3.30 \(\int \frac{(d+i c d x)^3 (a+b \tan ^{-1}(c x))}{x^7} \, dx\)

Optimal. Leaf size=214 \[ \frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{7 i b c^4 d^3}{15 x^2}+\frac{11 b c^3 d^3}{36 x^3}-\frac{3 i b c^2 d^3}{20 x^4}-\frac{11 b c^5 d^3}{12 x}+\frac{14}{15} i b c^6 d^3 \log (x)-\frac{1}{120} i b c^6 d^3 \log (-c x+i)-\frac{37}{40} i b c^6 d^3 \log (c x+i)-\frac{b c d^3}{30 x^5} \]

[Out]

-(b*c*d^3)/(30*x^5) - (((3*I)/20)*b*c^2*d^3)/x^4 + (11*b*c^3*d^3)/(36*x^3) + (((7*I)/15)*b*c^4*d^3)/x^2 - (11*
b*c^5*d^3)/(12*x) - (d^3*(a + b*ArcTan[c*x]))/(6*x^6) - (((3*I)/5)*c*d^3*(a + b*ArcTan[c*x]))/x^5 + (3*c^2*d^3
*(a + b*ArcTan[c*x]))/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x]))/x^3 + ((14*I)/15)*b*c^6*d^3*Log[x] - (I/12
0)*b*c^6*d^3*Log[I - c*x] - ((37*I)/40)*b*c^6*d^3*Log[I + c*x]

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Rubi [A]  time = 0.178493, antiderivative size = 214, 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, 1802} \[ \frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{7 i b c^4 d^3}{15 x^2}+\frac{11 b c^3 d^3}{36 x^3}-\frac{3 i b c^2 d^3}{20 x^4}-\frac{11 b c^5 d^3}{12 x}+\frac{14}{15} i b c^6 d^3 \log (x)-\frac{1}{120} i b c^6 d^3 \log (-c x+i)-\frac{37}{40} i b c^6 d^3 \log (c x+i)-\frac{b c d^3}{30 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^7,x]

[Out]

-(b*c*d^3)/(30*x^5) - (((3*I)/20)*b*c^2*d^3)/x^4 + (11*b*c^3*d^3)/(36*x^3) + (((7*I)/15)*b*c^4*d^3)/x^2 - (11*
b*c^5*d^3)/(12*x) - (d^3*(a + b*ArcTan[c*x]))/(6*x^6) - (((3*I)/5)*c*d^3*(a + b*ArcTan[c*x]))/x^5 + (3*c^2*d^3
*(a + b*ArcTan[c*x]))/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x]))/x^3 + ((14*I)/15)*b*c^6*d^3*Log[x] - (I/12
0)*b*c^6*d^3*Log[I - c*x] - ((37*I)/40)*b*c^6*d^3*Log[I + c*x]

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 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \frac{(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )}{x^7} \, dx &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{d^3 \left (-10-36 i c x+45 c^2 x^2+20 i c^3 x^3\right )}{60 x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{60} \left (b c d^3\right ) \int \frac{-10-36 i c x+45 c^2 x^2+20 i c^3 x^3}{x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{60} \left (b c d^3\right ) \int \left (-\frac{10}{x^6}-\frac{36 i c}{x^5}+\frac{55 c^2}{x^4}+\frac{56 i c^3}{x^3}-\frac{55 c^4}{x^2}-\frac{56 i c^5}{x}+\frac{i c^6}{2 (-i+c x)}+\frac{111 i c^6}{2 (i+c x)}\right ) \, dx\\ &=-\frac{b c d^3}{30 x^5}-\frac{3 i b c^2 d^3}{20 x^4}+\frac{11 b c^3 d^3}{36 x^3}+\frac{7 i b c^4 d^3}{15 x^2}-\frac{11 b c^5 d^3}{12 x}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{14}{15} i b c^6 d^3 \log (x)-\frac{1}{120} i b c^6 d^3 \log (i-c x)-\frac{37}{40} i b c^6 d^3 \log (i+c x)\\ \end{align*}

Mathematica [C]  time = 0.111934, size = 188, normalized size = 0.88 \[ \frac{d^3 \left (-2 b c x \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-c^2 x^2\right )+i \left (-15 i b c^3 x^3 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-c^2 x^2\right )+20 a c^3 x^3-45 i a c^2 x^2-36 a c x+10 i a+28 b c^4 x^4-9 b c^2 x^2+56 b c^6 x^6 \log (x)-28 b c^6 x^6 \log \left (c^2 x^2+1\right )+20 b c^3 x^3 \tan ^{-1}(c x)-45 i b c^2 x^2 \tan ^{-1}(c x)-36 b c x \tan ^{-1}(c x)+10 i b \tan ^{-1}(c x)\right )\right )}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^7,x]

[Out]

(d^3*(-2*b*c*x*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)] + I*((10*I)*a - 36*a*c*x - (45*I)*a*c^2*x^2 - 9*b*
c^2*x^2 + 20*a*c^3*x^3 + 28*b*c^4*x^4 + (10*I)*b*ArcTan[c*x] - 36*b*c*x*ArcTan[c*x] - (45*I)*b*c^2*x^2*ArcTan[
c*x] + 20*b*c^3*x^3*ArcTan[c*x] - (15*I)*b*c^3*x^3*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 56*b*c^6*x^6
*Log[x] - 28*b*c^6*x^6*Log[1 + c^2*x^2])))/(60*x^6)

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Maple [A]  time = 0.039, size = 215, normalized size = 1. \begin{align*}{\frac{3\,{c}^{2}{d}^{3}a}{4\,{x}^{4}}}-{\frac{{d}^{3}a}{6\,{x}^{6}}}+{\frac{14\,i}{15}}{c}^{6}{d}^{3}b\ln \left ( cx \right ) +{\frac{{\frac{i}{3}}{c}^{3}{d}^{3}a}{{x}^{3}}}+{\frac{3\,b{c}^{2}{d}^{3}\arctan \left ( cx \right ) }{4\,{x}^{4}}}-{\frac{b{d}^{3}\arctan \left ( cx \right ) }{6\,{x}^{6}}}+{\frac{{\frac{7\,i}{15}}b{c}^{4}{d}^{3}}{{x}^{2}}}-{\frac{7\,i}{15}}{c}^{6}{d}^{3}b\ln \left ({c}^{2}{x}^{2}+1 \right ) -{\frac{{\frac{3\,i}{5}}c{d}^{3}b\arctan \left ( cx \right ) }{{x}^{5}}}-{\frac{11\,b{c}^{6}{d}^{3}\arctan \left ( cx \right ) }{12}}-{\frac{{\frac{3\,i}{5}}c{d}^{3}a}{{x}^{5}}}+{\frac{{\frac{i}{3}}{c}^{3}{d}^{3}b\arctan \left ( cx \right ) }{{x}^{3}}}-{\frac{{\frac{3\,i}{20}}b{c}^{2}{d}^{3}}{{x}^{4}}}-{\frac{bc{d}^{3}}{30\,{x}^{5}}}+{\frac{11\,b{c}^{3}{d}^{3}}{36\,{x}^{3}}}-{\frac{11\,b{c}^{5}{d}^{3}}{12\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^7,x)

[Out]

3/4*c^2*d^3*a/x^4-1/6*d^3*a/x^6+14/15*I*c^6*d^3*b*ln(c*x)+1/3*I*c^3*d^3*a/x^3+3/4*c^2*d^3*b*arctan(c*x)/x^4-1/
6*d^3*b*arctan(c*x)/x^6+7/15*I*b*c^4*d^3/x^2-7/15*I*c^6*d^3*b*ln(c^2*x^2+1)-3/5*I*c*d^3*b*arctan(c*x)/x^5-11/1
2*b*c^6*d^3*arctan(c*x)-3/5*I*c*d^3*a/x^5+1/3*I*c^3*d^3*b*arctan(c*x)/x^3-3/20*I*b*c^2*d^3/x^4-1/30*b*c*d^3/x^
5+11/36*b*c^3*d^3/x^3-11/12*b*c^5*d^3/x

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Maxima [A]  time = 1.49242, size = 335, normalized size = 1.57 \begin{align*} -\frac{1}{6} i \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} c - \frac{2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{3} d^{3} - \frac{1}{4} \,{\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac{3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac{3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c^{2} d^{3} - \frac{3}{20} i \,{\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac{2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac{4 \, \arctan \left (c x\right )}{x^{5}}\right )} b c d^{3} - \frac{1}{90} \,{\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac{15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac{15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d^{3} + \frac{i \, a c^{3} d^{3}}{3 \, x^{3}} + \frac{3 \, a c^{2} d^{3}}{4 \, x^{4}} - \frac{3 i \, a c d^{3}}{5 \, x^{5}} - \frac{a d^{3}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^7,x, algorithm="maxima")

[Out]

-1/6*I*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*b*c^3*d^3 - 1/4*((3*c^3*arctan(c*
x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x)/x^4)*b*c^2*d^3 - 3/20*I*((2*c^4*log(c^2*x^2 + 1) - 2*c^4*log(x^2)
- (2*c^2*x^2 - 1)/x^4)*c + 4*arctan(c*x)/x^5)*b*c*d^3 - 1/90*((15*c^5*arctan(c*x) + (15*c^4*x^4 - 5*c^2*x^2 +
3)/x^5)*c + 15*arctan(c*x)/x^6)*b*d^3 + 1/3*I*a*c^3*d^3/x^3 + 3/4*a*c^2*d^3/x^4 - 3/5*I*a*c*d^3/x^5 - 1/6*a*d^
3/x^6

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Fricas [A]  time = 3.16693, size = 481, normalized size = 2.25 \begin{align*} \frac{336 i \, b c^{6} d^{3} x^{6} \log \left (x\right ) - 333 i \, b c^{6} d^{3} x^{6} \log \left (\frac{c x + i}{c}\right ) - 3 i \, b c^{6} d^{3} x^{6} \log \left (\frac{c x - i}{c}\right ) - 330 \, b c^{5} d^{3} x^{5} + 168 i \, b c^{4} d^{3} x^{4} +{\left (120 i \, a + 110 \, b\right )} c^{3} d^{3} x^{3} + 54 \,{\left (5 \, a - i \, b\right )} c^{2} d^{3} x^{2} +{\left (-216 i \, a - 12 \, b\right )} c d^{3} x - 60 \, a d^{3} -{\left (60 \, b c^{3} d^{3} x^{3} - 135 i \, b c^{2} d^{3} x^{2} - 108 \, b c d^{3} x + 30 i \, b d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{360 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^7,x, algorithm="fricas")

[Out]

1/360*(336*I*b*c^6*d^3*x^6*log(x) - 333*I*b*c^6*d^3*x^6*log((c*x + I)/c) - 3*I*b*c^6*d^3*x^6*log((c*x - I)/c)
- 330*b*c^5*d^3*x^5 + 168*I*b*c^4*d^3*x^4 + (120*I*a + 110*b)*c^3*d^3*x^3 + 54*(5*a - I*b)*c^2*d^3*x^2 + (-216
*I*a - 12*b)*c*d^3*x - 60*a*d^3 - (60*b*c^3*d^3*x^3 - 135*I*b*c^2*d^3*x^2 - 108*b*c*d^3*x + 30*I*b*d^3)*log(-(
c*x + I)/(c*x - I)))/x^6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)**3*(a+b*atan(c*x))/x**7,x)

[Out]

Timed out

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Giac [A]  time = 1.69019, size = 292, normalized size = 1.36 \begin{align*} -\frac{3 \, b c^{6} d^{3} i x^{6} \log \left (c i x + 1\right ) + 333 \, b c^{6} d^{3} i x^{6} \log \left (-c i x + 1\right ) - 336 \, b c^{6} d^{3} i x^{6} \log \left (x\right ) + 330 \, b c^{5} d^{3} x^{5} - 168 \, b c^{4} d^{3} i x^{4} - 120 \, b c^{3} d^{3} i x^{3} \arctan \left (c x\right ) - 120 \, a c^{3} d^{3} i x^{3} - 110 \, b c^{3} d^{3} x^{3} + 54 \, b c^{2} d^{3} i x^{2} - 270 \, b c^{2} d^{3} x^{2} \arctan \left (c x\right ) - 270 \, a c^{2} d^{3} x^{2} + 216 \, b c d^{3} i x \arctan \left (c x\right ) + 216 \, a c d^{3} i x + 12 \, b c d^{3} x + 60 \, b d^{3} \arctan \left (c x\right ) + 60 \, a d^{3}}{360 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^7,x, algorithm="giac")

[Out]

-1/360*(3*b*c^6*d^3*i*x^6*log(c*i*x + 1) + 333*b*c^6*d^3*i*x^6*log(-c*i*x + 1) - 336*b*c^6*d^3*i*x^6*log(x) +
330*b*c^5*d^3*x^5 - 168*b*c^4*d^3*i*x^4 - 120*b*c^3*d^3*i*x^3*arctan(c*x) - 120*a*c^3*d^3*i*x^3 - 110*b*c^3*d^
3*x^3 + 54*b*c^2*d^3*i*x^2 - 270*b*c^2*d^3*x^2*arctan(c*x) - 270*a*c^2*d^3*x^2 + 216*b*c*d^3*i*x*arctan(c*x) +
216*a*c*d^3*i*x + 12*b*c*d^3*x + 60*b*d^3*arctan(c*x) + 60*a*d^3)/x^6