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

Optimal result

Integrand size = 16, antiderivative size = 84 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx=\frac {a b}{c x}+\frac {b^2 \cot ^{-1}\left (\frac {x}{c}\right )}{c x}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 x^2}-\frac {b^2 \log \left (1+\frac {c^2}{x^2}\right )}{2 c^2} \]

[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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4948, 4946, 5036, 4930, 266, 5004} \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx=-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 x^2}+\frac {a b}{c x}-\frac {b^2 \log \left (\frac {c^2}{x^2}+1\right )}{2 c^2}+\frac {b^2 \cot ^{-1}\left (\frac {x}{c}\right )}{c x} \]

[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*} \text {integral}& = -\text {Subst}\left (\int x (a+b \arctan (c x))^2 \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 x^2}+(b c) \text {Subst}\left (\int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 x^2}+\frac {b \text {Subst}\left (\int (a+b \arctan (c x)) \, dx,x,\frac {1}{x}\right )}{c}-\frac {b \text {Subst}\left (\int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {a b}{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 \text {Subst}\left (\int \arctan (c x) \, dx,x,\frac {1}{x}\right )}{c} \\ & = \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}-b^2 \text {Subst}\left (\int \frac {x}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx=-\frac {a^2 c^2-2 a b c x+2 b c (a c-b x) \arctan \left (\frac {c}{x}\right )+b^2 \left (c^2+x^2\right ) \arctan \left (\frac {c}{x}\right )^2-2 a b x^2 \arctan \left (\frac {x}{c}\right )-2 b^2 x^2 \log (x)+b^2 x^2 \log \left (c^2+x^2\right )}{2 c^2 x^2} \]

[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)

Maple [A] (verified)

Time = 3.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.23

method	result	size
parts	\(-\frac {a^{2}}{2 x^{2}}-\frac {b^{2} \left (\frac {c^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 x^{2}}+\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {c \arctan \left (\frac {c}{x}\right )}{x}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}\right )}{c^{2}}-\frac {a b \arctan \left (\frac {c}{x}\right )}{x^{2}}+\frac {a b}{c x}+\frac {a b \arctan \left (\frac {x}{c}\right )}{c^{2}}\)	\(103\)
derivativedivides	\(-\frac {\frac {a^{2} c^{2}}{2 x^{2}}+b^{2} \left (\frac {c^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 x^{2}}+\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {c \arctan \left (\frac {c}{x}\right )}{x}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}\right )+2 a b \left (\frac {c^{2} \arctan \left (\frac {c}{x}\right )}{2 x^{2}}-\frac {c}{2 x}+\frac {\arctan \left (\frac {c}{x}\right )}{2}\right )}{c^{2}}\)	\(106\)
default	\(-\frac {\frac {a^{2} c^{2}}{2 x^{2}}+b^{2} \left (\frac {c^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 x^{2}}+\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {c \arctan \left (\frac {c}{x}\right )}{x}+\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}\right )+2 a b \left (\frac {c^{2} \arctan \left (\frac {c}{x}\right )}{2 x^{2}}-\frac {c}{2 x}+\frac {\arctan \left (\frac {c}{x}\right )}{2}\right )}{c^{2}}\)	\(106\)
parallelrisch	\(\frac {-x^{2} \arctan \left (\frac {c}{x}\right )^{2} b^{2}-\arctan \left (\frac {c}{x}\right )^{2} b^{2} c^{2}+2 b^{2} \ln \left (x \right ) x^{2}-b^{2} \ln \left (c^{2}+x^{2}\right ) x^{2}-2 x^{2} \arctan \left (\frac {c}{x}\right ) a b +2 x \arctan \left (\frac {c}{x}\right ) b^{2} c -2 \arctan \left (\frac {c}{x}\right ) a b \,c^{2}+2 a b c x -a^{2} c^{2}}{2 x^{2} c^{2}}\)	\(121\)
risch	\(\text {Expression too large to display}\)	\(59876\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx=\frac {2 \, a b x^{2} \arctan \left (\frac {x}{c}\right ) - b^{2} x^{2} \log \left (c^{2} + x^{2}\right ) + 2 \, b^{2} x^{2} \log \left (x\right ) - a^{2} c^{2} + 2 \, a b c x - {\left (b^{2} c^{2} + b^{2} x^{2}\right )} \arctan \left (\frac {c}{x}\right )^{2} - 2 \, {\left (a b c^{2} - b^{2} c x\right )} \arctan \left (\frac {c}{x}\right )}{2 \, c^{2} x^{2}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx=\begin {cases} - \frac {a^{2}}{2 x^{2}} - \frac {a b \operatorname {atan}{\left (\frac {c}{x} \right )}}{x^{2}} + \frac {a b}{c x} - \frac {a b \operatorname {atan}{\left (\frac {c}{x} \right )}}{c^{2}} - \frac {b^{2} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{2 x^{2}} + \frac {b^{2} \operatorname {atan}{\left (\frac {c}{x} \right )}}{c x} + \frac {b^{2} \log {\left (x \right )}}{c^{2}} - \frac {b^{2} \log {\left (c^{2} + x^{2} \right )}}{2 c^{2}} - \frac {b^{2} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{2 x^{2}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx={\left (c {\left (\frac {\arctan \left (\frac {x}{c}\right )}{c^{3}} + \frac {1}{c^{2} x}\right )} - \frac {\arctan \left (\frac {c}{x}\right )}{x^{2}}\right )} a b + \frac {1}{2} \, {\left (2 \, c {\left (\frac {\arctan \left (\frac {x}{c}\right )}{c^{3}} + \frac {1}{c^{2} x}\right )} \arctan \left (\frac {c}{x}\right ) + \frac {\arctan \left (\frac {x}{c}\right )^{2} - \log \left (c^{2} + x^{2}\right ) + 2 \, \log \left (x\right )}{c^{2}}\right )} b^{2} - \frac {b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]

[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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx=-\frac {b^{2} \arctan \left (\frac {c}{x}\right )^{2} + \frac {b^{2} c^{2} \arctan \left (\frac {c}{x}\right )^{2}}{x^{2}} + \frac {2 \, a b c^{2} \arctan \left (\frac {c}{x}\right )}{x^{2}} - \frac {2 \, b^{2} c \arctan \left (\frac {c}{x}\right )}{x} + i \, a b \log \left (\frac {i \, c}{x} - 1\right ) + b^{2} \log \left (\frac {i \, c}{x} - 1\right ) - i \, a b \log \left (-\frac {i \, c}{x} - 1\right ) + b^{2} \log \left (-\frac {i \, c}{x} - 1\right ) + \frac {a^{2} c^{2}}{x^{2}} - \frac {2 \, a b c}{x}}{2 \, c^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 2.93 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{x^3} \, dx=\frac {b^2\,\ln \left (x\right )-\frac {b^2\,\ln \left (x+c\,1{}\mathrm {i}\right )}{2}-\frac {b^2\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{2}+\frac {b^2\,\ln \left (\frac {1}{-x+c\,1{}\mathrm {i}}\right )}{2}+\frac {a\,b\,\ln \left (x+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\frac {a\,b\,\ln \left (-x+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}}{c^2}-\frac {\frac {a^2\,c^2}{2}-x\,\left (c\,\mathrm {atan}\left (\frac {c}{x}\right )\,b^2+a\,c\,b\right )+\frac {b^2\,c^2\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{2}+a\,b\,c^2\,\mathrm {atan}\left (\frac {c}{x}\right )}{c^2\,x^2} \]

[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)