Integrand size = 12, antiderivative size = 109 \[ \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {4 b \sqrt {1-c x}}{45 c^6 \sqrt {\frac {1}{1+c x}}}-\frac {2 b x^2 \sqrt {1-c x}}{45 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \]
1/6*x^6*(a+b*arcsech(c*x))-4/45*b*(-c*x+1)^(1/2)/c^6/(1/(c*x+1))^(1/2)-2/4 5*b*x^2*(-c*x+1)^(1/2)/c^4/(1/(c*x+1))^(1/2)-1/30*b*x^4*(-c*x+1)^(1/2)/c^2 /(1/(c*x+1))^(1/2)
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89 \[ \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {a x^6}{6}+b \sqrt {\frac {1-c x}{1+c x}} \left (-\frac {4}{45 c^6}-\frac {4 x}{45 c^5}-\frac {2 x^2}{45 c^4}-\frac {2 x^3}{45 c^3}-\frac {x^4}{30 c^2}-\frac {x^5}{30 c}\right )+\frac {1}{6} b x^6 \text {sech}^{-1}(c x) \]
(a*x^6)/6 + b*Sqrt[(1 - c*x)/(1 + c*x)]*(-4/(45*c^6) - (4*x)/(45*c^5) - (2 *x^2)/(45*c^4) - (2*x^3)/(45*c^3) - x^4/(30*c^2) - x^5/(30*c)) + (b*x^6*Ar cSech[c*x])/6
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6837, 111, 27, 111, 27, 83}
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.
\(\displaystyle \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
\(\Big \downarrow \) 6837 |
\(\displaystyle \frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^5}{\sqrt {1-c x} \sqrt {c x+1}}dx+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {\int -\frac {4 x^3}{\sqrt {1-c x} \sqrt {c x+1}}dx}{5 c^2}-\frac {x^4 \sqrt {1-c x} \sqrt {c x+1}}{5 c^2}\right )+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4 \int \frac {x^3}{\sqrt {1-c x} \sqrt {c x+1}}dx}{5 c^2}-\frac {x^4 \sqrt {1-c x} \sqrt {c x+1}}{5 c^2}\right )+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4 \left (-\frac {\int -\frac {2 x}{\sqrt {1-c x} \sqrt {c x+1}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c x} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c x} \sqrt {c x+1}}{5 c^2}\right )+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {1-c x} \sqrt {c x+1}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c x} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c x} \sqrt {c x+1}}{5 c^2}\right )+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4 \left (-\frac {2 \sqrt {1-c x} \sqrt {c x+1}}{3 c^4}-\frac {x^2 \sqrt {1-c x} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c x} \sqrt {c x+1}}{5 c^2}\right )\) |
(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(-1/5*(x^4*Sqrt[1 - c*x]*Sqrt[1 + c* x])/c^2 + (4*((-2*Sqrt[1 - c*x]*Sqrt[1 + c*x])/(3*c^4) - (x^2*Sqrt[1 - c*x ]*Sqrt[1 + c*x])/(3*c^2)))/(5*c^2)))/6 + (x^6*(a + b*ArcSech[c*x]))/6
3.1.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Si mp[(d*x)^(m + 1)*((a + b*ArcSech[c*x])/(d*(m + 1))), x] + Simp[b*(Sqrt[1 + c*x]/(m + 1))*Sqrt[1/(1 + c*x)] Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c*x]) , x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71
method | result | size |
parts | \(\frac {a \,x^{6}}{6}+\frac {b \left (\frac {c^{6} x^{6} \operatorname {arcsech}\left (c x \right )}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90}\right )}{c^{6}}\) | \(77\) |
derivativedivides | \(\frac {\frac {a \,c^{6} x^{6}}{6}+b \left (\frac {c^{6} x^{6} \operatorname {arcsech}\left (c x \right )}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90}\right )}{c^{6}}\) | \(81\) |
default | \(\frac {\frac {a \,c^{6} x^{6}}{6}+b \left (\frac {c^{6} x^{6} \operatorname {arcsech}\left (c x \right )}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90}\right )}{c^{6}}\) | \(81\) |
1/6*a*x^6+b/c^6*(1/6*c^6*x^6*arcsech(c*x)-1/90*(-(c*x-1)/c/x)^(1/2)*c*x*(( c*x+1)/c/x)^(1/2)*(3*c^4*x^4+4*c^2*x^2+8))
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {15 \, b c^{5} x^{6} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 15 \, a c^{5} x^{6} - {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{90 \, c^{5}} \]
1/90*(15*b*c^5*x^6*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 1 5*a*c^5*x^6 - (3*b*c^4*x^5 + 4*b*c^2*x^3 + 8*b*x)*sqrt(-(c^2*x^2 - 1)/(c^2 *x^2)))/c^5
Time = 0.71 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\begin {cases} \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {asech}{\left (c x \right )}}{6} - \frac {b x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {2 b x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {4 b \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} & \text {for}\: c \neq 0 \\\frac {x^{6} \left (a + \infty b\right )}{6} & \text {otherwise} \end {cases} \]
Piecewise((a*x**6/6 + b*x**6*asech(c*x)/6 - b*x**4*sqrt(-c**2*x**2 + 1)/(3 0*c**2) - 2*b*x**2*sqrt(-c**2*x**2 + 1)/(45*c**4) - 4*b*sqrt(-c**2*x**2 + 1)/(45*c**6), Ne(c, 0)), (x**6*(a + oo*b)/6, True))
Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{6} \, a x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b \]
1/6*a*x^6 + 1/90*(15*x^6*arcsech(c*x) - (3*c^4*x^5*(1/(c^2*x^2) - 1)^(5/2) - 10*c^2*x^3*(1/(c^2*x^2) - 1)^(3/2) + 15*x*sqrt(1/(c^2*x^2) - 1))/c^5)*b
\[ \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5} \,d x } \]
Timed out. \[ \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]