3.1.94 \(\int \frac {(d+i c d x)^3 (a+b \text {ArcTan}(c x))^2}{x^7} \, dx\) [94]

Optimal. Leaf size=513 \[ -\frac {b^2 c^2 d^3}{60 x^4}-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {61 b^2 c^4 d^3}{180 x^2}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {37}{30} i b^2 c^6 d^3 \text {ArcTan}(c x)-\frac {b c d^3 (a+b \text {ArcTan}(c x))}{15 x^5}-\frac {3 i b c^2 d^3 (a+b \text {ArcTan}(c x))}{10 x^4}+\frac {11 b c^3 d^3 (a+b \text {ArcTan}(c x))}{18 x^3}+\frac {14 i b c^4 d^3 (a+b \text {ArcTan}(c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \text {ArcTan}(c x))}{6 x}-\frac {d^3 (a+b \text {ArcTan}(c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \text {ArcTan}(c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \text {ArcTan}(c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \text {ArcTan}(c x))^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1+c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,i c x)+\frac {37}{40} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \]

[Out]

-1/60*b^2*c^2*d^3/x^4+28/15*I*a*b*c^6*d^3*ln(x)+61/180*b^2*c^4*d^3/x^2-1/10*I*b^2*c^3*d^3/x^3-3/5*I*c*d^3*(a+b
*arctan(c*x))^2/x^5-1/15*b*c*d^3*(a+b*arctan(c*x))/x^5+1/3*I*c^3*d^3*(a+b*arctan(c*x))^2/x^3+11/18*b*c^3*d^3*(
a+b*arctan(c*x))/x^3+37/30*I*b^2*c^5*d^3/x-11/6*b*c^5*d^3*(a+b*arctan(c*x))/x-1/6*d^3*(a+b*arctan(c*x))^2/x^6+
37/20*I*b*c^6*d^3*(a+b*arctan(c*x))*ln(2/(1-I*c*x))+3/4*c^2*d^3*(a+b*arctan(c*x))^2/x^4+37/30*I*b^2*c^6*d^3*ar
ctan(c*x)+1/60*I*b*c^6*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))+113/45*b^2*c^6*d^3*ln(x)-3/10*I*b*c^2*d^3*(a+b*ar
ctan(c*x))/x^4+14/15*I*b*c^4*d^3*(a+b*arctan(c*x))/x^2-113/90*b^2*c^6*d^3*ln(c^2*x^2+1)-14/15*b^2*c^6*d^3*poly
log(2,-I*c*x)+14/15*b^2*c^6*d^3*polylog(2,I*c*x)+37/40*b^2*c^6*d^3*polylog(2,1-2/(1-I*c*x))-1/120*b^2*c^6*d^3*
polylog(2,1-2/(1+I*c*x))

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {45, 4994, 4946, 272, 46, 331, 209, 36, 29, 31, 4940, 2438, 4964, 2449, 2352} \begin {gather*} \frac {37}{20} i b c^6 d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))+\frac {1}{60} i b c^6 d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))-\frac {11 b c^5 d^3 (a+b \text {ArcTan}(c x))}{6 x}+\frac {14 i b c^4 d^3 (a+b \text {ArcTan}(c x))}{15 x^2}+\frac {i c^3 d^3 (a+b \text {ArcTan}(c x))^2}{3 x^3}+\frac {11 b c^3 d^3 (a+b \text {ArcTan}(c x))}{18 x^3}+\frac {3 c^2 d^3 (a+b \text {ArcTan}(c x))^2}{4 x^4}-\frac {3 i b c^2 d^3 (a+b \text {ArcTan}(c x))}{10 x^4}-\frac {d^3 (a+b \text {ArcTan}(c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \text {ArcTan}(c x))^2}{5 x^5}-\frac {b c d^3 (a+b \text {ArcTan}(c x))}{15 x^5}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{30} i b^2 c^6 d^3 \text {ArcTan}(c x)-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {i b^2 c^3 d^3}{10 x^3}-\frac {b^2 c^2 d^3}{60 x^4}-\frac {113}{90} b^2 c^6 d^3 \log \left (c^2 x^2+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^7,x]

[Out]

-1/60*(b^2*c^2*d^3)/x^4 - ((I/10)*b^2*c^3*d^3)/x^3 + (61*b^2*c^4*d^3)/(180*x^2) + (((37*I)/30)*b^2*c^5*d^3)/x
+ ((37*I)/30)*b^2*c^6*d^3*ArcTan[c*x] - (b*c*d^3*(a + b*ArcTan[c*x]))/(15*x^5) - (((3*I)/10)*b*c^2*d^3*(a + b*
ArcTan[c*x]))/x^4 + (11*b*c^3*d^3*(a + b*ArcTan[c*x]))/(18*x^3) + (((14*I)/15)*b*c^4*d^3*(a + b*ArcTan[c*x]))/
x^2 - (11*b*c^5*d^3*(a + b*ArcTan[c*x]))/(6*x) - (d^3*(a + b*ArcTan[c*x])^2)/(6*x^6) - (((3*I)/5)*c*d^3*(a + b
*ArcTan[c*x])^2)/x^5 + (3*c^2*d^3*(a + b*ArcTan[c*x])^2)/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x])^2)/x^3 +
((28*I)/15)*a*b*c^6*d^3*Log[x] + (113*b^2*c^6*d^3*Log[x])/45 + ((37*I)/20)*b*c^6*d^3*(a + b*ArcTan[c*x])*Log[
2/(1 - I*c*x)] + (I/60)*b*c^6*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (113*b^2*c^6*d^3*Log[1 + c^2*x^2])/
90 - (14*b^2*c^6*d^3*PolyLog[2, (-I)*c*x])/15 + (14*b^2*c^6*d^3*PolyLog[2, I*c*x])/15 + (37*b^2*c^6*d^3*PolyLo
g[2, 1 - 2/(1 - I*c*x)])/40 - (b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/120

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 45

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 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x
^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4994

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTan[c*x])^p, u, x] - Dist[b*c*p, Int[ExpandIntegrand[(a +
b*ArcTan[c*x])^(p - 1), u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c
^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && 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 )^2}{x^7} \, dx &=-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\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 {11 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{12 x^4}+\frac {14 i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^3}-\frac {11 c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{12 x^2}-\frac {14 i c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x}+\frac {i c^6 d^3 \left (a+b \tan ^{-1}(c x)\right )}{120 (-i+c x)}+\frac {37 i c^6 d^3 \left (a+b \tan ^{-1}(c x)\right )}{40 (i+c x)}\right ) \, dx\\ &=-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b c d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^6} \, dx+\frac {1}{5} \left (6 i b c^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^5} \, dx-\frac {1}{6} \left (11 b c^3 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^4} \, dx-\frac {1}{15} \left (28 i b c^4 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx+\frac {1}{6} \left (11 b c^5 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx+\frac {1}{15} \left (28 i b c^6 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx-\frac {1}{60} \left (i b c^7 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{-i+c x} \, dx-\frac {1}{20} \left (37 i b c^7 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i+c x} \, dx\\ &=-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+\frac {1}{15} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^5 \left (1+c^2 x^2\right )} \, dx+\frac {1}{10} \left (3 i b^2 c^3 d^3\right ) \int \frac {1}{x^4 \left (1+c^2 x^2\right )} \, dx-\frac {1}{18} \left (11 b^2 c^4 d^3\right ) \int \frac {1}{x^3 \left (1+c^2 x^2\right )} \, dx-\frac {1}{15} \left (14 i b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx+\frac {1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac {\log (1+i c x)}{x} \, dx+\frac {1}{6} \left (11 b^2 c^6 d^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\frac {1}{60} \left (i b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\frac {1}{20} \left (37 i b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {14 i b^2 c^5 d^3}{15 x}-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i c x)+\frac {1}{30} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{10} \left (3 i b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{60} \left (b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{20} \left (37 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )+\frac {1}{15} \left (14 i b^2 c^7 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {14}{15} i b^2 c^6 d^3 \tan ^{-1}(c x)-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+\frac {1}{30} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^3}-\frac {c^2}{x^2}+\frac {c^4}{x}-\frac {c^6}{1+c^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c^2}{x}+\frac {c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{10} \left (3 i b^2 c^7 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx-\frac {1}{12} \left (11 b^2 c^8 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 d^3}{60 x^4}-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {61 b^2 c^4 d^3}{180 x^2}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x)-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1+c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )\\ \end {align*}

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Mathematica [A]
time = 1.01, size = 401, normalized size = 0.78 \begin {gather*} \frac {d^3 \left (-30 a^2-108 i a^2 c x-12 a b c x+135 a^2 c^2 x^2-54 i a b c^2 x^2-3 b^2 c^2 x^2+60 i a^2 c^3 x^3+110 a b c^3 x^3-18 i b^2 c^3 x^3+168 i a b c^4 x^4+61 b^2 c^4 x^4-330 a b c^5 x^5+222 i b^2 c^5 x^5+64 b^2 c^6 x^6+3 b^2 (-i+c x)^4 \left (-10+4 i c x+c^2 x^2\right ) \text {ArcTan}(c x)^2+2 b \text {ArcTan}(c x) \left (b c x \left (-6-27 i c x+55 c^2 x^2+84 i c^3 x^3-165 c^4 x^4+111 i c^5 x^5\right )-3 a \left (10+36 i c x-45 c^2 x^2-20 i c^3 x^3+55 c^6 x^6\right )+168 i b c^6 x^6 \log \left (1-e^{2 i \text {ArcTan}(c x)}\right )\right )+336 i a b c^6 x^6 \log (c x)+452 b^2 c^6 x^6 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )-168 i a b c^6 x^6 \log \left (1+c^2 x^2\right )+168 b^2 c^6 x^6 \text {PolyLog}\left (2,e^{2 i \text {ArcTan}(c x)}\right )\right )}{180 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^7,x]

[Out]

(d^3*(-30*a^2 - (108*I)*a^2*c*x - 12*a*b*c*x + 135*a^2*c^2*x^2 - (54*I)*a*b*c^2*x^2 - 3*b^2*c^2*x^2 + (60*I)*a
^2*c^3*x^3 + 110*a*b*c^3*x^3 - (18*I)*b^2*c^3*x^3 + (168*I)*a*b*c^4*x^4 + 61*b^2*c^4*x^4 - 330*a*b*c^5*x^5 + (
222*I)*b^2*c^5*x^5 + 64*b^2*c^6*x^6 + 3*b^2*(-I + c*x)^4*(-10 + (4*I)*c*x + c^2*x^2)*ArcTan[c*x]^2 + 2*b*ArcTa
n[c*x]*(b*c*x*(-6 - (27*I)*c*x + 55*c^2*x^2 + (84*I)*c^3*x^3 - 165*c^4*x^4 + (111*I)*c^5*x^5) - 3*a*(10 + (36*
I)*c*x - 45*c^2*x^2 - (20*I)*c^3*x^3 + 55*c^6*x^6) + (168*I)*b*c^6*x^6*Log[1 - E^((2*I)*ArcTan[c*x])]) + (336*
I)*a*b*c^6*x^6*Log[c*x] + 452*b^2*c^6*x^6*Log[(c*x)/Sqrt[1 + c^2*x^2]] - (168*I)*a*b*c^6*x^6*Log[1 + c^2*x^2]
+ 168*b^2*c^6*x^6*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(180*x^6)

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Maple [A]
time = 0.87, size = 797, normalized size = 1.55 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x,method=_RETURNVERBOSE)

[Out]

c^6*(3/2*d^3*a*b*arctan(c*x)/c^4/x^4+1/3*I*d^3*b^2*arctan(c*x)^2/c^3/x^3-3/5*I*d^3*b^2*arctan(c*x)^2/c^5/x^5-3
/10*I*d^3*b^2*arctan(c*x)/c^4/x^4+14/15*I*d^3*b^2*arctan(c*x)/c^2/x^2-3/10*I*d^3*a*b/c^4/x^4+14/15*I*d^3*a*b/c
^2/x^2-1/3*d^3*a*b*arctan(c*x)/c^6/x^6+11/18*d^3*a*b/c^3/x^3+11/18*d^3*b^2*arctan(c*x)/c^3/x^3+3/4*d^3*b^2*arc
tan(c*x)^2/c^4/x^4+37/30*I*d^3*b^2*arctan(c*x)-1/60*d^3*b^2/c^4/x^4+61/180*d^3*b^2/c^2/x^2+14/15*d^3*b^2*ln(c*
x)*ln(1-I*c*x)-14/15*d^3*b^2*ln(c*x)*ln(1+I*c*x)+a^2*d^3*(-1/6/c^6/x^6+1/3*I/c^3/x^3+3/4/c^4/x^4-3/5*I/c^5/x^5
)-7/15*d^3*b^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))-11/6*d^3*a*b*arctan(c*x)-7/15*d^3*b^2*ln(c*x+I)*ln(c^2*x^2+1)+7/15
*d^3*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))+7/15*d^3*b^2*ln(c*x-I)*ln(c^2*x^2+1)+113/45*d^3*b^2*ln(c*x)-14/15*d^3*b^2
*dilog(1+I*c*x)+14/15*d^3*b^2*dilog(1-I*c*x)+7/15*d^3*b^2*dilog(1/2*I*(c*x-I))-7/15*d^3*b^2*dilog(-1/2*I*(c*x+
I))-11/12*d^3*b^2*arctan(c*x)^2+7/30*d^3*b^2*ln(c*x+I)^2-7/30*d^3*b^2*ln(c*x-I)^2+2/3*I*d^3*a*b*arctan(c*x)/c^
3/x^3-6/5*I*d^3*a*b*arctan(c*x)/c^5/x^5-14/15*I*d^3*b^2*arctan(c*x)*ln(c^2*x^2+1)-1/15*d^3*a*b/c^5/x^5-1/10*I*
d^3*b^2/c^3/x^3+37/30*I*d^3*b^2/c/x-1/15*d^3*b^2*arctan(c*x)/c^5/x^5-1/6*d^3*b^2*arctan(c*x)^2/c^6/x^6+28/15*I
*d^3*b^2*arctan(c*x)*ln(c*x)-14/15*I*d^3*a*b*ln(c^2*x^2+1)+28/15*I*d^3*a*b*ln(c*x)-113/90*b^2*d^3*ln(c^2*x^2+1
)-11/6*d^3*a*b/c/x-11/6*d^3*b^2*arctan(c*x)/c/x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x, algorithm="maxima")

[Out]

-1/3*I*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*a*b*c^3*d^3 - 1/2*((3*c^3*arctan(
c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x)/x^4)*a*b*c^2*d^3 - 3/10*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)*a*b*c*d^3 - 1/45*((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)*a*b*d^3 - 1/180*(4*(15*c^5*arctan(c*x) + (15*c^4*x^4 - 5*c^2*x^2 + 3)/x^
5)*c*arctan(c*x) - (30*c^4*x^4*arctan(c*x)^2 - 46*c^4*x^4*log(c^2*x^2 + 1) + 92*c^4*x^4*log(x) + 16*c^2*x^2 -
3)*c^2/x^4)*b^2*d^3 + 1/3*I*a^2*c^3*d^3/x^3 + 3/4*a^2*c^2*d^3/x^4 - 3/5*I*a^2*c*d^3/x^5 - 1/6*b^2*d^3*arctan(c
*x)^2/x^6 - 1/6*a^2*d^3/x^6 - 1/960*(960*I*x^5*integrate(1/240*(180*(b^2*c^5*d^3*x^4 - 2*b^2*c^3*d^3*x^2 - 3*b
^2*c*d^3)*arctan(c*x)^2 + 15*(b^2*c^5*d^3*x^4 - 2*b^2*c^3*d^3*x^2 - 3*b^2*c*d^3)*log(c^2*x^2 + 1)^2 + 2*(65*b^
2*c^4*d^3*x^3 - 36*b^2*c^2*d^3*x)*arctan(c*x) - (20*b^2*c^5*d^3*x^4 - 81*b^2*c^3*d^3*x^2 + 180*(b^2*c^4*d^3*x^
3 + b^2*c^2*d^3*x)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^8 + x^6), x) + 960*x^5*integrate(1/240*(540*(b^2*c^4*
d^3*x^3 + b^2*c^2*d^3*x)*arctan(c*x)^2 + 45*(b^2*c^4*d^3*x^3 + b^2*c^2*d^3*x)*log(c^2*x^2 + 1)^2 - 2*(20*b^2*c
^5*d^3*x^4 - 81*b^2*c^3*d^3*x^2)*arctan(c*x) - (65*b^2*c^4*d^3*x^3 - 36*b^2*c^2*d^3*x - 60*(b^2*c^5*d^3*x^4 -
2*b^2*c^3*d^3*x^2 - 3*b^2*c*d^3)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^8 + x^6), x) - 4*(20*I*b^2*c^3*d^3*x^2
+ 45*b^2*c^2*d^3*x - 36*I*b^2*c*d^3)*arctan(c*x)^2 + 4*(20*b^2*c^3*d^3*x^2 - 45*I*b^2*c^2*d^3*x - 36*b^2*c*d^3
)*arctan(c*x)*log(c^2*x^2 + 1) + (20*I*b^2*c^3*d^3*x^2 + 45*b^2*c^2*d^3*x - 36*I*b^2*c*d^3)*log(c^2*x^2 + 1)^2
)/x^5

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x, algorithm="fricas")

[Out]

1/240*(240*x^6*integral(1/60*(-60*I*a^2*c^5*d^3*x^5 - 180*a^2*c^4*d^3*x^4 + 120*I*a^2*c^3*d^3*x^3 - 120*a^2*c^
2*d^3*x^2 + 180*I*a^2*c*d^3*x + 60*a^2*d^3 + (60*a*b*c^5*d^3*x^5 - 20*(9*I*a*b - b^2)*c^4*d^3*x^4 - 15*(8*a*b
+ 3*I*b^2)*c^3*d^3*x^3 - 12*(10*I*a*b + 3*b^2)*c^2*d^3*x^2 - 10*(18*a*b - I*b^2)*c*d^3*x + 60*I*a*b*d^3)*log(-
(c*x + I)/(c*x - I)))/(c^2*x^9 + x^7), x) + (-20*I*b^2*c^3*d^3*x^3 - 45*b^2*c^2*d^3*x^2 + 36*I*b^2*c*d^3*x + 1
0*b^2*d^3)*log(-(c*x + I)/(c*x - I))^2)/x^6

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)**3*(a+b*atan(c*x))**2/x**7,x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^7,x)

[Out]

int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^7, x)

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