3.1.81 \(\int \frac {(d+i c d x)^2 (a+b \text {ArcTan}(c x))^2}{x^2} \, dx\) [81]

Optimal. Leaf size=317 \[ -2 i c d^2 (a+b \text {ArcTan}(c x))^2-\frac {d^2 (a+b \text {ArcTan}(c x))^2}{x}-c^2 d^2 x (a+b \text {ArcTan}(c x))^2+4 i c d^2 (a+b \text {ArcTan}(c x))^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-2 b c d^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )+2 b c d^2 (a+b \text {ArcTan}(c x)) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+2 b c d^2 (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-2 b c d^2 (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-i b^2 c d^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+i b^2 c d^2 \text {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]

[Out]

-2*I*c*d^2*(a+b*arctan(c*x))^2-d^2*(a+b*arctan(c*x))^2/x-c^2*d^2*x*(a+b*arctan(c*x))^2-4*I*c*d^2*(a+b*arctan(c
*x))^2*arctanh(-1+2/(1+I*c*x))-2*b*c*d^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))+2*b*c*d^2*(a+b*arctan(c*x))*ln(2-2/
(1-I*c*x))-I*b^2*c*d^2*polylog(2,-1+2/(1-I*c*x))-I*b^2*c*d^2*polylog(2,1-2/(1+I*c*x))+2*b*c*d^2*(a+b*arctan(c*
x))*polylog(2,1-2/(1+I*c*x))-2*b*c*d^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-I*b^2*c*d^2*polylog(3,1-2/(
1+I*c*x))+I*b^2*c*d^2*polylog(3,-1+2/(1+I*c*x))

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Rubi [A]
time = 0.46, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4996, 4930, 5040, 4964, 2449, 2352, 4946, 5044, 4988, 2497, 4942, 5108, 5004, 5114, 6745} \begin {gather*} -c^2 d^2 x (a+b \text {ArcTan}(c x))^2+2 b c d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) (a+b \text {ArcTan}(c x))-2 b c d^2 \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) (a+b \text {ArcTan}(c x))-2 i c d^2 (a+b \text {ArcTan}(c x))^2-\frac {d^2 (a+b \text {ArcTan}(c x))^2}{x}-2 b c d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))+2 b c d^2 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))+4 i c d^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))^2-i b^2 c d^2 \text {Li}_2\left (\frac {2}{1-i c x}-1\right )-i b^2 c d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )-i b^2 c d^2 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )+i b^2 c d^2 \text {Li}_3\left (\frac {2}{i c x+1}-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*I)*c*d^2*(a + b*ArcTan[c*x])^2 - (d^2*(a + b*ArcTan[c*x])^2)/x - c^2*d^2*x*(a + b*ArcTan[c*x])^2 + (4*I)*c
*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] - 2*b*c*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] + 2*b
*c*d^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] - I*b^2*c*d^2*P
olyLog[2, 1 - 2/(1 + I*c*x)] + 2*b*c*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - 2*b*c*d^2*(a + b*
ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - I*b^2*c*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)] + I*b^2*c*d^2*PolyLog[
3, -1 + 2/(1 + I*c*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 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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4930

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

Rule 4942

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

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 4988

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 4996

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 5004

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

Rule 5040

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

Rule 5044

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

Rule 5108

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

Rule 5114

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {(d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx &=\int \left (-c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2}+\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^2 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx+\left (2 i c d^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx-\left (c^2 d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-c^2 d^2 x \left (a+b \tan ^{-1}(c x)\right )^2+4 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+\left (2 b c d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx-\left (8 i b c^2 d^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (2 b c^3 d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-c^2 d^2 x \left (a+b \tan ^{-1}(c x)\right )^2+4 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+\left (2 i b c d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx+\left (4 i b c^2 d^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (4 i b c^2 d^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (2 b c^2 d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx\\ &=-2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-c^2 d^2 x \left (a+b \tan ^{-1}(c x)\right )^2+4 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )+\left (2 b^2 c^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (2 b^2 c^2 d^2\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx-\left (2 b^2 c^2 d^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (2 b^2 c^2 d^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-c^2 d^2 x \left (a+b \tan ^{-1}(c x)\right )^2+4 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )-i b^2 c d^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )+i b^2 c d^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )-\left (2 i b^2 c d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )\\ &=-2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-c^2 d^2 x \left (a+b \tan ^{-1}(c x)\right )^2+4 i c d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )-i b^2 c d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-2 b c d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )-i b^2 c d^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )+i b^2 c d^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 378, normalized size = 1.19 \begin {gather*} -\frac {d^2 \left (12 a^2-b^2 c \pi ^3 x+12 a^2 c^2 x^2+24 a b \text {ArcTan}(c x)+24 a b c^2 x^2 \text {ArcTan}(c x)+12 b^2 \text {ArcTan}(c x)^2+12 b^2 c^2 x^2 \text {ArcTan}(c x)^2+16 b^2 c x \text {ArcTan}(c x)^3-24 i b^2 c x \text {ArcTan}(c x)^2 \log \left (1-e^{-2 i \text {ArcTan}(c x)}\right )-24 b^2 c x \text {ArcTan}(c x) \log \left (1-e^{2 i \text {ArcTan}(c x)}\right )+24 b^2 c x \text {ArcTan}(c x) \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )+24 i b^2 c x \text {ArcTan}(c x)^2 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )-24 i a^2 c x \log (c x)-24 a b c x \log (c x)+24 b^2 c x \text {ArcTan}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcTan}(c x)}\right )+12 b^2 c x (-i+2 \text {ArcTan}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )+12 i b^2 c x \text {PolyLog}\left (2,e^{2 i \text {ArcTan}(c x)}\right )+24 a b c x \text {PolyLog}(2,-i c x)-24 a b c x \text {PolyLog}(2,i c x)-12 i b^2 c x \text {PolyLog}\left (3,e^{-2 i \text {ArcTan}(c x)}\right )+12 i b^2 c x \text {PolyLog}\left (3,-e^{2 i \text {ArcTan}(c x)}\right )\right )}{12 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(d^2*(12*a^2 - b^2*c*Pi^3*x + 12*a^2*c^2*x^2 + 24*a*b*ArcTan[c*x] + 24*a*b*c^2*x^2*ArcTan[c*x] + 12*b^2*
ArcTan[c*x]^2 + 12*b^2*c^2*x^2*ArcTan[c*x]^2 + 16*b^2*c*x*ArcTan[c*x]^3 - (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 -
E^((-2*I)*ArcTan[c*x])] - 24*b^2*c*x*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + 24*b^2*c*x*ArcTan[c*x]*Log[
1 + E^((2*I)*ArcTan[c*x])] + (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*a^2*c*x*Log[
c*x] - 24*a*b*c*x*Log[c*x] + 24*b^2*c*x*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 12*b^2*c*x*(-I + 2*Ar
cTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (12*I)*b^2*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])] + 24*a*b*c*x*
PolyLog[2, (-I)*c*x] - 24*a*b*c*x*PolyLog[2, I*c*x] - (12*I)*b^2*c*x*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (12*
I)*b^2*c*x*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/x

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.90, size = 11833, normalized size = 37.33

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(11833\)
default	\(\text {Expression too large to display}\)	\(11833\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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)^2*(a+b*arctan(c*x))^2/x^2,x, algorithm="maxima")

[Out]

-a^2*c^2*d^2*x - (2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*c*d^2 + 2*I*a^2*c*d^2*log(x) - (c*(log(c^2*x^2 + 1
) - log(x^2)) + 2*arctan(c*x)/x)*a*b*d^2 - a^2*d^2/x + 1/16*(8*b^2*c^2*d^2*x^2 - 2*b^2*c^2*d^2*x*integrate(4*a
rctan(c*x)^2 + log(c^2*x^2 + 1)^2, x) + 4*I*b^2*c^2*d^2*x*integrate(-1/4*(8*(c^2*x^2 + 1)*c*x*arctan(c*x)^2 -
2*(c^2*x^2 + 1)*c*x*log(c^2*x^2 + 1)^2 + 8*(c^2*x^2 + 1)*arctan(c*x)*log(c^2*x^2 + 1) - (4*(c^2*x^2 + 1)^(3/2)
*arctan(c*x)*cos(2*arctan(c*x))*log(c^2*x^2 + 1) + 4*sqrt(c^2*x^2 + 1)*arctan(c*x)*log(c^2*x^2 + 1) + (4*(c^2*
x^2 + 1)^(3/2)*arctan(c*x)^2 - (c^2*x^2 + 1)^(3/2)*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*sqrt(c^2*x^2 + 1))/
((c^2*x^2 + 1)^3*cos(2*arctan(c*x))^2 + (c^2*x^2 + 1)^3*sin(2*arctan(c*x))^2 + c^2*x^2 + 2*(c^2*x^2 + 1)^2*cos
(2*arctan(c*x)) + 4*(c^2*x^2 + 1)^2 - 4*((c^2*x^2 + 1)^(3/2)*c*x*sin(2*arctan(c*x)) + (c^2*x^2 + 1)^(3/2)*cos(
2*arctan(c*x)) + sqrt(c^2*x^2 + 1))*sqrt(c^2*x^2 + 1) + 1), x) - 4*b^2*c^2*d^2*x*integrate(1/4*(8*(c^2*x^2 + 1
)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 8*(c^2*x^2 + 1)*arctan(c*x)^2 + 2*(c^2*x^2 + 1)*log(c^2*x^2 + 1)^2 - (4*(
c^2*x^2 + 1)^(3/2)*arctan(c*x)*log(c^2*x^2 + 1)*sin(2*arctan(c*x)) - 4*sqrt(c^2*x^2 + 1)*arctan(c*x)^2 + sqrt(
c^2*x^2 + 1)*log(c^2*x^2 + 1)^2 - (4*(c^2*x^2 + 1)^(3/2)*arctan(c*x)^2 - (c^2*x^2 + 1)^(3/2)*log(c^2*x^2 + 1)^
2)*cos(2*arctan(c*x)))*sqrt(c^2*x^2 + 1))/((c^2*x^2 + 1)^3*cos(2*arctan(c*x))^2 + (c^2*x^2 + 1)^3*sin(2*arctan
(c*x))^2 + c^2*x^2 + 2*(c^2*x^2 + 1)^2*cos(2*arctan(c*x)) + 4*(c^2*x^2 + 1)^2 - 4*((c^2*x^2 + 1)^(3/2)*c*x*sin
(2*arctan(c*x)) + (c^2*x^2 + 1)^(3/2)*cos(2*arctan(c*x)) + sqrt(c^2*x^2 + 1))*sqrt(c^2*x^2 + 1) + 1), x) - 4*b
^2*c*d^2*x*(I*gamma(3, -log(I*c*x + 1)) - 2*I) + 8*b^2*c*d^2*x*integrate(arctan(c*x)*log(c^2*x^2 + 1)/x, x) +
4*I*b^2*c*d^2*x*integrate((4*arctan(c*x)^2 + log(c^2*x^2 + 1)^2)/x, x) - 2*I*b^2*c*d^2*x*integrate(-(4*arctan(
c*x)^2 - log(c^2*x^2 + 1)^2)/x, x) + 4*I*b^2*c*d^2*x*integrate(log(c^2*x^2 + 1)/x, x) - 8*(-8*I*a*b - b^2)*c*d
^2*x*integrate(arctan(c*x)/x, x) + 2*b^2*d^2*x*integrate((4*arctan(c*x)^2 + log(c^2*x^2 + 1)^2)/x^2, x) - 4*(b
^2*c^2*d^2*x^2 + b^2*d^2)*arctan(c*x)^2 + (b^2*c^2*d^2*x^2 + b^2*d^2)*log(c^2*x^2 + 1)^2 - 8*(-I*b^2*c^2*d^2*x
^2 + b^2*c*d^2*x)*arctan(c*x) - 4*(b^2*c^2*d^2*x^2 + I*b^2*c*d^2*x + (I*b^2*c^2*d^2*x^2 + I*b^2*d^2)*arctan(c*
x))*log(c^2*x^2 + 1))/x

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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)^2*(a+b*arctan(c*x))^2/x^2,x, algorithm="fricas")

[Out]

integral(-1/4*(4*a^2*c^2*d^2*x^2 - 8*I*a^2*c*d^2*x - 4*a^2*d^2 - (b^2*c^2*d^2*x^2 - 2*I*b^2*c*d^2*x - b^2*d^2)
*log(-(c*x + I)/(c*x - I))^2 + 4*(I*a*b*c^2*d^2*x^2 + 2*a*b*c*d^2*x - I*a*b*d^2)*log(-(c*x + I)/(c*x - I)))/x^
2, x)

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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)**2*(a+b*atan(c*x))**2/x**2,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)^2*(a+b*arctan(c*x))^2/x^2,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 )}^2}{x^2} \,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)^2)/x^2,x)

[Out]

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

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