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

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

[Out]

-1/30*b*c*d^3/x^5-3/20*I*b*c^2*d^3/x^4+11/36*b*c^3*d^3/x^3+7/15*I*b*c^4*d^3/x^2-11/12*b*c^5*d^3/x-1/6*d^3*(a+b
*arctan(c*x))/x^6-3/5*I*c*d^3*(a+b*arctan(c*x))/x^5+3/4*c^2*d^3*(a+b*arctan(c*x))/x^4+1/3*I*c^3*d^3*(a+b*arcta
n(c*x))/x^3+14/15*I*b*c^6*d^3*ln(x)-1/120*I*b*c^6*d^3*ln(I-c*x)-37/40*I*b*c^6*d^3*ln(I+c*x)

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Rubi [A]  time = 0.18, antiderivative size = 214, normalized size of antiderivative = 1.00, 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 12

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

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]))

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*}

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Mathematica [C]  time = 0.12, size = 188, normalized size = 0.88 \[ \frac {d^3 \left (-2 b c x \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-c^2 x^2\right )+i \left (20 a c^3 x^3-45 i a c^2 x^2-36 a c x+10 i a+56 b c^6 x^6 \log (x)+28 b c^4 x^4+20 b c^3 x^3 \tan ^{-1}(c x)-9 b c^2 x^2-45 i b c^2 x^2 \tan ^{-1}(c x)-28 b c^6 x^6 \log \left (c^2 x^2+1\right )-15 i b c^3 x^3 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )-36 b c x \tan ^{-1}(c x)+10 i b \tan ^{-1}(c x)\right )\right )}{60 x^6} \]

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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fricas [A]  time = 0.55, size = 198, normalized size = 0.93 \[ \frac {336 i \, b c^{6} d^{3} x^{6} \log \relax (x) - 333 i \, b c^{6} d^{3} x^{6} \log \left (\frac {c x + i}{c}\right ) - 3 i \, b c^{6} d^{3} x^{6} \log \left (\frac {c x - i}{c}\right ) - 330 \, b c^{5} d^{3} x^{5} + 168 i \, b c^{4} d^{3} x^{4} + {\left (120 i \, a + 110 \, b\right )} c^{3} d^{3} x^{3} + 54 \, {\left (5 \, a - i \, b\right )} c^{2} d^{3} x^{2} + {\left (-216 i \, a - 12 \, b\right )} c d^{3} x - 60 \, a d^{3} - {\left (60 \, b c^{3} d^{3} x^{3} - 135 i \, b c^{2} d^{3} x^{2} - 108 \, b c d^{3} x + 30 i \, b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{360 \, x^{6}} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [A]  time = 0.05, size = 215, normalized size = 1.00 \[ -\frac {7 i c^{6} d^{3} b \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {3 c^{2} d^{3} a}{4 x^{4}}-\frac {3 i c \,d^{3} b \arctan \left (c x \right )}{5 x^{5}}-\frac {d^{3} a}{6 x^{6}}+\frac {i c^{3} d^{3} b \arctan \left (c x \right )}{3 x^{3}}+\frac {3 c^{2} d^{3} b \arctan \left (c x \right )}{4 x^{4}}+\frac {i c^{3} d^{3} a}{3 x^{3}}-\frac {d^{3} b \arctan \left (c x \right )}{6 x^{6}}-\frac {3 i b \,c^{2} d^{3}}{20 x^{4}}+\frac {14 i c^{6} d^{3} b \ln \left (c x \right )}{15}+\frac {7 i b \,c^{4} d^{3}}{15 x^{2}}-\frac {b c \,d^{3}}{30 x^{5}}+\frac {11 b \,c^{3} d^{3}}{36 x^{3}}-\frac {11 b \,c^{5} d^{3}}{12 x}-\frac {3 i c \,d^{3} a}{5 x^{5}}-\frac {11 b \,c^{6} d^{3} \arctan \left (c x \right )}{12} \]

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]

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

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maxima [A]  time = 0.41, size = 248, normalized size = 1.16 \[ -\frac {1}{6} i \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{3} d^{3} - \frac {1}{4} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c^{2} d^{3} - \frac {3}{20} i \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b c d^{3} - \frac {1}{90} \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d^{3} + \frac {i \, a c^{3} d^{3}}{3 \, x^{3}} + \frac {3 \, a c^{2} d^{3}}{4 \, x^{4}} - \frac {3 i \, a c d^{3}}{5 \, x^{5}} - \frac {a d^{3}}{6 \, x^{6}} \]

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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mupad [B]  time = 1.06, size = 192, normalized size = 0.90 \[ -\frac {\frac {d^3\,\left (30\,a+30\,b\,\mathrm {atan}\left (c\,x\right )\right )}{180}+\frac {d^3\,x\,\left (a\,c\,108{}\mathrm {i}+6\,b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,108{}\mathrm {i}\right )}{180}-\frac {d^3\,x^3\,\left (a\,c^3\,60{}\mathrm {i}+55\,b\,c^3+b\,c^3\,\mathrm {atan}\left (c\,x\right )\,60{}\mathrm {i}\right )}{180}-\frac {d^3\,x^2\,\left (135\,a\,c^2+135\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,27{}\mathrm {i}\right )}{180}+\frac {11\,b\,c^5\,d^3\,x^5}{12}-\frac {b\,c^4\,d^3\,x^4\,7{}\mathrm {i}}{15}}{x^6}-\frac {d^3\,\left (\frac {165\,b\,c^9\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )}{{\left (c^2\right )}^{3/2}}+b\,c^6\,\ln \left (c^2\,x^2+1\right )\,84{}\mathrm {i}-b\,c^6\,\ln \relax (x)\,168{}\mathrm {i}\right )}{180} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atan(c*x))*(d + c*d*x*1i)^3)/x^7,x)

[Out]

- ((d^3*(30*a + 30*b*atan(c*x)))/180 + (d^3*x*(a*c*108i + 6*b*c + b*c*atan(c*x)*108i))/180 - (d^3*x^3*(a*c^3*6
0i + 55*b*c^3 + b*c^3*atan(c*x)*60i))/180 - (d^3*x^2*(135*a*c^2 - b*c^2*27i + 135*b*c^2*atan(c*x)))/180 - (b*c
^4*d^3*x^4*7i)/15 + (11*b*c^5*d^3*x^5)/12)/x^6 - (d^3*(b*c^6*log(c^2*x^2 + 1)*84i - b*c^6*log(x)*168i + (165*b
*c^9*atan((c^2*x)/(c^2)^(1/2)))/(c^2)^(3/2)))/180

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sympy [A]  time = 84.08, size = 347, normalized size = 1.62 \[ \frac {14 i b c^{6} d^{3} \log {\left (1385945 b^{2} c^{13} d^{6} x \right )}}{15} - \frac {i b c^{6} d^{3} \log {\left (1385945 b^{2} c^{13} d^{6} x - 1385945 i b^{2} c^{12} d^{6} \right )}}{120} - \frac {37 i b c^{6} d^{3} \log {\left (1385945 b^{2} c^{13} d^{6} x + 1385945 i b^{2} c^{12} d^{6} \right )}}{40} + \frac {\left (- 20 b c^{3} d^{3} x^{3} + 45 i b c^{2} d^{3} x^{2} + 36 b c d^{3} x - 10 i b d^{3}\right ) \log {\left (- i c x + 1 \right )}}{120 x^{6}} + \frac {\left (20 b c^{3} d^{3} x^{3} - 45 i b c^{2} d^{3} x^{2} - 36 b c d^{3} x + 10 i b d^{3}\right ) \log {\left (i c x + 1 \right )}}{120 x^{6}} - \frac {30 a d^{3} + 165 b c^{5} d^{3} x^{5} - 84 i b c^{4} d^{3} x^{4} + x^{3} \left (- 60 i a c^{3} d^{3} - 55 b c^{3} d^{3}\right ) + x^{2} \left (- 135 a c^{2} d^{3} + 27 i b c^{2} d^{3}\right ) + x \left (108 i a c d^{3} + 6 b c d^{3}\right )}{180 x^{6}} \]

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]

14*I*b*c**6*d**3*log(1385945*b**2*c**13*d**6*x)/15 - I*b*c**6*d**3*log(1385945*b**2*c**13*d**6*x - 1385945*I*b
**2*c**12*d**6)/120 - 37*I*b*c**6*d**3*log(1385945*b**2*c**13*d**6*x + 1385945*I*b**2*c**12*d**6)/40 + (-20*b*
c**3*d**3*x**3 + 45*I*b*c**2*d**3*x**2 + 36*b*c*d**3*x - 10*I*b*d**3)*log(-I*c*x + 1)/(120*x**6) + (20*b*c**3*
d**3*x**3 - 45*I*b*c**2*d**3*x**2 - 36*b*c*d**3*x + 10*I*b*d**3)*log(I*c*x + 1)/(120*x**6) - (30*a*d**3 + 165*
b*c**5*d**3*x**5 - 84*I*b*c**4*d**3*x**4 + x**3*(-60*I*a*c**3*d**3 - 55*b*c**3*d**3) + x**2*(-135*a*c**2*d**3
+ 27*I*b*c**2*d**3) + x*(108*I*a*c*d**3 + 6*b*c*d**3))/(180*x**6)

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