Integrand size = 12, antiderivative size = 98 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx=\frac {b c \sqrt {1+\frac {1}{c^2 x^2}}}{36 x^5}-\frac {5 b c^3 \sqrt {1+\frac {1}{c^2 x^2}}}{144 x^3}+\frac {5 b c^5 \sqrt {1+\frac {1}{c^2 x^2}}}{96 x}-\frac {5}{96} b c^6 \text {csch}^{-1}(c x)-\frac {a+b \text {csch}^{-1}(c x)}{6 x^6} \]
-5/96*b*c^6*arccsch(c*x)+1/6*(-a-b*arccsch(c*x))/x^6+1/36*b*c*(1+1/c^2/x^2 )^(1/2)/x^5-5/144*b*c^3*(1+1/c^2/x^2)^(1/2)/x^3+5/96*b*c^5*(1+1/c^2/x^2)^( 1/2)/x
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx=-\frac {a}{6 x^6}+b \left (\frac {c}{36 x^5}-\frac {5 c^3}{144 x^3}+\frac {5 c^5}{96 x}\right ) \sqrt {\frac {1+c^2 x^2}{c^2 x^2}}-\frac {b \text {csch}^{-1}(c x)}{6 x^6}-\frac {5}{96} b c^6 \text {arcsinh}\left (\frac {1}{c x}\right ) \]
-1/6*a/x^6 + b*(c/(36*x^5) - (5*c^3)/(144*x^3) + (5*c^5)/(96*x))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/(6*x^6) - (5*b*c^6*ArcSinh[1/(c*x) ])/96
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6838, 858, 262, 262, 262, 222}
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.
\(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx\) |
\(\Big \downarrow \) 6838 |
\(\displaystyle -\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^8}dx}{6 c}-\frac {a+b \text {csch}^{-1}(c x)}{6 x^6}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^6}d\frac {1}{x}}{6 c}-\frac {a+b \text {csch}^{-1}(c x)}{6 x^6}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {b \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 x^5}-\frac {5}{6} c^2 \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^4}d\frac {1}{x}\right )}{6 c}-\frac {a+b \text {csch}^{-1}(c x)}{6 x^6}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {b \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 x^5}-\frac {5}{6} c^2 \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{4 x^3}-\frac {3}{4} c^2 \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}d\frac {1}{x}\right )\right )}{6 c}-\frac {a+b \text {csch}^{-1}(c x)}{6 x^6}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {b \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 x^5}-\frac {5}{6} c^2 \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{4 x^3}-\frac {3}{4} c^2 \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{2 x}-\frac {1}{2} c^2 \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}\right )\right )\right )}{6 c}-\frac {a+b \text {csch}^{-1}(c x)}{6 x^6}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {b \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 x^5}-\frac {5}{6} c^2 \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{4 x^3}-\frac {3}{4} c^2 \left (\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{2 x}-\frac {1}{2} c^3 \text {arcsinh}\left (\frac {1}{c x}\right )\right )\right )\right )}{6 c}-\frac {a+b \text {csch}^{-1}(c x)}{6 x^6}\) |
-1/6*(a + b*ArcCsch[c*x])/x^6 + (b*((c^2*Sqrt[1 + 1/(c^2*x^2)])/(6*x^5) - (5*c^2*((c^2*Sqrt[1 + 1/(c^2*x^2)])/(4*x^3) - (3*c^2*((c^2*Sqrt[1 + 1/(c^2 *x^2)])/(2*x) - (c^3*ArcSinh[1/(c*x)])/2))/4))/6))/(6*c)
3.1.14.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Si mp[(d*x)^(m + 1)*((a + b*ArcCsch[c*x])/(d*(m + 1))), x] + Simp[b*(d/(c*(m + 1))) Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.39 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38
method | result | size |
parts | \(-\frac {a}{6 x^{6}}+b \,c^{6} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{6 c^{6} x^{6}}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) c^{6} x^{6}-15 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\right )}{288 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{7} x^{7}}\right )\) | \(135\) |
derivativedivides | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}+b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{6 c^{6} x^{6}}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) c^{6} x^{6}-15 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\right )}{288 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{7} x^{7}}\right )\right )\) | \(139\) |
default | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}+b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{6 c^{6} x^{6}}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) c^{6} x^{6}-15 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\right )}{288 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{7} x^{7}}\right )\right )\) | \(139\) |
-1/6*a/x^6+b*c^6*(-1/6/c^6/x^6*arccsch(c*x)-1/288*(c^2*x^2+1)^(1/2)*(15*ar ctanh(1/(c^2*x^2+1)^(1/2))*c^6*x^6-15*(c^2*x^2+1)^(1/2)*c^4*x^4+10*(c^2*x^ 2+1)^(1/2)*c^2*x^2-8*(c^2*x^2+1)^(1/2))/((c^2*x^2+1)/c^2/x^2)^(1/2)/c^7/x^ 7)
Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx=-\frac {3 \, {\left (5 \, b c^{6} x^{6} + 16 \, b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (15 \, b c^{5} x^{5} - 10 \, b c^{3} x^{3} + 8 \, b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 48 \, a}{288 \, x^{6}} \]
-1/288*(3*(5*b*c^6*x^6 + 16*b)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1) /(c*x)) - (15*b*c^5*x^5 - 10*b*c^3*x^3 + 8*b*c*x)*sqrt((c^2*x^2 + 1)/(c^2* x^2)) + 48*a)/x^6
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{7}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (82) = 164\).
Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.89 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx=-\frac {1}{576} \, b {\left (\frac {15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - 15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right ) - \frac {2 \, {\left (15 \, c^{12} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, c^{10} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, c^{8} x \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} x^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - 1}}{c} + \frac {96 \, \operatorname {arcsch}\left (c x\right )}{x^{6}}\right )} - \frac {a}{6 \, x^{6}} \]
-1/576*b*((15*c^7*log(c*x*sqrt(1/(c^2*x^2) + 1) + 1) - 15*c^7*log(c*x*sqrt (1/(c^2*x^2) + 1) - 1) - 2*(15*c^12*x^5*(1/(c^2*x^2) + 1)^(5/2) - 40*c^10* x^3*(1/(c^2*x^2) + 1)^(3/2) + 33*c^8*x*sqrt(1/(c^2*x^2) + 1))/(c^6*x^6*(1/ (c^2*x^2) + 1)^3 - 3*c^4*x^4*(1/(c^2*x^2) + 1)^2 + 3*c^2*x^2*(1/(c^2*x^2) + 1) - 1))/c + 96*arccsch(c*x)/x^6) - 1/6*a/x^6
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^7} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^7} \,d x \]