Integrand size = 14, antiderivative size = 65 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx=-\frac {2 b}{3 c x}-\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}+\frac {b \text {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}} \]
-2/3*b/c/x-1/3*b*arctan(x/c^(1/2))/c^(3/2)+1/3*(-a-b*arctanh(c/x^2))/x^3+1 /3*b*arctanh(x/c^(1/2))/c^(3/2)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {2 b}{3 c x}-\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {b \log \left (\sqrt {c}-x\right )}{6 c^{3/2}}+\frac {b \log \left (\sqrt {c}+x\right )}{6 c^{3/2}} \]
-1/3*a/x^3 - (2*b)/(3*c*x) - (b*ArcTan[x/Sqrt[c]])/(3*c^(3/2)) - (b*ArcTan h[c/x^2])/(3*x^3) - (b*Log[Sqrt[c] - x])/(6*c^(3/2)) + (b*Log[Sqrt[c] + x] )/(6*c^(3/2))
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6452, 795, 847, 25, 827, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle -\frac {2}{3} b c \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^6}dx-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle -\frac {2}{3} b c \int \frac {1}{x^2 \left (x^4-c^2\right )}dx-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {2}{3} b c \left (\frac {\int -\frac {x^2}{c^2-x^4}dx}{c^2}+\frac {1}{c^2 x}\right )-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{3} b c \left (\frac {1}{c^2 x}-\frac {\int \frac {x^2}{c^2-x^4}dx}{c^2}\right )-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {2}{3} b c \left (\frac {1}{c^2 x}-\frac {\frac {1}{2} \int \frac {1}{c-x^2}dx-\frac {1}{2} \int \frac {1}{x^2+c}dx}{c^2}\right )-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2}{3} b c \left (\frac {1}{c^2 x}-\frac {\frac {1}{2} \int \frac {1}{c-x^2}dx-\frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{2 \sqrt {c}}}{c^2}\right )-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {2}{3} b c \left (\frac {1}{c^2 x}-\frac {\frac {\text {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{2 \sqrt {c}}}{c^2}\right )\) |
-1/3*(a + b*ArcTanh[c/x^2])/x^3 - (2*b*c*(1/(c^2*x) - (-1/2*ArcTan[x/Sqrt[ c]]/Sqrt[c] + ArcTanh[x/Sqrt[c]]/(2*Sqrt[c]))/c^2))/3
3.2.69.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[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]
Time = 0.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85
| method | result | size |
| parts | \(-\frac {a}{3 x^{3}}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{3 x^{3}}-\frac {2 b}{3 c x}+\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}-\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\) | \(55\) |
| derivativedivides | \(-\frac {a}{3 x^{3}}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{3 x^{3}}-\frac {2 b}{3 c x}+\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}+\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}\) | \(57\) |
| default | \(-\frac {a}{3 x^{3}}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{3 x^{3}}-\frac {2 b}{3 c x}+\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}+\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}\) | \(57\) |
| risch | \(-\frac {b \ln \left (x^{2}+c \right )}{6 x^{3}}-\frac {-i b \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}+4 a +2 i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi \,\operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}-2 i b \pi }{12 x^{3}}+\frac {b \ln \left (-x^{2}+c \right )}{6 x^{3}}-\frac {2 b}{3 c x}+\frac {b \,\operatorname {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}-\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\) | \(320\) |
-1/3*a/x^3-1/3*b/x^3*arctanh(c/x^2)-2/3*b/c/x+1/3*b/c^(3/2)*arctanh(1/x*c^ (1/2))-1/3*b*arctan(x/c^(1/2))/c^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (49) = 98\).
Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.91 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx=\left [-\frac {2 \, b \sqrt {c} x^{3} \arctan \left (\frac {x}{\sqrt {c}}\right ) - b \sqrt {c} x^{3} \log \left (\frac {x^{2} + 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}, -\frac {2 \, b \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + b \sqrt {-c} x^{3} \log \left (\frac {x^{2} + 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}\right ] \]
[-1/6*(2*b*sqrt(c)*x^3*arctan(x/sqrt(c)) - b*sqrt(c)*x^3*log((x^2 + 2*sqrt (c)*x + c)/(x^2 - c)) + 4*b*c*x^2 + b*c^2*log((x^2 + c)/(x^2 - c)) + 2*a*c ^2)/(c^2*x^3), -1/6*(2*b*sqrt(-c)*x^3*arctan(sqrt(-c)*x/c) + b*sqrt(-c)*x^ 3*log((x^2 + 2*sqrt(-c)*x - c)/(x^2 + c)) + 4*b*c*x^2 + b*c^2*log((x^2 + c )/(x^2 - c)) + 2*a*c^2)/(c^2*x^3)]
Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (60) = 120\).
Time = 4.82 (sec) , antiderivative size = 1046, normalized size of antiderivative = 16.09 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx=\begin {cases} - \frac {a}{3 x^{3}} & \text {for}\: c = 0 \\- \frac {a - \infty b}{3 x^{3}} & \text {for}\: c = - x^{2} \\- \frac {a + \infty b}{3 x^{3}} & \text {for}\: c = x^{2} \\\frac {2 a c^{\frac {17}{2}} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 a c^{\frac {13}{2}} x^{4} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {2 b c^{\frac {17}{2}} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{\frac {15}{2}} x^{3} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{\frac {15}{2}} x^{3} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {4 b c^{\frac {15}{2}} x^{2} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 b c^{\frac {13}{2}} x^{4} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{\frac {11}{2}} x^{7} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{\frac {11}{2}} x^{7} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {4 b c^{\frac {11}{2}} x^{6} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {2 b c^{7} x^{3} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{7} x^{3} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{7} x^{3} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {2 b c^{7} x^{3} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 b c^{5} x^{7} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{5} x^{7} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{5} x^{7} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 b c^{5} x^{7} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} & \text {otherwise} \end {cases} \]
Piecewise((-a/(3*x**3), Eq(c, 0)), (-(a - oo*b)/(3*x**3), Eq(c, -x**2)), ( -(a + oo*b)/(3*x**3), Eq(c, x**2)), (2*a*c**(17/2)*sqrt(-c)/(-6*c**(17/2)* x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 2*a*c**(13/2)*x**4*sqrt(-c)/( -6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + 2*b*c**(17/2)*sq rt(-c)*atanh(c/x**2)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(- c)) + b*c**(15/2)*x**3*log(x - sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c **(13/2)*x**7*sqrt(-c)) - b*c**(15/2)*x**3*log(x + sqrt(-c))/(-6*c**(17/2) *x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + 4*b*c**(15/2)*x**2*sqrt(-c)/ (-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 2*b*c**(13/2)*x **4*sqrt(-c)*atanh(c/x**2)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7* sqrt(-c)) - b*c**(11/2)*x**7*log(x - sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + b*c**(11/2)*x**7*log(x + sqrt(-c))/(-6*c** (17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 4*b*c**(11/2)*x**6*sqr t(-c)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + 2*b*c**7* x**3*sqrt(-c)*log(-sqrt(c) + x)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)* x**7*sqrt(-c)) - b*c**7*x**3*sqrt(-c)*log(x - sqrt(-c))/(-6*c**(17/2)*x**3 *sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - b*c**7*x**3*sqrt(-c)*log(x + sqrt (-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + 2*b*c**7* x**3*sqrt(-c)*atanh(c/x**2)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7 *sqrt(-c)) - 2*b*c**5*x**7*sqrt(-c)*log(-sqrt(c) + x)/(-6*c**(17/2)*x**...
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx=-\frac {1}{6} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {4}{c^{2} x}\right )} + \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \]
-1/6*(c*(2*arctan(x/sqrt(c))/c^(5/2) + log((x - sqrt(c))/(x + sqrt(c)))/c^ (5/2) + 4/(c^2*x)) + 2*arctanh(c/x^2)/x^3)*b - 1/3*a/x^3
Time = 0.32 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx=-\frac {b \arctan \left (\frac {x}{\sqrt {-c}}\right )}{3 \, \sqrt {-c} c} - \frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 \, c^{\frac {3}{2}}} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{6 \, x^{3}} - \frac {2 \, b x^{2} + a c}{3 \, c x^{3}} \]
-1/3*b*arctan(x/sqrt(-c))/(sqrt(-c)*c) - 1/3*b*arctan(x/sqrt(c))/c^(3/2) - 1/6*b*log((x^2 + c)/(x^2 - c))/x^3 - 1/3*(2*b*x^2 + a*c)/(c*x^3)
Time = 3.60 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.06 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^4} \, dx=\frac {b\,\ln \left (x^2-c\right )}{6\,x^3}-\frac {2\,b}{3\,c\,x}-\frac {b\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )}{3\,c^{3/2}}-\frac {b\,\ln \left (x^2+c\right )}{6\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{3\,c^{3/2}} \]