Integrand size = 21, antiderivative size = 221 \[ \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}-\frac {b \left (3 c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 \sqrt {e}}-\frac {b d^{3/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e} \]
1/3*(e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/e-1/3*b*d^(3/2)*arctanh((e*x^2+d)^( 1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/e-1/6*b*( 3*c^2*d+e)*arctan(e^(1/2)*(-c^2*x^2+1)^(1/2)/c/(e*x^2+d)^(1/2))*(1/(c*x+1) )^(1/2)*(c*x+1)^(1/2)/c^3/e^(1/2)-1/6*b*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(- c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/c^2
Time = 22.76 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.39 \[ \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {\sqrt {d+e x^2} \left (-b e \sqrt {\frac {1-c x}{1+c x}} (1+c x)+2 a c^2 \left (d+e x^2\right )+2 b c^2 \left (d+e x^2\right ) \text {sech}^{-1}(c x)\right )}{6 c^2 e}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {-c^2 d-e} \sqrt {e} \left (3 c^2 d+e\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \arcsin \left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+2 c^5 d^{3/2} \sqrt {-d-e x^2} \arctan \left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{6 c^5 e (-1+c x) \sqrt {d+e x^2}} \]
(Sqrt[d + e*x^2]*(-(b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) + 2*a*c^2*(d + e*x^2) + 2*b*c^2*(d + e*x^2)*ArcSech[c*x]))/(6*c^2*e) + (b*Sqrt[(1 - c*x )/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*(3*c ^2*d + e)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c ^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 2*c^5*d^(3/2)*Sqrt[-d - e*x^2] *ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^5*e*(-1 + c*x )*Sqrt[d + e*x^2])
Time = 0.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6853, 2036, 354, 113, 27, 175, 66, 104, 218, 220}
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.
| \(\displaystyle \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6853 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (e x^2+d\right )^{3/2}}{x \sqrt {1-c x} \sqrt {c x+1}}dx}{3 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 2036 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (e x^2+d\right )^{3/2}}{x \sqrt {1-c^2 x^2}}dx}{3 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (e x^2+d\right )^{3/2}}{x^2 \sqrt {1-c^2 x^2}}dx^2}{6 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 113 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {\int -\frac {2 c^2 d^2+e \left (3 d c^2+e\right ) x^2}{2 x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{c^2}-\frac {e \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{c^2}\right )}{6 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {2 c^2 d^2+e \left (3 d c^2+e\right ) x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{c^2}\right )}{6 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {2 c^2 d^2 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+e \left (3 c^2 d+e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{c^2}\right )}{6 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {2 c^2 d^2 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+2 e \left (3 c^2 d+e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{c^2}\right )}{6 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4 c^2 d^2 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}+2 e \left (3 c^2 d+e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{c^2}\right )}{6 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4 c^2 d^2 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}-\frac {2 \sqrt {e} \left (3 c^2 d+e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{c^2}\right )}{6 e}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {-\frac {2 \sqrt {e} \left (3 c^2 d+e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}-4 c^2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{c^2}\right )}{6 e}\) |
((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e) + (b*Sqrt[(1 + c*x)^(-1)]*S qrt[1 + c*x]*(-((e*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/c^2) + ((-2*Sqrt[e]* (3*c^2*d + e)*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/c - 4*c^2*d^(3/2)*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*c^ 2)))/(6*e)
3.2.32.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
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] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p _.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2 *x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && E qQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && Gt Q[a2, 0]))
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSech[c*x])/(2*e*(p + 1))), x] + Simp[b*(Sqrt[1 + c*x]/(2*e*(p + 1)))*Sqrt[1/(1 + c*x)] Int[(d + e*x ^2)^(p + 1)/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e , p}, x] && NeQ[p, -1]
\[\int x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (131) = 262\).
Time = 0.45 (sec) , antiderivative size = 1382, normalized size of antiderivative = 6.25 \[ \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
[1/24*(2*b*c^3*d^(3/2)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4*((c^3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt(- (c^2*x^2 - 1)/(c^2*x^2)) + 8*d^2)/x^4) - (3*b*c^2*d + b*e)*sqrt(-e)*log(8* c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 - 4*(2*c^4*e *x^3 + (c^4*d - c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt(-(c^2*x^2 - 1)/(c^ 2*x^2)) + e^2) + 8*(b*c^3*e*x^2 + b*c^3*d)*sqrt(e*x^2 + d)*log((c*x*sqrt(- (c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(2*a*c^3*e*x^2 - b*c^2*e*x*sqrt(- (c^2*x^2 - 1)/(c^2*x^2)) + 2*a*c^3*d)*sqrt(e*x^2 + d))/(c^3*e), 1/12*(b*c^ 3*d^(3/2)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4 *((c^3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt(-(c^2*x^2 - 1) /(c^2*x^2)) + 8*d^2)/x^4) - (3*b*c^2*d + b*e)*sqrt(e)*arctan(1/2*(2*c^2*e* x^3 + (c^2*d - e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2) )/(c^2*e^2*x^4 + (c^2*d*e - e^2)*x^2 - d*e)) + 4*(b*c^3*e*x^2 + b*c^3*d)*s qrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(2* a*c^3*e*x^2 - b*c^2*e*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 2*a*c^3*d)*sqrt(e *x^2 + d))/(c^3*e), -1/24*(4*b*c^3*sqrt(-d)*d*arctan(-1/2*((c^3*d - c*e)*x ^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(-d)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))/(c^2 *d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) + (3*b*c^2*d + b*e)*sqrt(-e)*log(8* c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 - 4*(2*c^4*e *x^3 + (c^4*d - c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt(-(c^2*x^2 - 1)/...
\[ \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
\[ \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x \,d x } \]
1/3*((e*x^2 + d)^(3/2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/e - 3*integra te(1/3*sqrt(e*x^2 + d)*(6*(c^2*e*x^2 - e)*x*log(sqrt(x)) + 3*(c^2*e*x^2*lo g(c) - e*log(c))*x + (6*(c^2*e*x^2 - e)*x*log(sqrt(x)) + ((3*e*log(c) + e) *c^2*x^2 + c^2*d - 3*e*log(c))*x)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)) )/(c^2*e*x^2 + (c^2*e*x^2 - e)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)) - e), x))*b + 1/3*(e*x^2 + d)^(3/2)*a/e
\[ \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x \,d x } \]
Timed out. \[ \int x \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]