Integrand size = 14, antiderivative size = 48 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=-\frac {b}{12 c x^4}-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}-\frac {b \log \left (1-\frac {c^2}{x^4}\right )}{12 c^3} \] Output:
-1/12*b/c/x^4-1/6*(a+b*arctanh(c/x^2))/x^6-1/12*b*ln(1-c^2/x^4)/c^3
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=-\frac {a}{6 x^6}-\frac {b}{12 c x^4}-\frac {b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}+\frac {b \log (x)}{3 c^3}-\frac {b \log \left (-c^2+x^4\right )}{12 c^3} \] Input:
Integrate[(a + b*ArcTanh[c/x^2])/x^7,x]
Output:
-1/6*a/x^6 - b/(12*c*x^4) - (b*ArcTanh[c/x^2])/(6*x^6) + (b*Log[x])/(3*c^3 ) - (b*Log[-c^2 + x^4])/(12*c^3)
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6452, 795, 798, 25, 54, 2009}
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.
\(\displaystyle \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle -\frac {1}{3} b c \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^9}dx-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}\) |
\(\Big \downarrow \) 795 |
\(\displaystyle -\frac {1}{3} b c \int \frac {1}{x^5 \left (x^4-c^2\right )}dx-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{12} b c \int -\frac {1}{x^8 \left (c^2-x^4\right )}dx^4-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{12} b c \int \frac {1}{x^8 \left (c^2-x^4\right )}dx^4-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{12} b c \int \left (\frac {1}{c^4 x^4}+\frac {1}{c^2 x^8}+\frac {1}{c^4 \left (c^2-x^4\right )}\right )dx^4-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{6 x^6}-\frac {1}{12} b c \left (-\frac {\log \left (x^4\right )}{c^4}+\frac {1}{c^2 x^4}+\frac {\log \left (c^2-x^4\right )}{c^4}\right )\) |
Input:
Int[(a + b*ArcTanh[c/x^2])/x^7,x]
Output:
-1/6*(a + b*ArcTanh[c/x^2])/x^6 - (b*c*(1/(c^2*x^4) - Log[x^4]/c^4 + Log[c ^2 - x^4]/c^4))/12
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 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_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
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 = 0.40 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {a}{6 x^{6}}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{6 x^{6}}-\frac {b}{12 c \,x^{4}}-\frac {b \ln \left (\frac {c^{2}}{x^{4}}-1\right )}{12 c^{3}}\) | \(45\) |
default | \(-\frac {a}{6 x^{6}}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{6 x^{6}}-\frac {b}{12 c \,x^{4}}-\frac {b \ln \left (\frac {c^{2}}{x^{4}}-1\right )}{12 c^{3}}\) | \(45\) |
parts | \(-\frac {a}{6 x^{6}}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{6 x^{6}}-\frac {b}{12 c \,x^{4}}-\frac {b \ln \left (\frac {c^{2}}{x^{4}}-1\right )}{12 c^{3}}\) | \(45\) |
parallelrisch | \(\frac {4 b \ln \left (x \right ) x^{6}-2 \ln \left (x^{2}-c \right ) x^{6} b -2 x^{6} \operatorname {arctanh}\left (\frac {c}{x^{2}}\right ) b -b \,c^{2} x^{2}-2 \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right ) b \,c^{3}-2 a \,c^{3}}{12 x^{6} c^{3}}\) | \(71\) |
risch | \(-\frac {b \ln \left (x^{2}+c \right )}{12 x^{6}}-\frac {-2 i \pi b \,c^{3}-i \pi b \,c^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i \pi b \,c^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-i \pi b \,c^{3} {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}+2 i \pi b \,c^{3} {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i \pi b \,c^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i \pi b \,c^{3} \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}-i \pi b \,c^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )-i \pi b \,c^{3} {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}+i \pi b \,c^{3} \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-8 b \ln \left (x \right ) x^{6}+2 b \ln \left (-x^{4}+c^{2}\right ) x^{6}-2 b \ln \left (-x^{2}+c \right ) c^{3}+2 b \,c^{2} x^{2}+4 a \,c^{3}}{24 c^{3} x^{6}}\) | \(356\) |
Input:
int((a+b*arctanh(c/x^2))/x^7,x,method=_RETURNVERBOSE)
Output:
-1/6*a/x^6-1/6*b/x^6*arctanh(c/x^2)-1/12*b/c/x^4-1/12*b/c^3*ln(c^2/x^4-1)
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.40 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=-\frac {b x^{6} \log \left (x^{4} - c^{2}\right ) - 4 \, b x^{6} \log \left (x\right ) + b c^{2} x^{2} + b c^{3} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{3}}{12 \, c^{3} x^{6}} \] Input:
integrate((a+b*arctanh(c/x^2))/x^7,x, algorithm="fricas")
Output:
-1/12*(b*x^6*log(x^4 - c^2) - 4*b*x^6*log(x) + b*c^2*x^2 + b*c^3*log((x^2 + c)/(x^2 - c)) + 2*a*c^3)/(c^3*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (42) = 84\).
Time = 7.97 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.96 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=\begin {cases} - \frac {a}{6 x^{6}} - \frac {b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{6 x^{6}} - \frac {b}{12 c x^{4}} + \frac {b \log {\left (x \right )}}{3 c^{3}} - \frac {b \log {\left (x - \sqrt {- c} \right )}}{6 c^{3}} - \frac {b \log {\left (x + \sqrt {- c} \right )}}{6 c^{3}} + \frac {b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{6 c^{3}} & \text {for}\: c \neq 0 \\- \frac {a}{6 x^{6}} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*atanh(c/x**2))/x**7,x)
Output:
Piecewise((-a/(6*x**6) - b*atanh(c/x**2)/(6*x**6) - b/(12*c*x**4) + b*log( x)/(3*c**3) - b*log(x - sqrt(-c))/(6*c**3) - b*log(x + sqrt(-c))/(6*c**3) + b*atanh(c/x**2)/(6*c**3), Ne(c, 0)), (-a/(6*x**6), True))
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=-\frac {1}{12} \, {\left (c {\left (\frac {\log \left (x^{4} - c^{2}\right )}{c^{4}} - \frac {\log \left (x^{4}\right )}{c^{4}} + \frac {1}{c^{2} x^{4}}\right )} + \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x^{6}}\right )} b - \frac {a}{6 \, x^{6}} \] Input:
integrate((a+b*arctanh(c/x^2))/x^7,x, algorithm="maxima")
Output:
-1/12*(c*(log(x^4 - c^2)/c^4 - log(x^4)/c^4 + 1/(c^2*x^4)) + 2*arctanh(c/x ^2)/x^6)*b - 1/6*a/x^6
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.35 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=-\frac {b \log \left (x^{4} - c^{2}\right )}{12 \, c^{3}} + \frac {b \log \left (x\right )}{3 \, c^{3}} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{12 \, x^{6}} - \frac {b x^{2} + 2 \, a c}{12 \, c x^{6}} \] Input:
integrate((a+b*arctanh(c/x^2))/x^7,x, algorithm="giac")
Output:
-1/12*b*log(x^4 - c^2)/c^3 + 1/3*b*log(x)/c^3 - 1/12*b*log((x^2 + c)/(x^2 - c))/x^6 - 1/12*(b*x^2 + 2*a*c)/(c*x^6)
Time = 3.79 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.38 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=\frac {b\,\ln \left (x\right )}{3\,c^3}-\frac {b\,\ln \left (x^4-c^2\right )}{12\,c^3}-\frac {b}{12\,c\,x^4}-\frac {a}{6\,x^6}-\frac {b\,\ln \left (x^2+c\right )}{12\,x^6}+\frac {b\,\ln \left (x^2-c\right )}{12\,x^6} \] Input:
int((a + b*atanh(c/x^2))/x^7,x)
Output:
(b*log(x))/(3*c^3) - (b*log(x^4 - c^2))/(12*c^3) - b/(12*c*x^4) - a/(6*x^6 ) - (b*log(c + x^2))/(12*x^6) + (b*log(x^2 - c))/(12*x^6)
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.42 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^7} \, dx=\frac {-2 \mathit {atanh} \left (\frac {c}{x^{2}}\right ) b \,c^{3}+2 \mathit {atanh} \left (\frac {c}{x^{2}}\right ) b \,x^{6}-2 \,\mathrm {log}\left (x^{2}+c \right ) b \,x^{6}+4 \,\mathrm {log}\left (x \right ) b \,x^{6}-2 a \,c^{3}-b \,c^{2} x^{2}}{12 c^{3} x^{6}} \] Input:
int((a+b*atanh(c/x^2))/x^7,x)
Output:
( - 2*atanh(c/x**2)*b*c**3 + 2*atanh(c/x**2)*b*x**6 - 2*log(c + x**2)*b*x* *6 + 4*log(x)*b*x**6 - 2*a*c**3 - b*c**2*x**2)/(12*c**3*x**6)