Integrand size = 19, antiderivative size = 217 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{4 d^2 e}-\frac {b \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}} \]
1/4*(-a-b*arcsech(c*x))/e/(e*x^2+d)^2+1/4*b*arctanh((-c^2*x^2+1)^(1/2))*(1 /(c*x+1))^(1/2)*(c*x+1)^(1/2)/d^2/e-1/8*b*(3*c^2*d+2*e)*arctanh(e^(1/2)*(- c^2*x^2+1)^(1/2)/(c^2*d+e)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/d^2/(c^2 *d+e)^(3/2)/e^(1/2)-1/8*b*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/ 2)/d/(c^2*d+e)/(e*x^2+d)
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.24 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}-\frac {2 \sqrt {\frac {1-c x}{1+c x}} (b+b c x)}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \text {sech}^{-1}(c x)}{e \left (d+e x^2\right )^2}-\frac {4 b \log (x)}{d^2 e}+\frac {4 b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d^2 e}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 d^2 \sqrt {e} \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d+e} x \sqrt {\frac {1-c x}{1+c x}}\right )}{b \left (3 c^2 d+2 e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}\right )}{d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 d^2 \sqrt {e} \sqrt {c^2 d+e} \left (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d+e} x \sqrt {\frac {1-c x}{1+c x}}\right )}{b \left (3 c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}\right ) \]
((-4*a)/(e*(d + e*x^2)^2) - (2*Sqrt[(1 - c*x)/(1 + c*x)]*(b + b*c*x))/(d*( c^2*d + e)*(d + e*x^2)) - (4*b*ArcSech[c*x])/(e*(d + e*x^2)^2) - (4*b*Log[ x])/(d^2*e) + (4*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/ (1 + c*x)]])/(d^2*e) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(3*c^2*d + 2*e)*((-I )*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(c^2*d + e)^(3/2)) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt [c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x) /(1 + c*x)]))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*( c^2*d + e)^(3/2)))/16
Time = 0.53 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6853, 2036, 354, 114, 27, 174, 73, 221}
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.
| \(\displaystyle \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6853 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \left (e x^2+d\right )^2}dx}{4 e}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 2036 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x \sqrt {1-c^2 x^2} \left (e x^2+d\right )^2}dx}{4 e}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (e x^2+d\right )^2}dx^2}{8 e}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {-e x^2 c^2+2 d c^2+2 e}{2 x^2 \sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx^2}{d \left (c^2 d+e\right )}+\frac {e \sqrt {1-c^2 x^2}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {2 \left (d c^2+e\right )-c^2 e x^2}{x^2 \sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx^2}{2 d \left (c^2 d+e\right )}+\frac {e \sqrt {1-c^2 x^2}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {2 \left (c^2 d+e\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2}{d}-\frac {e \left (3 c^2 d+2 e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx^2}{d}}{2 d \left (c^2 d+e\right )}+\frac {e \sqrt {1-c^2 x^2}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {2 e \left (3 c^2 d+2 e\right ) \int \frac {1}{-\frac {e x^4}{c^2}+d+\frac {e}{c^2}}d\sqrt {1-c^2 x^2}}{c^2 d}-\frac {4 \left (c^2 d+e\right ) \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2 d}}{2 d \left (c^2 d+e\right )}+\frac {e \sqrt {1-c^2 x^2}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {2 \sqrt {e} \left (3 c^2 d+2 e\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{d \sqrt {c^2 d+e}}-\frac {4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \left (c^2 d+e\right )}{d}}{2 d \left (c^2 d+e\right )}+\frac {e \sqrt {1-c^2 x^2}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}\right )}{8 e}\) |
-1/4*(a + b*ArcSech[c*x])/(e*(d + e*x^2)^2) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt [1 + c*x]*((e*Sqrt[1 - c^2*x^2])/(d*(c^2*d + e)*(d + e*x^2)) + ((-4*(c^2*d + e)*ArcTanh[Sqrt[1 - c^2*x^2]])/d + (2*Sqrt[e]*(3*c^2*d + 2*e)*ArcTanh[( Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(d*Sqrt[c^2*d + e]))/(2*d*(c^ 2*d + e))))/(8*e)
3.2.25.3.1 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_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1317\) vs. \(2(186)=372\).
Time = 5.10 (sec) , antiderivative size = 1318, normalized size of antiderivative = 6.07
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1318\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1329\) |
| default | \(\text {Expression too large to display}\) | \(1329\) |
-1/4*a/e/(e*x^2+d)^2+b/c^2*(-1/4*c^6/e/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)-1/ 16*c^3*e^2*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(4*((c^2*d+e)/e)^(1/ 2)*arctanh(1/(-c^2*x^2+1)^(1/2))*c^6*d^2*e*x^2+4*((c^2*d+e)/e)^(1/2)*arcta nh(1/(-c^2*x^2+1)^(1/2))*c^6*d^3-3*ln(-2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1) ^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1/2)))*x^2*c^6*d^2*e- 3*ln(-2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/ (-c*e*x+(-c^2*d*e)^(1/2)))*c^6*d^3-3*ln(2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1 )^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^6*d^2*e*x^2- 3*ln(2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/( c*e*x+(-c^2*d*e)^(1/2)))*c^6*d^3+8*((c^2*d+e)/e)^(1/2)*arctanh(1/(-c^2*x^2 +1)^(1/2))*c^4*d*e^2*x^2+8*((c^2*d+e)/e)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2 ))*c^4*d^2*e-2*(-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*c^4*d^2*e-5*ln(-2*(( (c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(- c^2*d*e)^(1/2)))*x^2*c^4*d*e^2-5*ln(-2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^( 1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2*e-5*ln(2 *(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*e*x+ (-c^2*d*e)^(1/2)))*c^4*d*e^2*x^2-5*ln(2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^ (1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2*e+4*((c^ 2*d+e)/e)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))*e^3*c^2*x^2+4*((c^2*d+e)/e)^ (1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))*c^2*d*e^2-2*(-c^2*x^2+1)^(1/2)*((c^...
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (135) = 270\).
Time = 0.37 (sec) , antiderivative size = 1232, normalized size of antiderivative = 5.68 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/16*(4*a*c^4*d^4 + 2*(4*a + b)*c^2*d^3*e + 2*(2*a + b)*d^2*e^2 + 2*(b*c ^2*d*e^3 + b*e^4)*x^4 + 4*(b*c^2*d^2*e^2 + b*d*e^3)*x^2 - (3*b*c^2*d^3 + ( 3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^2*d^2*e + 2*b*d*e^2)*x ^2)*sqrt(c^2*d*e + e^2)*log((c^4*d^2 + 4*c^2*d*e - (c^4*d*e + 2*c^2*e^2)*x ^2 + 4*(c^3*d*e + c*e^2)*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 4*e^2 + 2*(c^2 *e*x^2 - c^2*d - (c^3*d + 2*c*e)*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 2*e)*s qrt(c^2*d*e + e^2))/(e*x^2 + d)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^ 2 + (b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2 *d^2*e^2 + b*d*e^3)*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c ^2*x^2)) + 1)/(c*x)) + 2*((b*c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^6*e + 2*c^2*d^5*e^ 2 + d^4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4)*x^2), -1/8*(2*a*c^4*d^4 + (4*a + b)*c^2*d^3*e + (2*a + b)*d^2*e^2 + (b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^2*d^2*e^2 + b*d*e ^3)*x^2 + (3*b*c^2*d^3 + (3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3* b*c^2*d^2*e + 2*b*d*e^2)*x^2)*sqrt(-c^2*d*e - e^2)*arctan((sqrt(-c^2*d*e - e^2)*c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - sqrt(-c^2*d*e - e^2)*(e*x^2 + d))/((c^2*d*e + e^2)*x^2)) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + ( b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d...
Timed out. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]