3.42 \(\int \frac {(d+i c d x)^4 (a+b \tan ^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=243 \[ -\frac {c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}+\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac {2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^6}-\frac {176}{105} b c^7 d^4 \log (x)+\frac {1}{210} b c^7 d^4 \log (-c x+i)+\frac {117}{70} b c^7 d^4 \log (c x+i)-\frac {5 i b c^6 d^4}{3 x}-\frac {88 b c^5 d^4}{105 x^2}+\frac {5 i b c^4 d^4}{9 x^3}+\frac {47 b c^3 d^4}{140 x^4}-\frac {2 i b c^2 d^4}{15 x^5}-\frac {b c d^4}{42 x^6} \]

[Out]

-1/42*b*c*d^4/x^6-2/15*I*b*c^2*d^4/x^5+47/140*b*c^3*d^4/x^4+5/9*I*b*c^4*d^4/x^3-88/105*b*c^5*d^4/x^2-5/3*I*b*c
^6*d^4/x-1/7*d^4*(a+b*arctan(c*x))/x^7-2/3*I*c*d^4*(a+b*arctan(c*x))/x^6+6/5*c^2*d^4*(a+b*arctan(c*x))/x^5+I*c
^3*d^4*(a+b*arctan(c*x))/x^4-1/3*c^4*d^4*(a+b*arctan(c*x))/x^3-176/105*b*c^7*d^4*ln(x)+1/210*b*c^7*d^4*ln(I-c*
x)+117/70*b*c^7*d^4*ln(I+c*x)

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Rubi [A]  time = 0.20, antiderivative size = 243, 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} \[ -\frac {c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}+\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac {2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^6}-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac {88 b c^5 d^4}{105 x^2}+\frac {5 i b c^4 d^4}{9 x^3}+\frac {47 b c^3 d^4}{140 x^4}-\frac {2 i b c^2 d^4}{15 x^5}-\frac {5 i b c^6 d^4}{3 x}-\frac {176}{105} b c^7 d^4 \log (x)+\frac {1}{210} b c^7 d^4 \log (-c x+i)+\frac {117}{70} b c^7 d^4 \log (c x+i)-\frac {b c d^4}{42 x^6} \]

Antiderivative was successfully verified.

[In]

Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^8,x]

[Out]

-(b*c*d^4)/(42*x^6) - (((2*I)/15)*b*c^2*d^4)/x^5 + (47*b*c^3*d^4)/(140*x^4) + (((5*I)/9)*b*c^4*d^4)/x^3 - (88*
b*c^5*d^4)/(105*x^2) - (((5*I)/3)*b*c^6*d^4)/x - (d^4*(a + b*ArcTan[c*x]))/(7*x^7) - (((2*I)/3)*c*d^4*(a + b*A
rcTan[c*x]))/x^6 + (6*c^2*d^4*(a + b*ArcTan[c*x]))/(5*x^5) + (I*c^3*d^4*(a + b*ArcTan[c*x]))/x^4 - (c^4*d^4*(a
+ b*ArcTan[c*x]))/(3*x^3) - (176*b*c^7*d^4*Log[x])/105 + (b*c^7*d^4*Log[I - c*x])/210 + (117*b*c^7*d^4*Log[I
+ c*x])/70

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)^4 \left (a+b \tan ^{-1}(c x)\right )}{x^8} \, dx &=-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac {2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^6}+\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac {i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d^4 \left (-15-70 i c x+126 c^2 x^2+105 i c^3 x^3-35 c^4 x^4\right )}{105 x^7 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac {2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^6}+\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac {i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{105} \left (b c d^4\right ) \int \frac {-15-70 i c x+126 c^2 x^2+105 i c^3 x^3-35 c^4 x^4}{x^7 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac {2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^6}+\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac {i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{105} \left (b c d^4\right ) \int \left (-\frac {15}{x^7}-\frac {70 i c}{x^6}+\frac {141 c^2}{x^5}+\frac {175 i c^3}{x^4}-\frac {176 c^4}{x^3}-\frac {175 i c^5}{x^2}+\frac {176 c^6}{x}-\frac {c^7}{2 (-i+c x)}-\frac {351 c^7}{2 (i+c x)}\right ) \, dx\\ &=-\frac {b c d^4}{42 x^6}-\frac {2 i b c^2 d^4}{15 x^5}+\frac {47 b c^3 d^4}{140 x^4}+\frac {5 i b c^4 d^4}{9 x^3}-\frac {88 b c^5 d^4}{105 x^2}-\frac {5 i b c^6 d^4}{3 x}-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac {2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^6}+\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac {i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {176}{105} b c^7 d^4 \log (x)+\frac {1}{210} b c^7 d^4 \log (i-c x)+\frac {117}{70} b c^7 d^4 \log (i+c x)\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 293, normalized size = 1.21 \[ -\frac {a c^4 d^4}{3 x^3}+\frac {i a c^3 d^4}{x^4}+\frac {6 a c^2 d^4}{5 x^5}-\frac {2 i a c d^4}{3 x^6}-\frac {a d^4}{7 x^7}-\frac {176}{105} b c^7 d^4 \log (x)-\frac {88 b c^5 d^4}{105 x^2}-\frac {b c^4 d^4 \tan ^{-1}(c x)}{3 x^3}+\frac {47 b c^3 d^4}{140 x^4}+\frac {i b c^3 d^4 \tan ^{-1}(c x)}{x^4}-\frac {2 i b c^2 d^4 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-c^2 x^2\right )}{15 x^5}+\frac {6 b c^2 d^4 \tan ^{-1}(c x)}{5 x^5}+\frac {88}{105} b c^7 d^4 \log \left (c^2 x^2+1\right )+\frac {i b c^4 d^4 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )}{3 x^3}-\frac {b d^4 \tan ^{-1}(c x)}{7 x^7}-\frac {b c d^4}{42 x^6}-\frac {2 i b c d^4 \tan ^{-1}(c x)}{3 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^8,x]

[Out]

-1/7*(a*d^4)/x^7 - (((2*I)/3)*a*c*d^4)/x^6 - (b*c*d^4)/(42*x^6) + (6*a*c^2*d^4)/(5*x^5) + (I*a*c^3*d^4)/x^4 +
(47*b*c^3*d^4)/(140*x^4) - (a*c^4*d^4)/(3*x^3) - (88*b*c^5*d^4)/(105*x^2) - (b*d^4*ArcTan[c*x])/(7*x^7) - (((2
*I)/3)*b*c*d^4*ArcTan[c*x])/x^6 + (6*b*c^2*d^4*ArcTan[c*x])/(5*x^5) + (I*b*c^3*d^4*ArcTan[c*x])/x^4 - (b*c^4*d
^4*ArcTan[c*x])/(3*x^3) - (((2*I)/15)*b*c^2*d^4*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)])/x^5 + ((I/3)*b*c
^4*d^4*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/x^3 - (176*b*c^7*d^4*Log[x])/105 + (88*b*c^7*d^4*Log[1 +
c^2*x^2])/105

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fricas [A]  time = 0.54, size = 230, normalized size = 0.95 \[ -\frac {2112 \, b c^{7} d^{4} x^{7} \log \relax (x) - 2106 \, b c^{7} d^{4} x^{7} \log \left (\frac {c x + i}{c}\right ) - 6 \, b c^{7} d^{4} x^{7} \log \left (\frac {c x - i}{c}\right ) + 2100 i \, b c^{6} d^{4} x^{6} + 1056 \, b c^{5} d^{4} x^{5} + 140 \, {\left (3 \, a - 5 i \, b\right )} c^{4} d^{4} x^{4} - {\left (1260 i \, a + 423 \, b\right )} c^{3} d^{4} x^{3} - 168 \, {\left (9 \, a - i \, b\right )} c^{2} d^{4} x^{2} - {\left (-840 i \, a - 30 \, b\right )} c d^{4} x + 180 \, a d^{4} - {\left (-210 i \, b c^{4} d^{4} x^{4} - 630 \, b c^{3} d^{4} x^{3} + 756 i \, b c^{2} d^{4} x^{2} + 420 \, b c d^{4} x - 90 i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{1260 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/1260*(2112*b*c^7*d^4*x^7*log(x) - 2106*b*c^7*d^4*x^7*log((c*x + I)/c) - 6*b*c^7*d^4*x^7*log((c*x - I)/c) +
2100*I*b*c^6*d^4*x^6 + 1056*b*c^5*d^4*x^5 + 140*(3*a - 5*I*b)*c^4*d^4*x^4 - (1260*I*a + 423*b)*c^3*d^4*x^3 - 1
68*(9*a - I*b)*c^2*d^4*x^2 - (-840*I*a - 30*b)*c*d^4*x + 180*a*d^4 - (-210*I*b*c^4*d^4*x^4 - 630*b*c^3*d^4*x^3
+ 756*I*b*c^2*d^4*x^2 + 420*b*c*d^4*x - 90*I*b*d^4)*log(-(c*x + I)/(c*x - I)))/x^7

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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)^4*(a+b*arctan(c*x))/x^8,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 255, normalized size = 1.05 \[ -\frac {d^{4} a}{7 x^{7}}-\frac {c^{4} d^{4} a}{3 x^{3}}+\frac {i c^{3} d^{4} b \arctan \left (c x \right )}{x^{4}}+\frac {6 c^{2} d^{4} a}{5 x^{5}}-\frac {5 i c^{7} d^{4} b \arctan \left (c x \right )}{3}-\frac {d^{4} b \arctan \left (c x \right )}{7 x^{7}}-\frac {c^{4} d^{4} b \arctan \left (c x \right )}{3 x^{3}}+\frac {i c^{3} d^{4} a}{x^{4}}+\frac {6 c^{2} d^{4} b \arctan \left (c x \right )}{5 x^{5}}-\frac {2 i c \,d^{4} b \arctan \left (c x \right )}{3 x^{6}}+\frac {5 i b \,c^{4} d^{4}}{9 x^{3}}-\frac {2 i c \,d^{4} a}{3 x^{6}}-\frac {2 i b \,c^{2} d^{4}}{15 x^{5}}-\frac {b c \,d^{4}}{42 x^{6}}+\frac {47 b \,c^{3} d^{4}}{140 x^{4}}-\frac {88 b \,c^{5} d^{4}}{105 x^{2}}-\frac {176 c^{7} d^{4} b \ln \left (c x \right )}{105}+\frac {88 c^{7} d^{4} b \ln \left (c^{2} x^{2}+1\right )}{105}-\frac {5 i b \,c^{6} d^{4}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^8,x)

[Out]

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

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maxima [A]  time = 0.41, size = 329, normalized size = 1.35 \[ \frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{4} d^{4} - \frac {1}{3} i \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c^{3} d^{4} + \frac {3}{10} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b c^{2} d^{4} - \frac {2}{45} i \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b c d^{4} + \frac {1}{84} \, {\left ({\left (6 \, c^{6} \log \left (c^{2} x^{2} + 1\right ) - 6 \, c^{6} \log \left (x^{2}\right ) - \frac {6 \, c^{4} x^{4} - 3 \, c^{2} x^{2} + 2}{x^{6}}\right )} c - \frac {12 \, \arctan \left (c x\right )}{x^{7}}\right )} b d^{4} - \frac {a c^{4} d^{4}}{3 \, x^{3}} + \frac {i \, a c^{3} d^{4}}{x^{4}} + \frac {6 \, a c^{2} d^{4}}{5 \, x^{5}} - \frac {2 i \, a c d^{4}}{3 \, x^{6}} - \frac {a d^{4}}{7 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.17, size = 317, normalized size = 1.30 \[ \frac {88\,b\,c^7\,d^4\,\ln \left (c^2\,x^2+1\right )}{105}-\frac {\frac {a\,d^4}{7}+\frac {b\,d^4\,\mathrm {atan}\left (c\,x\right )}{7}+\frac {88\,b\,c^7\,d^4\,x^7}{105}+\frac {b\,c^8\,d^4\,x^8\,5{}\mathrm {i}}{3}+\frac {c\,d^4\,x\,\left (b+a\,28{}\mathrm {i}\right )}{42}+\frac {c^6\,d^4\,x^6\,\left (3\,a+b\,10{}\mathrm {i}\right )}{9}-\frac {c^4\,d^4\,x^4\,\left (39\,a+b\,19{}\mathrm {i}\right )}{45}-\frac {c^2\,d^4\,x^2\,\left (111\,a-b\,14{}\mathrm {i}\right )}{105}-\frac {c^3\,d^4\,x^3\,\left (131\,b+a\,140{}\mathrm {i}\right )}{420}-c^5\,d^4\,x^5\,\left (-\frac {211\,b}{420}+a\,1{}\mathrm {i}\right )-\frac {37\,b\,c^2\,d^4\,x^2\,\mathrm {atan}\left (c\,x\right )}{35}-\frac {b\,c^3\,d^4\,x^3\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i}}{3}-\frac {13\,b\,c^4\,d^4\,x^4\,\mathrm {atan}\left (c\,x\right )}{15}-b\,c^5\,d^4\,x^5\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i}+\frac {b\,c^6\,d^4\,x^6\,\mathrm {atan}\left (c\,x\right )}{3}+\frac {b\,c\,d^4\,x\,\mathrm {atan}\left (c\,x\right )\,2{}\mathrm {i}}{3}}{c^2\,x^9+x^7}-\frac {176\,b\,c^7\,d^4\,\ln \relax (x)}{105}-\frac {b\,c^{10}\,d^4\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )\,5{}\mathrm {i}}{3\,{\left (c^2\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atan(c*x))*(d + c*d*x*1i)^4)/x^8,x)

[Out]

(88*b*c^7*d^4*log(c^2*x^2 + 1))/105 - ((a*d^4)/7 + (b*d^4*atan(c*x))/7 + (88*b*c^7*d^4*x^7)/105 + (b*c^8*d^4*x
^8*5i)/3 + (c*d^4*x*(a*28i + b))/42 + (c^6*d^4*x^6*(3*a + b*10i))/9 - (c^4*d^4*x^4*(39*a + b*19i))/45 - (c^2*d
^4*x^2*(111*a - b*14i))/105 - (c^3*d^4*x^3*(a*140i + 131*b))/420 - c^5*d^4*x^5*(a*1i - (211*b)/420) - (37*b*c^
2*d^4*x^2*atan(c*x))/35 - (b*c^3*d^4*x^3*atan(c*x)*1i)/3 - (13*b*c^4*d^4*x^4*atan(c*x))/15 - b*c^5*d^4*x^5*ata
n(c*x)*1i + (b*c^6*d^4*x^6*atan(c*x))/3 + (b*c*d^4*x*atan(c*x)*2i)/3)/(x^7 + c^2*x^9) - (176*b*c^7*d^4*log(x))
/105 - (b*c^10*d^4*atan((c^2*x)/(c^2)^(1/2))*5i)/(3*(c^2)^(3/2))

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sympy [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)**4*(a+b*atan(c*x))/x**8,x)

[Out]

Timed out

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