Integrand size = 14, antiderivative size = 114 \[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{2 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {1}{2 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{2 b^3}-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{2 b^3} \]
1/2/b^2/x/(a+b*arcsech(c*x))-1/2*c*cosh(a/b)*Shi(a/b+arcsech(c*x))/b^3+1/2 *c*Chi(a/b+arcsech(c*x))*sinh(a/b)/b^3+1/2*(c*x+1)*((-c*x+1)/(c*x+1))^(1/2 )/b/x/(a+b*arcsech(c*x))^2
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {\frac {b^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {b}{a x+b x \text {sech}^{-1}(c x)}+c \left (\text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )}{2 b^3} \]
((b^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x*(a + b*ArcSech[c*x])^2) + b/ (a*x + b*x*ArcSech[c*x]) + c*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]]))/(2*b^3)
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.24, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6839, 3042, 26, 3778, 3042, 3778, 26, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 6839 |
| \(\displaystyle -c \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x \left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c \int -\frac {i \sin \left (i \text {sech}^{-1}(c x)\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i c \int \frac {\sin \left (i \text {sech}^{-1}(c x)\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle i c \left (\frac {i \int \frac {1}{c x \left (a+b \text {sech}^{-1}(c x)\right )^2}d\text {sech}^{-1}(c x)}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i c \left (\frac {i \int \frac {\sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^2}d\text {sech}^{-1}(c x)}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {i \int -\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x \left (a+b \text {sech}^{-1}(c x)\right )}d\text {sech}^{-1}(c x)}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i c \left (\frac {i \left (\frac {\int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x \left (a+b \text {sech}^{-1}(c x)\right )}d\text {sech}^{-1}(c x)}{b}-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {\int -\frac {i \sin \left (i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {i \int \frac {\sin \left (i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {i \left (\cosh \left (\frac {a}{b}\right ) \int \frac {i \sinh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {i \left (i \cosh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {i \left (i \cosh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {i \left (\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {i \left (\frac {i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}-i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle i c \left (\frac {i \left (-\frac {1}{b c x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {i \left (\frac {i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}-\frac {i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}\right )}{b}\right )}{2 b}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 b c x \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
I*c*(((-1/2*I)*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(b*c*x*(a + b*ArcSech[ c*x])^2) + ((I/2)*(-(1/(b*c*x*(a + b*ArcSech[c*x]))) - (I*(((-I)*CoshInteg ral[a/b + ArcSech[c*x]]*Sinh[a/b])/b + (I*Cosh[a/b]*SinhIntegral[a/b + Arc Sech[c*x]])/b))/b))/b)
3.1.66.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ -(c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G tQ[n, 0] || LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(104)=208\).
Time = 0.61 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.14
| method | result | size |
| derivativedivides | \(c \left (-\frac {\left (\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-1\right ) \left (b \,\operatorname {arcsech}\left (c x \right )+a -b \right )}{4 c x \,b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{4 b^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{4 b c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{4 b^{2} c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{4 b^{3}}\right )\) | \(244\) |
| default | \(c \left (-\frac {\left (\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-1\right ) \left (b \,\operatorname {arcsech}\left (c x \right )+a -b \right )}{4 c x \,b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{4 b^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{4 b c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{4 b^{2} c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{4 b^{3}}\right )\) | \(244\) |
c*(-1/4*((-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)-1)*(b*arcsech(c*x)+a -b)/c/x/b^2/(b^2*arcsech(c*x)^2+2*a*b*arcsech(c*x)+a^2)-1/4/b^3*exp(a/b)*E i(1,a/b+arcsech(c*x))+1/4/b*((-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)+ 1)/c/x/(a+b*arcsech(c*x))^2+1/4/b^2*((-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x )^(1/2)+1)/c/x/(a+b*arcsech(c*x))+1/4/b^3*exp(-a/b)*Ei(1,-arcsech(c*x)-a/b ))
\[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]
integral(1/(b^3*x^2*arcsech(c*x)^3 + 3*a*b^2*x^2*arcsech(c*x)^2 + 3*a^2*b* x^2*arcsech(c*x) + a^3*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]
-1/2*((b*c^6*(log(c) - 1) - a*c^6)*x^7 - 3*(b*c^4*(log(c) - 1) - a*c^4)*x^ 5 - (b*c^2*x^3 - (b*c^4*log(c) - a*c^4)*x^5 + (b*(log(c) - 1) - a)*x - (b* c^4*x^5 - b*x)*log(x))*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + 3*(b*c^2*(log(c) - 1) - a*c^2)*x^3 - (2*b*c^4*x^5 + (b*c^2*(3*log(c) - 5) - 3*a*c^2)*x^3 - 3*(b*(log(c) - 1) - a)*x + 3*(b*c^2*x^3 - b*x)*log(x))*(c*x + 1)*(c*x - 1 ) + ((b*c^6*(log(c) - 1) - a*c^6)*x^7 - (b*c^4*(4*log(c) - 5) - 4*a*c^4)*x ^5 + (b*c^2*(6*log(c) - 7) - 6*a*c^2)*x^3 - 3*(b*(log(c) - 1) - a)*x + (b* c^6*x^7 - 4*b*c^4*x^5 + 6*b*c^2*x^3 - 3*b*x)*log(x))*sqrt(c*x + 1)*sqrt(-c *x + 1) - (b*(log(c) - 1) - a)*x - (b*c^6*x^7 - 3*b*c^4*x^5 + 3*b*c^2*x^3 + (b*c^4*x^5 - b*x)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - 3*(b*c^2*x^3 - b*x) *(c*x + 1)*(c*x - 1) + (b*c^6*x^7 - 4*b*c^4*x^5 + 6*b*c^2*x^3 - 3*b*x)*sqr t(c*x + 1)*sqrt(-c*x + 1) - b*x)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + ( b*c^6*x^7 - 3*b*c^4*x^5 + 3*b*c^2*x^3 - b*x)*log(x))/((b^4*c^6*x^6 - 3*b^4 *c^4*x^4 + 3*b^4*c^2*x^2 - b^4)*x^2*log(x)^2 - (b^4*x^2*log(x)^2 + 2*(b^4* log(c) - a*b^3)*x^2*log(x) + (b^4*log(c)^2 - 2*a*b^3*log(c) + a^2*b^2)*x^2 )*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + 2*((b^4*c^6*log(c) - a*b^3*c^6)*x^6 - 3*(b^4*c^4*log(c) - a*b^3*c^4)*x^4 - b^4*log(c) + a*b^3 + 3*(b^4*c^2*log( c) - a*b^3*c^2)*x^2)*x^2*log(x) - 3*((b^4*c^2*x^2 - b^4)*x^2*log(x)^2 - 2* (b^4*log(c) - a*b^3 - (b^4*c^2*log(c) - a*b^3*c^2)*x^2)*x^2*log(x) - (b^4* log(c)^2 - 2*a*b^3*log(c) + a^2*b^2 - (b^4*c^2*log(c)^2 - 2*a*b^3*c^2*l...
\[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]