Integrand size = 14, antiderivative size = 213 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx=\frac {14 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{1+c x}\right )^{3/2} (1+c x)^3}{27 x^3}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3} \]
2/27*b^3*((-c*x+1)/(c*x+1))^(3/2)*(c*x+1)^3/x^3-2/9*b^2*(a+b*arcsech(c*x)) /x^3-4/3*b^2*c^2*(a+b*arcsech(c*x))/x-1/3*(a+b*arcsech(c*x))^3/x^3+14/9*b^ 3*c^2*(c*x+1)*((-c*x+1)/(c*x+1))^(1/2)/x+1/3*b*(c*x+1)*(a+b*arcsech(c*x))^ 2*((-c*x+1)/(c*x+1))^(1/2)/x^3+2/3*b*c^2*(c*x+1)*(a+b*arcsech(c*x))^2*((-c *x+1)/(c*x+1))^(1/2)/x
Time = 0.41 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx=\frac {-9 a^3-6 a b^2 \left (1+6 c^2 x^2\right )+9 a^2 b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )+2 b^3 \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+20 c^2 x^2+20 c^3 x^3\right )-3 b \left (9 a^2+2 b^2 \left (1+6 c^2 x^2\right )-6 a b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )\right ) \text {sech}^{-1}(c x)+9 b^2 \left (-3 a+b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )\right ) \text {sech}^{-1}(c x)^2-9 b^3 \text {sech}^{-1}(c x)^3}{27 x^3} \]
(-9*a^3 - 6*a*b^2*(1 + 6*c^2*x^2) + 9*a^2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3) + 2*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 20*c^2*x^2 + 20*c^3*x^3) - 3*b*(9*a^2 + 2*b^2*(1 + 6*c^2*x^2) - 6*a*b*Sqr t[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3))*ArcSech[c*x] + 9 *b^2*(-3*a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3) )*ArcSech[c*x]^2 - 9*b^3*ArcSech[c*x]^3)/(27*x^3)
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.29, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6839, 5970, 3042, 3792, 3042, 3113, 2009, 3777, 26, 3042, 26, 3777, 3042, 3117}
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.
\(\displaystyle \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx\) |
\(\Big \downarrow \) 6839 |
\(\displaystyle -c^3 \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{c^3 x^3}d\text {sech}^{-1}(c x)\) |
\(\Big \downarrow \) 5970 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{c^3 x^3}d\text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}d\text {sech}^{-1}(c x)+\frac {2}{9} b^2 \int \frac {1}{c^3 x^3}d\text {sech}^{-1}(c x)-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(c x)+\frac {2}{9} b^2 \int \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(c x)-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}\right )\right )\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(c x)+\frac {2}{9} i b^2 \int \left (\frac {(1-c x) (c x+1)}{c^2 x^2}+1\right )d\left (-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(c x)-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \int -\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}d\text {sech}^{-1}(c x)\right )-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 b \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}d\text {sech}^{-1}(c x)\right )-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 b \int -i \left (a+b \text {sech}^{-1}(c x)\right ) \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}+2 i b \int \left (a+b \text {sech}^{-1}(c x)\right ) \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}+2 i b \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-i b \int \frac {1}{c x}d\text {sech}^{-1}(c x)\right )\right )-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}+2 i b \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-i b \int \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(c x)\right )\right )-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}+\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}+2 i b \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-\frac {i b \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )+\frac {2}{9} i b^2 \left (-\frac {i \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{3 c^3 x^3}-\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )\right )\right )\) |
-(c^3*((a + b*ArcSech[c*x])^3/(3*c^3*x^3) - b*(((2*I)/9)*b^2*(((-I)*Sqrt[( 1 - c*x)/(1 + c*x)]*(1 + c*x))/(c*x) - ((I/3)*((1 - c*x)/(1 + c*x))^(3/2)* (1 + c*x)^3)/(c^3*x^3)) - (2*b*(a + b*ArcSech[c*x]))/(9*c^3*x^3) + (Sqrt[( 1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(3*c^3*x^3) + (2*((S qrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(c*x) + (2*I)*b *(((-I)*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (I*(a + b*ArcSech[c *x]))/(c*x))))/3)))
3.1.49.3.1 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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
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. \(386\) vs. \(2(191)=382\).
Time = 0.83 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{3 c^{2} x^{2}}-\frac {4 \,\operatorname {arcsech}\left (c x \right )}{3 c x}+\frac {40 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27}-\frac {2 \,\operatorname {arcsech}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+3 b \,a^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\) | \(387\) |
default | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{3 c^{2} x^{2}}-\frac {4 \,\operatorname {arcsech}\left (c x \right )}{3 c x}+\frac {40 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27}-\frac {2 \,\operatorname {arcsech}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+3 b \,a^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\) | \(387\) |
parts | \(-\frac {a^{3}}{3 x^{3}}+b^{3} c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{3 c^{2} x^{2}}-\frac {4 \,\operatorname {arcsech}\left (c x \right )}{3 c x}+\frac {40 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27}-\frac {2 \,\operatorname {arcsech}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+3 b \,a^{2} c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\) | \(389\) |
c^3*(-1/3*a^3/c^3/x^3+b^3*(-1/3/c^3/x^3*arcsech(c*x)^3+2/3*(-(c*x-1)/c/x)^ (1/2)*((c*x+1)/c/x)^(1/2)*arcsech(c*x)^2+1/3/c^2/x^2*(-(c*x-1)/c/x)^(1/2)* ((c*x+1)/c/x)^(1/2)*arcsech(c*x)^2-4/3/c/x*arcsech(c*x)+40/27*(-(c*x-1)/c/ x)^(1/2)*((c*x+1)/c/x)^(1/2)-2/9/c^3/x^3*arcsech(c*x)+2/27*(-(c*x-1)/c/x)^ (1/2)*((c*x+1)/c/x)^(1/2)/c^2/x^2)+3*a*b^2*(-1/3/c^3/x^3*arcsech(c*x)^2+4/ 9*arcsech(c*x)*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+2/9/c^2/x^2*(-(c*x -1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*arcsech(c*x)-4/9/c/x-2/27/c^3/x^3)+3*b* a^2*(-1/3/c^3/x^3*arcsech(c*x)+1/9*(-(c*x-1)/c/x)^(1/2)/c^2/x^2*((c*x+1)/c /x)^(1/2)*(2*c^2*x^2+1)))
Time = 0.27 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx=-\frac {36 \, a b^{2} c^{2} x^{2} + 9 \, b^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} + 9 \, a^{3} + 6 \, a b^{2} + 9 \, {\left (3 \, a b^{2} - {\left (2 \, b^{3} c^{3} x^{3} + b^{3} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} + 9 \, a^{2} b + 2 \, b^{3} - 6 \, {\left (2 \, a b^{2} c^{3} x^{3} + a b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (9 \, a^{2} b + 20 \, b^{3}\right )} c^{3} x^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{27 \, x^{3}} \]
-1/27*(36*a*b^2*c^2*x^2 + 9*b^3*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^3 + 9*a^3 + 6*a*b^2 + 9*(3*a*b^2 - (2*b^3*c^3*x^3 + b^3*c*x)*sqr t(-(c^2*x^2 - 1)/(c^2*x^2)))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/ (c*x))^2 + 3*(12*b^3*c^2*x^2 + 9*a^2*b + 2*b^3 - 6*(2*a*b^2*c^3*x^3 + a*b^ 2*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x ^2)) + 1)/(c*x)) - (2*(9*a^2*b + 20*b^3)*c^3*x^3 + (9*a^2*b + 2*b^3)*c*x)* sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/x^3
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \]
1/3*a^2*b*((c^4*(1/(c^2*x^2) - 1)^(3/2) + 3*c^4*sqrt(1/(c^2*x^2) - 1))/c - 3*arcsech(c*x)/x^3) - 1/3*a^3/x^3 + integrate(b^3*log(sqrt(1/(c*x) + 1)*s qrt(1/(c*x) - 1) + 1/(c*x))^3/x^4 + 3*a*b^2*log(sqrt(1/(c*x) + 1)*sqrt(1/( c*x) - 1) + 1/(c*x))^2/x^4, x)
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \]