3.2.46 \(\int \frac {(a+b \text {ArcTan}(\frac {c}{x}))^2}{x^3} \, dx\) [146]

Optimal. Leaf size=84 \[ \frac {a b}{c x}+\frac {b^2 \cot ^{-1}\left (\frac {x}{c}\right )}{c x}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 x^2}-\frac {b^2 \log \left (1+\frac {c^2}{x^2}\right )}{2 c^2} \]

[Out]

a*b/c/x+b^2*arccot(x/c)/c/x-1/2*(a+b*arccot(x/c))^2/c^2-1/2*(a+b*arccot(x/c))^2/x^2-1/2*b^2*ln(1+c^2/x^2)/c^2

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Rubi [A]
time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4948, 4946, 5036, 4930, 266, 5004} \begin {gather*} -\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 x^2}+\frac {a b}{c x}-\frac {b^2 \log \left (\frac {c^2}{x^2}+1\right )}{2 c^2}+\frac {b^2 \cot ^{-1}\left (\frac {x}{c}\right )}{c x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c/x])^2/x^3,x]

[Out]

(a*b)/(c*x) + (b^2*ArcCot[x/c])/(c*x) - (a + b*ArcCot[x/c])^2/(2*c^2) - (a + b*ArcCot[x/c])^2/(2*x^2) - (b^2*L
og[1 + c^2/x^2])/(2*c^2)

Rule 266

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

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 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 4948

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

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 5036

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )^2}{x^3} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 x^3}+\frac {b \left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log \left (1+\frac {i c}{x}\right )}{2 x^3}-\frac {b^2 \log ^2\left (1+\frac {i c}{x}\right )}{4 x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{x^3} \, dx+\frac {1}{2} b \int \frac {\left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+\frac {i c}{x}\right )}{x^3} \, dx\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int x (2 a+i b \log (1-i c x))^2 \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{2} b \int \left (-\frac {2 i a \log \left (1+\frac {i c}{x}\right )}{x^3}+\frac {b \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3}\right ) \, dx+\frac {1}{4} b^2 \text {Subst}\left (\int x \log ^2(1+i c x) \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int \left (-\frac {i (2 a+i b \log (1-i c x))^2}{c}+\frac {i (1-i c x) (2 a+i b \log (1-i c x))^2}{c}\right ) \, dx,x,\frac {1}{x}\right )\right )-(i a b) \int \frac {\log \left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {1}{4} b^2 \text {Subst}\left (\int \left (\frac {i \log ^2(1+i c x)}{c}-\frac {i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{2} b^2 \int \frac {\log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{x^3} \, dx\\ &=-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+(i a b) \text {Subst}\left (\int x \log (1+i c x) \, dx,x,\frac {1}{x}\right )-\frac {1}{2} b^2 \int \frac {c \log \left (1-\frac {i c}{x}\right )}{2 (c-i x) x^3} \, dx-\frac {1}{2} b^2 \int \frac {c \log \left (1+\frac {i c}{x}\right )}{2 (c+i x) x^3} \, dx+\frac {i \text {Subst}\left (\int (2 a+i b \log (1-i c x))^2 \, dx,x,\frac {1}{x}\right )}{4 c}-\frac {i \text {Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^2 \, dx,x,\frac {1}{x}\right )}{4 c}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \log ^2(1+i c x) \, dx,x,\frac {1}{x}\right )}{4 c}-\frac {\left (i b^2\right ) \text {Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}-\frac {\text {Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-\frac {i c}{x}\right )}{4 c^2}+\frac {\text {Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}+\frac {1}{2} (a b c) \text {Subst}\left (\int \frac {x^2}{1+i c x} \, dx,x,\frac {1}{x}\right )-\frac {1}{4} \left (b^2 c\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{(c-i x) x^3} \, dx-\frac {1}{4} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{(c+i x) x^3} \, dx\\ &=-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {(i b) \text {Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-\frac {i c}{x}\right )}{4 c^2}+\frac {(i b) \text {Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-\frac {i c}{x}\right )}{2 c^2}+\frac {b^2 \text {Subst}\left (\int x \log (x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+\frac {i c}{x}\right )}{2 c^2}+\frac {1}{2} (a b c) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {i x}{c}+\frac {i}{c^2 (-i+c x)}\right ) \, dx,x,\frac {1}{x}\right )-\frac {1}{4} \left (b^2 c\right ) \int \left (-\frac {i \log \left (1-\frac {i c}{x}\right )}{c^3 (c-i x)}+\frac {\log \left (1-\frac {i c}{x}\right )}{c x^3}+\frac {i \log \left (1-\frac {i c}{x}\right )}{c^2 x^2}-\frac {\log \left (1-\frac {i c}{x}\right )}{c^3 x}\right ) \, dx-\frac {1}{4} \left (b^2 c\right ) \int \left (\frac {i \log \left (1+\frac {i c}{x}\right )}{c^3 (c+i x)}+\frac {\log \left (1+\frac {i c}{x}\right )}{c x^3}-\frac {i \log \left (1+\frac {i c}{x}\right )}{c^2 x^2}-\frac {\log \left (1+\frac {i c}{x}\right )}{c^3 x}\right ) \, dx\\ &=-\frac {b^2 \left (1-\frac {i c}{x}\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {i a b}{4 x^2}+\frac {3 a b}{2 c x}+\frac {i b^2}{2 c x}+\frac {i a b \log \left (i-\frac {c}{x}\right )}{2 c^2}-\frac {i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{8 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{2 c^2}+\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {1}{4} b^2 \int \frac {\log \left (1-\frac {i c}{x}\right )}{x^3} \, dx-\frac {1}{4} b^2 \int \frac {\log \left (1+\frac {i c}{x}\right )}{x^3} \, dx+\frac {\left (i b^2\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{c-i x} \, dx}{4 c^2}-\frac {\left (i b^2\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{c+i x} \, dx}{4 c^2}+\frac {b^2 \int \frac {\log \left (1-\frac {i c}{x}\right )}{x} \, dx}{4 c^2}+\frac {b^2 \int \frac {\log \left (1+\frac {i c}{x}\right )}{x} \, dx}{4 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-\frac {i c}{x}\right )}{2 c^2}-\frac {\left (i b^2\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{x^2} \, dx}{4 c}+\frac {\left (i b^2\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{x^2} \, dx}{4 c}\\ &=-\frac {b^2 \left (1-\frac {i c}{x}\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {i a b}{4 x^2}+\frac {3 a b}{2 c x}+\frac {i a b \log \left (i-\frac {c}{x}\right )}{2 c^2}-\frac {b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{2 c^2}-\frac {i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{8 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{2 c^2}+\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}+\frac {1}{4} b^2 \text {Subst}\left (\int x \log (1-i c x) \, dx,x,\frac {1}{x}\right )+\frac {1}{4} b^2 \text {Subst}\left (\int x \log (1+i c x) \, dx,x,\frac {1}{x}\right )+\frac {\left (i b^2\right ) \int \frac {\log (c-i x)}{\left (1-\frac {i c}{x}\right ) x^2} \, dx}{4 c}-\frac {\left (i b^2\right ) \int \frac {\log (c+i x)}{\left (1+\frac {i c}{x}\right ) x^2} \, dx}{4 c}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \log (1-i c x) \, dx,x,\frac {1}{x}\right )}{4 c}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \log (1+i c x) \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=-\frac {b^2 \left (1-\frac {i c}{x}\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {i a b}{4 x^2}+\frac {3 a b}{2 c x}+\frac {i a b \log \left (i-\frac {c}{x}\right )}{2 c^2}-\frac {b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{2 c^2}+\frac {b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{8 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{2 c^2}+\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}+\frac {b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+\frac {i c}{x}\right )}{4 c^2}+\frac {\left (i b^2\right ) \int \left (\frac {\log (c-i x)}{c (c+i x)}+\frac {i \log (c-i x)}{c x}\right ) \, dx}{4 c}-\frac {\left (i b^2\right ) \int \left (\frac {\log (c+i x)}{c (c-i x)}-\frac {i \log (c+i x)}{c x}\right ) \, dx}{4 c}+\frac {1}{8} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1-i c x} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1+i c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b^2 \left (1-\frac {i c}{x}\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {i a b}{4 x^2}+\frac {3 a b}{2 c x}+\frac {i a b \log \left (i-\frac {c}{x}\right )}{2 c^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{8 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}+\frac {b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}+\frac {\left (i b^2\right ) \int \frac {\log (c-i x)}{c+i x} \, dx}{4 c^2}-\frac {\left (i b^2\right ) \int \frac {\log (c+i x)}{c-i x} \, dx}{4 c^2}-\frac {b^2 \int \frac {\log (c-i x)}{x} \, dx}{4 c^2}-\frac {b^2 \int \frac {\log (c+i x)}{x} \, dx}{4 c^2}-\frac {1}{8} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {i x}{c}+\frac {i}{c^2 (-i+c x)}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}+\frac {i x}{c}-\frac {i}{c^2 (i+c x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b^2 \left (1-\frac {i c}{x}\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {i a b}{4 x^2}-\frac {b^2}{8 x^2}+\frac {3 a b}{2 c x}+\frac {i a b \log \left (i-\frac {c}{x}\right )}{2 c^2}+\frac {b^2 \log \left (i-\frac {c}{x}\right )}{8 c^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{8 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}+\frac {b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {b^2 \log \left (i+\frac {c}{x}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )}{4 c^2}-\frac {b^2 \log (c+i x) \log \left (-\frac {i x}{c}\right )}{4 c^2}-\frac {b^2 \log (c-i x) \log \left (\frac {i x}{c}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}-\frac {\left (i b^2\right ) \int \frac {\log \left (\frac {c-i x}{2 c}\right )}{c+i x} \, dx}{4 c^2}+\frac {\left (i b^2\right ) \int \frac {\log \left (\frac {c+i x}{2 c}\right )}{c-i x} \, dx}{4 c^2}+\frac {\left (i b^2\right ) \int \frac {\log \left (-\frac {i x}{c}\right )}{c+i x} \, dx}{4 c^2}-\frac {\left (i b^2\right ) \int \frac {\log \left (\frac {i x}{c}\right )}{c-i x} \, dx}{4 c^2}\\ &=-\frac {b^2 \left (1-\frac {i c}{x}\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {i a b}{4 x^2}-\frac {b^2}{8 x^2}+\frac {3 a b}{2 c x}+\frac {i a b \log \left (i-\frac {c}{x}\right )}{2 c^2}+\frac {b^2 \log \left (i-\frac {c}{x}\right )}{8 c^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{8 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}+\frac {b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {b^2 \log \left (i+\frac {c}{x}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )}{4 c^2}-\frac {b^2 \log (c+i x) \log \left (-\frac {i x}{c}\right )}{4 c^2}-\frac {b^2 \log (c-i x) \log \left (\frac {i x}{c}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \text {Li}_2\left (1-\frac {i x}{c}\right )}{4 c^2}-\frac {b^2 \text {Li}_2\left (1+\frac {i x}{c}\right )}{4 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-i x\right )}{4 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+i x\right )}{4 c^2}\\ &=-\frac {b^2 \left (1-\frac {i c}{x}\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2}{16 c^2}-\frac {i a b}{4 x^2}-\frac {b^2}{8 x^2}+\frac {3 a b}{2 c x}+\frac {i a b \log \left (i-\frac {c}{x}\right )}{2 c^2}+\frac {b^2 \log \left (i-\frac {c}{x}\right )}{8 c^2}-\frac {3 b^2 \left (1-\frac {i c}{x}\right ) \log \left (1-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \log \left (1-\frac {i c}{x}\right )}{8 x^2}-\frac {i b \left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )}{8 c^2}-\frac {\left (1-\frac {i c}{x}\right ) \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{4 c^2}+\frac {\left (1-\frac {i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2}{8 c^2}-\frac {3 b^2 \left (1+\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log \left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {i a b \log \left (1+\frac {i c}{x}\right )}{2 x^2}+\frac {b^2 \log \left (1+\frac {i c}{x}\right )}{8 x^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )}{4 x^2}+\frac {b^2 \left (1+\frac {i c}{x}\right ) \log ^2\left (1+\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {i c}{x}\right )^2 \log ^2\left (1+\frac {i c}{x}\right )}{8 c^2}+\frac {b^2 \log \left (i+\frac {c}{x}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac {b^2 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac {b^2 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )}{4 c^2}-\frac {b^2 \log (c+i x) \log \left (-\frac {i x}{c}\right )}{4 c^2}-\frac {b^2 \log (c-i x) \log \left (\frac {i x}{c}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c-i x}{2 c}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c+i x}{2 c}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {i c}{x}\right )}{4 c^2}+\frac {b^2 \text {Li}_2\left (\frac {i c}{x}\right )}{4 c^2}-\frac {b^2 \text {Li}_2\left (1-\frac {i x}{c}\right )}{4 c^2}-\frac {b^2 \text {Li}_2\left (1+\frac {i x}{c}\right )}{4 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 99, normalized size = 1.18 \begin {gather*} -\frac {a^2 c^2-2 a b c x+2 b c (a c-b x) \text {ArcTan}\left (\frac {c}{x}\right )+b^2 \left (c^2+x^2\right ) \text {ArcTan}\left (\frac {c}{x}\right )^2-2 a b x^2 \text {ArcTan}\left (\frac {x}{c}\right )-2 b^2 x^2 \log (x)+b^2 x^2 \log \left (c^2+x^2\right )}{2 c^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a^2*c^2 - 2*a*b*c*x + 2*b*c*(a*c - b*x)*ArcTan[c/x] + b^2*(c^2 + x^2)*ArcTan[c/x]^2 - 2*a*b*x^2*ArcTan[x
/c] - 2*b^2*x^2*Log[x] + b^2*x^2*Log[c^2 + x^2])/(c^2*x^2)

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Maple [A]
time = 0.94, size = 112, normalized size = 1.33

method	result	size
derivativedivides	\(-\frac {\frac {a^{2} c^{2}}{2 x^{2}}+\frac {b^{2} c^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 x^{2}}+\frac {b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {b^{2} \arctan \left (\frac {c}{x}\right ) c}{x}+\frac {b^{2} \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {a b \,c^{2} \arctan \left (\frac {c}{x}\right )}{x^{2}}+a b \arctan \left (\frac {c}{x}\right )-\frac {a b c}{x}}{c^{2}}\)	\(112\)
default	\(-\frac {\frac {a^{2} c^{2}}{2 x^{2}}+\frac {b^{2} c^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 x^{2}}+\frac {b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {b^{2} \arctan \left (\frac {c}{x}\right ) c}{x}+\frac {b^{2} \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {a b \,c^{2} \arctan \left (\frac {c}{x}\right )}{x^{2}}+a b \arctan \left (\frac {c}{x}\right )-\frac {a b c}{x}}{c^{2}}\)	\(112\)
risch	\(\text {Expression too large to display}\)	\(59876\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^2*(1/2*a^2*c^2/x^2+1/2*b^2*c^2/x^2*arctan(c/x)^2+1/2*b^2*arctan(c/x)^2-b^2*arctan(c/x)*c/x+1/2*b^2*ln(1+c
^2/x^2)+a*b*c^2/x^2*arctan(c/x)+a*b*arctan(c/x)-a*b*c/x)

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Maxima [A]
time = 0.50, size = 120, normalized size = 1.43 \begin {gather*} {\left (c {\left (\frac {\arctan \left (\frac {x}{c}\right )}{c^{3}} + \frac {1}{c^{2} x}\right )} - \frac {\arctan \left (\frac {c}{x}\right )}{x^{2}}\right )} a b + \frac {1}{2} \, {\left (2 \, c {\left (\frac {\arctan \left (\frac {x}{c}\right )}{c^{3}} + \frac {1}{c^{2} x}\right )} \arctan \left (\frac {c}{x}\right ) + \frac {\arctan \left (\frac {x}{c}\right )^{2} - \log \left (c^{2} + x^{2}\right ) + 2 \, \log \left (x\right )}{c^{2}}\right )} b^{2} - \frac {b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*(arctan(x/c)/c^3 + 1/(c^2*x)) - arctan(c/x)/x^2)*a*b + 1/2*(2*c*(arctan(x/c)/c^3 + 1/(c^2*x))*arctan(c/x) +
(arctan(x/c)^2 - log(c^2 + x^2) + 2*log(x))/c^2)*b^2 - 1/2*b^2*arctan(c/x)^2/x^2 - 1/2*a^2/x^2

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Fricas [A]
time = 0.76, size = 109, normalized size = 1.30 \begin {gather*} \frac {2 \, a b x^{2} \arctan \left (\frac {x}{c}\right ) - b^{2} x^{2} \log \left (c^{2} + x^{2}\right ) + 2 \, b^{2} x^{2} \log \left (x\right ) - a^{2} c^{2} + 2 \, a b c x - {\left (b^{2} c^{2} + b^{2} x^{2}\right )} \arctan \left (\frac {c}{x}\right )^{2} - 2 \, {\left (a b c^{2} - b^{2} c x\right )} \arctan \left (\frac {c}{x}\right )}{2 \, c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*b*x^2*arctan(x/c) - b^2*x^2*log(c^2 + x^2) + 2*b^2*x^2*log(x) - a^2*c^2 + 2*a*b*c*x - (b^2*c^2 + b^2*
x^2)*arctan(c/x)^2 - 2*(a*b*c^2 - b^2*c*x)*arctan(c/x))/(c^2*x^2)

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Sympy [A]
time = 0.37, size = 117, normalized size = 1.39 \begin {gather*} \begin {cases} - \frac {a^{2}}{2 x^{2}} - \frac {a b \operatorname {atan}{\left (\frac {c}{x} \right )}}{x^{2}} + \frac {a b}{c x} - \frac {a b \operatorname {atan}{\left (\frac {c}{x} \right )}}{c^{2}} - \frac {b^{2} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{2 x^{2}} + \frac {b^{2} \operatorname {atan}{\left (\frac {c}{x} \right )}}{c x} + \frac {b^{2} \log {\left (x \right )}}{c^{2}} - \frac {b^{2} \log {\left (c^{2} + x^{2} \right )}}{2 c^{2}} - \frac {b^{2} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2/(2*x**2) - a*b*atan(c/x)/x**2 + a*b/(c*x) - a*b*atan(c/x)/c**2 - b**2*atan(c/x)**2/(2*x**2) +
b**2*atan(c/x)/(c*x) + b**2*log(x)/c**2 - b**2*log(c**2 + x**2)/(2*c**2) - b**2*atan(c/x)**2/(2*c**2), Ne(c,
0)), (-a**2/(2*x**2), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 137, normalized size = 1.63 \begin {gather*} -\frac {b^{2} \arctan \left (\frac {c}{x}\right )^{2} + \frac {b^{2} c^{2} \arctan \left (\frac {c}{x}\right )^{2}}{x^{2}} + \frac {2 \, a b c^{2} \arctan \left (\frac {c}{x}\right )}{x^{2}} - \frac {2 \, b^{2} c \arctan \left (\frac {c}{x}\right )}{x} + i \, a b \log \left (\frac {i \, c}{x} - 1\right ) + b^{2} \log \left (\frac {i \, c}{x} - 1\right ) - i \, a b \log \left (-\frac {i \, c}{x} - 1\right ) + b^{2} \log \left (-\frac {i \, c}{x} - 1\right ) + \frac {a^{2} c^{2}}{x^{2}} - \frac {2 \, a b c}{x}}{2 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^2*arctan(c/x)^2 + b^2*c^2*arctan(c/x)^2/x^2 + 2*a*b*c^2*arctan(c/x)/x^2 - 2*b^2*c*arctan(c/x)/x + I*a*
b*log(I*c/x - 1) + b^2*log(I*c/x - 1) - I*a*b*log(-I*c/x - 1) + b^2*log(-I*c/x - 1) + a^2*c^2/x^2 - 2*a*b*c/x)
/c^2

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Mupad [B]
time = 2.75, size = 143, normalized size = 1.70 \begin {gather*} \frac {b^2\,\ln \left (x\right )-\frac {b^2\,\ln \left (x+c\,1{}\mathrm {i}\right )}{2}-\frac {b^2\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{2}+\frac {b^2\,\ln \left (\frac {1}{-x+c\,1{}\mathrm {i}}\right )}{2}+\frac {a\,b\,\ln \left (x+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\frac {a\,b\,\ln \left (-x+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}}{c^2}-\frac {\frac {a^2\,c^2}{2}-x\,\left (c\,\mathrm {atan}\left (\frac {c}{x}\right )\,b^2+a\,c\,b\right )+\frac {b^2\,c^2\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{2}+a\,b\,c^2\,\mathrm {atan}\left (\frac {c}{x}\right )}{c^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c/x))^2/x^3,x)

[Out]

(b^2*log(x) - (b^2*log(c*1i + x))/2 - (b^2*atan(c/x)^2)/2 + (b^2*log(1/(c*1i - x)))/2 + (a*b*log(c*1i + x)*1i)
/2 - (a*b*log(c*1i - x)*1i)/2)/c^2 - ((a^2*c^2)/2 - x*(b^2*c*atan(c/x) + a*b*c) + (b^2*c^2*atan(c/x)^2)/2 + a*
b*c^2*atan(c/x))/(c^2*x^2)

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