x+15 c^2 x^2\right )}{i+c x} \, dx}{105 c^2}\\ &=\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{\left (b d^4\right ) \int \left (-175 i+176 c x+175 i c^2 x^2-141 c^3 x^3-70 i c^4 x^4+15 c^5 x^5-\frac{176}{i+c x}\right ) \, dx}{105 c^2}\\ &=\frac{5 i b d^4 x}{3 c^2}-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3+\frac{47}{140} b c d^4 x^4+\frac{2}{15} i b c^2 d^4 x^5-\frac{1}{42} b c^3 d^4 x^6+\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}+\frac{176 b d^4 \log (i+c x)}{105 c^3}\\ \end{align*}

Mathematica [A]  time = 0.120197, size = 276, normalized size = 1.43 \[ \frac{1}{7} a c^4 d^4 x^7-\frac{2}{3} i a c^3 d^4 x^6-\frac{6}{5} a c^2 d^4 x^5+i a c d^4 x^4+\frac{1}{3} a d^4 x^3-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{88 b d^4 \log \left (c^2 x^2+1\right )}{105 c^3}+\frac{1}{7} b c^4 d^4 x^7 \tan ^{-1}(c x)-\frac{2}{3} i b c^3 d^4 x^6 \tan ^{-1}(c x)-\frac{6}{5} b c^2 d^4 x^5 \tan ^{-1}(c x)+\frac{5 i b d^4 x}{3 c^2}-\frac{5 i b d^4 \tan ^{-1}(c x)}{3 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}+i b c d^4 x^4 \tan ^{-1}(c x)+\frac{1}{3} b d^4 x^3 \tan ^{-1}(c x)-\frac{5}{9} i b d^4 x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]

[Out]

(((5*I)/3)*b*d^4*x)/c^2 - (88*b*d^4*x^2)/(105*c) + (a*d^4*x^3)/3 - ((5*I)/9)*b*d^4*x^3 + I*a*c*d^4*x^4 + (47*b
*c*d^4*x^4)/140 - (6*a*c^2*d^4*x^5)/5 + ((2*I)/15)*b*c^2*d^4*x^5 - ((2*I)/3)*a*c^3*d^4*x^6 - (b*c^3*d^4*x^6)/4
2 + (a*c^4*d^4*x^7)/7 - (((5*I)/3)*b*d^4*ArcTan[c*x])/c^3 + (b*d^4*x^3*ArcTan[c*x])/3 + I*b*c*d^4*x^4*ArcTan[c
*x] - (6*b*c^2*d^4*x^5*ArcTan[c*x])/5 - ((2*I)/3)*b*c^3*d^4*x^6*ArcTan[c*x] + (b*c^4*d^4*x^7*ArcTan[c*x])/7 +
(88*b*d^4*Log[1 + c^2*x^2])/(105*c^3)

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Maple [A]  time = 0.027, size = 237, normalized size = 1.2 \begin{align*}{\frac{{c}^{4}{d}^{4}a{x}^{7}}{7}}+{\frac{2\,i}{15}}b{c}^{2}{d}^{4}{x}^{5}-{\frac{6\,{c}^{2}{d}^{4}a{x}^{5}}{5}}+ic{d}^{4}b\arctan \left ( cx \right ){x}^{4}+{\frac{{d}^{4}a{x}^{3}}{3}}+{\frac{{c}^{4}{d}^{4}b\arctan \left ( cx \right ){x}^{7}}{7}}-{\frac{2\,i}{3}}{c}^{3}{d}^{4}b\arctan \left ( cx \right ){x}^{6}-{\frac{6\,{c}^{2}{d}^{4}b\arctan \left ( cx \right ){x}^{5}}{5}}-{\frac{{\frac{5\,i}{3}}{d}^{4}b\arctan \left ( cx \right ) }{{c}^{3}}}+{\frac{{d}^{4}b\arctan \left ( cx \right ){x}^{3}}{3}}+{\frac{{\frac{5\,i}{3}}{d}^{4}bx}{{c}^{2}}}-{\frac{b{c}^{3}{d}^{4}{x}^{6}}{42}}-{\frac{2\,i}{3}}{c}^{3}{d}^{4}a{x}^{6}+{\frac{47\,bc{d}^{4}{x}^{4}}{140}}+ic{d}^{4}a{x}^{4}-{\frac{88\,{d}^{4}b{x}^{2}}{105\,c}}+{\frac{88\,{d}^{4}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{105\,{c}^{3}}}-{\frac{5\,i}{9}}b{d}^{4}{x}^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x)

[Out]

1/7*c^4*d^4*a*x^7+2/15*I*b*c^2*d^4*x^5-6/5*c^2*d^4*a*x^5+I*c*d^4*b*arctan(c*x)*x^4+1/3*d^4*a*x^3+1/7*c^4*d^4*b
*arctan(c*x)*x^7-2/3*I*c^3*d^4*b*arctan(c*x)*x^6-6/5*c^2*d^4*b*arctan(c*x)*x^5-5/3*I/c^3*d^4*b*arctan(c*x)+1/3
*d^4*b*arctan(c*x)*x^3+5/3*I*b*d^4*x/c^2-1/42*b*c^3*d^4*x^6-2/3*I*c^3*d^4*a*x^6+47/140*b*c*d^4*x^4+I*c*d^4*a*x
^4-88/105*b*d^4*x^2/c+88/105/c^3*d^4*b*ln(c^2*x^2+1)-5/9*I*b*d^4*x^3

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Maxima [B]  time = 1.48581, size = 429, normalized size = 2.22 \begin{align*} \frac{1}{7} \, a c^{4} d^{4} x^{7} - \frac{2}{3} i \, a c^{3} d^{4} x^{6} - \frac{6}{5} \, a c^{2} d^{4} x^{5} + \frac{1}{84} \,{\left (12 \, x^{7} \arctan \left (c x\right ) - c{\left (\frac{2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac{6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{4} d^{4} + i \, a c d^{4} x^{4} - \frac{2}{45} i \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{3} d^{4} - \frac{3}{10} \,{\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^{2} d^{4} + \frac{1}{3} \, a d^{4} x^{3} + \frac{1}{3} i \,{\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 c d^{4} + \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 d^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

1/7*a*c^4*d^4*x^7 - 2/3*I*a*c^3*d^4*x^6 - 6/5*a*c^2*d^4*x^5 + 1/84*(12*x^7*arctan(c*x) - c*((2*c^4*x^6 - 3*c^2
*x^4 + 6*x^2)/c^6 - 6*log(c^2*x^2 + 1)/c^8))*b*c^4*d^4 + I*a*c*d^4*x^4 - 2/45*I*(15*x^6*arctan(c*x) - c*((3*c^
4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*b*c^3*d^4 - 3/10*(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^2*d^4 + 1/3*a*d^4*x^3 + 1/3*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4
+ 3*arctan(c*x)/c^5))*b*c*d^4 + 1/6*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*b*d^4

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Fricas [A]  time = 3.31694, size = 525, normalized size = 2.72 \begin{align*} \frac{180 \, a c^{7} d^{4} x^{7} +{\left (-840 i \, a - 30 \, b\right )} c^{6} d^{4} x^{6} - 168 \,{\left (9 \, a - i \, b\right )} c^{5} d^{4} x^{5} +{\left (1260 i \, a + 423 \, b\right )} c^{4} d^{4} x^{4} + 140 \,{\left (3 \, a - 5 i \, b\right )} c^{3} d^{4} x^{3} - 1056 \, b c^{2} d^{4} x^{2} + 2100 i \, b c d^{4} x + 2106 \, b d^{4} \log \left (\frac{c x + i}{c}\right ) + 6 \, b d^{4} \log \left (\frac{c x - i}{c}\right ) +{\left (90 i \, b c^{7} d^{4} x^{7} + 420 \, b c^{6} d^{4} x^{6} - 756 i \, b c^{5} d^{4} x^{5} - 630 \, b c^{4} d^{4} x^{4} + 210 i \, b c^{3} d^{4} x^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{1260 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

1/1260*(180*a*c^7*d^4*x^7 + (-840*I*a - 30*b)*c^6*d^4*x^6 - 168*(9*a - I*b)*c^5*d^4*x^5 + (1260*I*a + 423*b)*c
^4*d^4*x^4 + 140*(3*a - 5*I*b)*c^3*d^4*x^3 - 1056*b*c^2*d^4*x^2 + 2100*I*b*c*d^4*x + 2106*b*d^4*log((c*x + I)/
c) + 6*b*d^4*log((c*x - I)/c) + (90*I*b*c^7*d^4*x^7 + 420*b*c^6*d^4*x^6 - 756*I*b*c^5*d^4*x^5 - 630*b*c^4*d^4*
x^4 + 210*I*b*c^3*d^4*x^3)*log(-(c*x + I)/(c*x - I)))/c^3

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Sympy [A]  time = 4.20341, size = 325, normalized size = 1.68 \begin{align*} \frac{a c^{4} d^{4} x^{7}}{7} - \frac{88 b d^{4} x^{2}}{105 c} + \frac{5 i b d^{4} x}{3 c^{2}} + \frac{b d^{4} \left (\frac{\log{\left (x - \frac{i}{c} \right )}}{210} + \frac{117 \log{\left (x + \frac{i}{c} \right )}}{70}\right )}{c^{3}} + x^{6} \left (- \frac{2 i a c^{3} d^{4}}{3} - \frac{b c^{3} d^{4}}{42}\right ) + x^{5} \left (- \frac{6 a c^{2} d^{4}}{5} + \frac{2 i b c^{2} d^{4}}{15}\right ) + x^{4} \left (i a c d^{4} + \frac{47 b c d^{4}}{140}\right ) + x^{3} \left (\frac{a d^{4}}{3} - \frac{5 i b d^{4}}{9}\right ) + \left (- \frac{i b c^{4} d^{4} x^{7}}{14} - \frac{b c^{3} d^{4} x^{6}}{3} + \frac{3 i b c^{2} d^{4} x^{5}}{5} + \frac{b c d^{4} x^{4}}{2} - \frac{i b d^{4} x^{3}}{6}\right ) \log{\left (i c x + 1 \right )} + \left (\frac{i b c^{4} d^{4} x^{7}}{14} + \frac{b c^{3} d^{4} x^{6}}{3} - \frac{3 i b c^{2} d^{4} x^{5}}{5} - \frac{b c d^{4} x^{4}}{2} + \frac{i b d^{4} x^{3}}{6}\right ) \log{\left (- i c x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(d+I*c*d*x)**4*(a+b*atan(c*x)),x)

[Out]

a*c**4*d**4*x**7/7 - 88*b*d**4*x**2/(105*c) + 5*I*b*d**4*x/(3*c**2) + b*d**4*(log(x - I/c)/210 + 117*log(x + I
/c)/70)/c**3 + x**6*(-2*I*a*c**3*d**4/3 - b*c**3*d**4/42) + x**5*(-6*a*c**2*d**4/5 + 2*I*b*c**2*d**4/15) + x**
4*(I*a*c*d**4 + 47*b*c*d**4/140) + x**3*(a*d**4/3 - 5*I*b*d**4/9) + (-I*b*c**4*d**4*x**7/14 - b*c**3*d**4*x**6
/3 + 3*I*b*c**2*d**4*x**5/5 + b*c*d**4*x**4/2 - I*b*d**4*x**3/6)*log(I*c*x + 1) + (I*b*c**4*d**4*x**7/14 + b*c
**3*d**4*x**6/3 - 3*I*b*c**2*d**4*x**5/5 - b*c*d**4*x**4/2 + I*b*d**4*x**3/6)*log(-I*c*x + 1)

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Giac [A]  time = 1.20647, size = 333, normalized size = 1.73 \begin{align*} \frac{180 \, b c^{7} d^{4} x^{7} \arctan \left (c x\right ) + 180 \, a c^{7} d^{4} x^{7} - 840 \, b c^{6} d^{4} i x^{6} \arctan \left (c x\right ) - 840 \, a c^{6} d^{4} i x^{6} - 30 \, b c^{6} d^{4} x^{6} + 168 \, b c^{5} d^{4} i x^{5} - 1512 \, b c^{5} d^{4} x^{5} \arctan \left (c x\right ) - 1512 \, a c^{5} d^{4} x^{5} + 1260 \, b c^{4} d^{4} i x^{4} \arctan \left (c x\right ) + 1260 \, a c^{4} d^{4} i x^{4} + 423 \, b c^{4} d^{4} x^{4} - 700 \, b c^{3} d^{4} i x^{3} + 420 \, b c^{3} d^{4} x^{3} \arctan \left (c x\right ) + 420 \, a c^{3} d^{4} x^{3} - 1056 \, b c^{2} d^{4} x^{2} + 2100 \, b c d^{4} i x + 2106 \, b d^{4} \log \left (c x + i\right ) + 6 \, b d^{4} \log \left (c x - i\right )}{1260 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

1/1260*(180*b*c^7*d^4*x^7*arctan(c*x) + 180*a*c^7*d^4*x^7 - 840*b*c^6*d^4*i*x^6*arctan(c*x) - 840*a*c^6*d^4*i*
x^6 - 30*b*c^6*d^4*x^6 + 168*b*c^5*d^4*i*x^5 - 1512*b*c^5*d^4*x^5*arctan(c*x) - 1512*a*c^5*d^4*x^5 + 1260*b*c^
4*d^4*i*x^4*arctan(c*x) + 1260*a*c^4*d^4*i*x^4 + 423*b*c^4*d^4*x^4 - 700*b*c^3*d^4*i*x^3 + 420*b*c^3*d^4*x^3*a
rctan(c*x) + 420*a*c^3*d^4*x^3 - 1056*b*c^2*d^4*x^2 + 2100*b*c*d^4*i*x + 2106*b*d^4*log(c*x + i) + 6*b*d^4*log
(c*x - i))/c^3