Integrand size = 12, antiderivative size = 142 \[ \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {5 b x \sqrt {1-c x}}{112 c^6 \sqrt {\frac {1}{1+c x}}}-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{112 c^7} \]
1/7*x^7*(a+b*arcsech(c*x))-5/112*b*x*(-c*x+1)^(1/2)/c^6/(1/(c*x+1))^(1/2)- 5/168*b*x^3*(-c*x+1)^(1/2)/c^4/(1/(c*x+1))^(1/2)-1/42*b*x^5*(-c*x+1)^(1/2) /c^2/(1/(c*x+1))^(1/2)+5/112*b*arcsin(c*x)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2) /c^7
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01 \[ \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {a x^7}{7}+b \sqrt {\frac {1-c x}{1+c x}} \left (-\frac {5 x}{112 c^6}-\frac {5 x^2}{112 c^5}-\frac {5 x^3}{168 c^4}-\frac {5 x^4}{168 c^3}-\frac {x^5}{42 c^2}-\frac {x^6}{42 c}\right )+\frac {1}{7} b x^7 \text {sech}^{-1}(c x)+\frac {5 i b \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{112 c^7} \]
(a*x^7)/7 + b*Sqrt[(1 - c*x)/(1 + c*x)]*((-5*x)/(112*c^6) - (5*x^2)/(112*c ^5) - (5*x^3)/(168*c^4) - (5*x^4)/(168*c^3) - x^5/(42*c^2) - x^6/(42*c)) + (b*x^7*ArcSech[c*x])/7 + (((5*I)/112)*b*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x) /(1 + c*x)]*(1 + c*x)])/c^7
Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6837, 111, 27, 111, 27, 101, 25, 39, 223}
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.
| \(\displaystyle \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6837 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^6}{\sqrt {1-c x} \sqrt {c x+1}}dx+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {\int -\frac {5 x^4}{\sqrt {1-c x} \sqrt {c x+1}}dx}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 \int \frac {x^4}{\sqrt {1-c x} \sqrt {c x+1}}dx}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 \left (-\frac {\int -\frac {3 x^2}{\sqrt {1-c x} \sqrt {c x+1}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c x} \sqrt {c x+1}}{4 c^2}\right )}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 \left (\frac {3 \int \frac {x^2}{\sqrt {1-c x} \sqrt {c x+1}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c x} \sqrt {c x+1}}{4 c^2}\right )}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 \left (\frac {3 \left (-\frac {\int -\frac {1}{\sqrt {1-c x} \sqrt {c x+1}}dx}{2 c^2}-\frac {x \sqrt {1-c x} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c x} \sqrt {c x+1}}{4 c^2}\right )}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c x} \sqrt {c x+1}}dx}{2 c^2}-\frac {x \sqrt {1-c x} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c x} \sqrt {c x+1}}{4 c^2}\right )}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle \frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c x} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c x} \sqrt {c x+1}}{4 c^2}\right )}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c x} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c x} \sqrt {c x+1}}{4 c^2}\right )}{6 c^2}-\frac {x^5 \sqrt {1-c x} \sqrt {c x+1}}{6 c^2}\right )\) |
(x^7*(a + b*ArcSech[c*x]))/7 + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(-1/6 *(x^5*Sqrt[1 - c*x]*Sqrt[1 + c*x])/c^2 + (5*(-1/4*(x^3*Sqrt[1 - c*x]*Sqrt[ 1 + c*x])/c^2 + (3*(-1/2*(x*Sqrt[1 - c*x]*Sqrt[1 + c*x])/c^2 + ArcSin[c*x] /(2*c^3)))/(4*c^2)))/(6*c^2)))/7
3.1.19.3.1 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_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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 = 0.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94
| method | result | size |
| parts | \(\frac {a \,x^{7}}{7}+\frac {b \left (\frac {c^{7} x^{7} \operatorname {arcsech}\left (c x \right )}{7}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (8 \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}+10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+15 c x \sqrt {-c^{2} x^{2}+1}-15 \arcsin \left (c x \right )\right )}{336 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{7}}\) | \(134\) |
| derivativedivides | \(\frac {\frac {a \,c^{7} x^{7}}{7}+b \left (\frac {c^{7} x^{7} \operatorname {arcsech}\left (c x \right )}{7}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (8 \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}+10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+15 c x \sqrt {-c^{2} x^{2}+1}-15 \arcsin \left (c x \right )\right )}{336 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{7}}\) | \(138\) |
| default | \(\frac {\frac {a \,c^{7} x^{7}}{7}+b \left (\frac {c^{7} x^{7} \operatorname {arcsech}\left (c x \right )}{7}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (8 \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}+10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+15 c x \sqrt {-c^{2} x^{2}+1}-15 \arcsin \left (c x \right )\right )}{336 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{7}}\) | \(138\) |
1/7*a*x^7+b/c^7*(1/7*c^7*x^7*arcsech(c*x)-1/336*(-(c*x-1)/c/x)^(1/2)*c*x*( (c*x+1)/c/x)^(1/2)*(8*(-c^2*x^2+1)^(1/2)*c^5*x^5+10*c^3*x^3*(-c^2*x^2+1)^( 1/2)+15*c*x*(-c^2*x^2+1)^(1/2)-15*arcsin(c*x))/(-c^2*x^2+1)^(1/2))
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.29 \[ \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {48 \, a c^{7} x^{7} - 48 \, b c^{7} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 30 \, b \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + 48 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{336 \, c^{7}} \]
1/336*(48*a*c^7*x^7 - 48*b*c^7*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1 )/x) - 30*b*arctan((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) + 48*(b *c^7*x^7 - b*c^7)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (8 *b*c^6*x^6 + 10*b*c^4*x^4 + 15*b*c^2*x^2)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/ c^7
\[ \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^{6} \left (a + b \operatorname {asech}{\left (c x \right )}\right )\, dx \]
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{7} \, a x^{7} + \frac {1}{336} \, {\left (48 \, x^{7} \operatorname {arsech}\left (c x\right ) - \frac {\frac {15 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{6}}}{c}\right )} b \]
1/7*a*x^7 + 1/336*(48*x^7*arcsech(c*x) - ((15*(1/(c^2*x^2) - 1)^(5/2) + 40 *(1/(c^2*x^2) - 1)^(3/2) + 33*sqrt(1/(c^2*x^2) - 1))/(c^6*(1/(c^2*x^2) - 1 )^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^6*(1/(c^2*x^2) - 1) + c^6) + 15*arct an(sqrt(1/(c^2*x^2) - 1))/c^6)/c)*b
\[ \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{6} \,d x } \]
Timed out. \[ \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^6\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]