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

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

[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

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Rubi [A]  time = 0.195779, antiderivative size = 243, 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} \[ -\frac{c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}+\frac{6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^6}-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{88 b c^5 d^4}{105 x^2}+\frac{5 i b c^4 d^4}{9 x^3}+\frac{47 b c^3 d^4}{140 x^4}-\frac{2 i b c^2 d^4}{15 x^5}-\frac{5 i b c^6 d^4}{3 x}-\frac{176}{105} b c^7 d^4 \log (x)+\frac{1}{210} b c^7 d^4 \log (-c x+i)+\frac{117}{70} b c^7 d^4 \log (c x+i)-\frac{b c d^4}{42 x^6} \]

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 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)^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*}

Mathematica [C]  time = 0.0962663, size = 293, normalized size = 1.21 \[ \frac{i b c^4 d^4 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-c^2 x^2\right )}{3 x^3}-\frac{2 i b c^2 d^4 \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-c^2 x^2\right )}{15 x^5}-\frac{a c^4 d^4}{3 x^3}+\frac{i a c^3 d^4}{x^4}+\frac{6 a c^2 d^4}{5 x^5}-\frac{2 i a c d^4}{3 x^6}-\frac{a d^4}{7 x^7}-\frac{88 b c^5 d^4}{105 x^2}+\frac{47 b c^3 d^4}{140 x^4}+\frac{88}{105} b c^7 d^4 \log \left (c^2 x^2+1\right )-\frac{b c^4 d^4 \tan ^{-1}(c x)}{3 x^3}+\frac{i b c^3 d^4 \tan ^{-1}(c x)}{x^4}+\frac{6 b c^2 d^4 \tan ^{-1}(c x)}{5 x^5}-\frac{176}{105} b c^7 d^4 \log (x)-\frac{b c d^4}{42 x^6}-\frac{2 i b c d^4 \tan ^{-1}(c x)}{3 x^6}-\frac{b d^4 \tan ^{-1}(c x)}{7 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*d^4)/(7*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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Maple [A]  time = 0.036, size = 255, normalized size = 1.1 \begin{align*} -{\frac{{d}^{4}a}{7\,{x}^{7}}}-{\frac{{\frac{2\,i}{3}}c{d}^{4}a}{{x}^{6}}}-{\frac{{\frac{5\,i}{3}}b{c}^{6}{d}^{4}}{x}}+{\frac{6\,{c}^{2}{d}^{4}a}{5\,{x}^{5}}}-{\frac{{d}^{4}{c}^{4}a}{3\,{x}^{3}}}-{\frac{b{d}^{4}\arctan \left ( cx \right ) }{7\,{x}^{7}}}-{\frac{{\frac{2\,i}{15}}b{c}^{2}{d}^{4}}{{x}^{5}}}+{\frac{i{c}^{3}{d}^{4}b\arctan \left ( cx \right ) }{{x}^{4}}}+{\frac{6\,{c}^{2}{d}^{4}b\arctan \left ( cx \right ) }{5\,{x}^{5}}}-{\frac{b{c}^{4}{d}^{4}\arctan \left ( cx \right ) }{3\,{x}^{3}}}+{\frac{88\,{c}^{7}{d}^{4}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{105}}+{\frac{{\frac{5\,i}{9}}b{c}^{4}{d}^{4}}{{x}^{3}}}-{\frac{5\,i}{3}}{c}^{7}{d}^{4}b\arctan \left ( cx \right ) -{\frac{{\frac{2\,i}{3}}c{d}^{4}b\arctan \left ( cx \right ) }{{x}^{6}}}+{\frac{i{c}^{3}{d}^{4}a}{{x}^{4}}}-{\frac{bc{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 ( cx \right ) }{105}} \end{align*}

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

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Maxima [A]  time = 1.4912, size = 444, normalized size = 1.83 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 2.35951, size = 560, normalized size = 2.3 \begin{align*} -\frac{2112 \, b c^{7} d^{4} x^{7} \log \left (x\right ) - 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}} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 2.57516, size = 342, normalized size = 1.41 \begin{align*} \frac{2106 \, b c^{7} d^{4} x^{7} \log \left (c x + i\right ) + 6 \, b c^{7} d^{4} x^{7} \log \left (c x - i\right ) - 2112 \, b c^{7} d^{4} x^{7} \log \left (x\right ) - 2100 \, b c^{6} d^{4} i x^{6} - 1056 \, b c^{5} d^{4} x^{5} + 700 \, b c^{4} d^{4} i x^{4} - 420 \, b c^{4} d^{4} x^{4} \arctan \left (c x\right ) - 420 \, a c^{4} d^{4} x^{4} + 1260 \, b c^{3} d^{4} i x^{3} \arctan \left (c x\right ) + 1260 \, a c^{3} d^{4} i x^{3} + 423 \, b c^{3} d^{4} x^{3} - 168 \, b c^{2} d^{4} i x^{2} + 1512 \, b c^{2} d^{4} x^{2} \arctan \left (c x\right ) + 1512 \, a c^{2} d^{4} x^{2} - 840 \, b c d^{4} i x \arctan \left (c x\right ) - 840 \, a c d^{4} i x - 30 \, b c d^{4} x - 180 \, b d^{4} \arctan \left (c x\right ) - 180 \, a d^{4}}{1260 \, x^{7}} \end{align*}

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]

1/1260*(2106*b*c^7*d^4*x^7*log(c*x + i) + 6*b*c^7*d^4*x^7*log(c*x - i) - 2112*b*c^7*d^4*x^7*log(x) - 2100*b*c^
6*d^4*i*x^6 - 1056*b*c^5*d^4*x^5 + 700*b*c^4*d^4*i*x^4 - 420*b*c^4*d^4*x^4*arctan(c*x) - 420*a*c^4*d^4*x^4 + 1
260*b*c^3*d^4*i*x^3*arctan(c*x) + 1260*a*c^3*d^4*i*x^3 + 423*b*c^3*d^4*x^3 - 168*b*c^2*d^4*i*x^2 + 1512*b*c^2*
d^4*x^2*arctan(c*x) + 1512*a*c^2*d^4*x^2 - 840*b*c*d^4*i*x*arctan(c*x) - 840*a*c*d^4*i*x - 30*b*c*d^4*x - 180*
b*d^4*arctan(c*x) - 180*a*d^4)/x^7