3.1.33 \(\int x^3 (a+b \text {sech}^{-1}(c x))^2 \, dx\) [33]

3.1.33.1 Optimal result

Integrand size = 14, antiderivative size = 124 \[ \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=-\frac {b^2 x^2}{12 c^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^4}-\frac {b x^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{6 c^2}+\frac {1}{4} x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{3 c^4} \]

output

-1/12*b^2*x^2/c^2+1/4*x^4*(a+b*arcsech(c*x))^2-1/3*b^2*ln(x)/c^4-1/3*b*(c* 
x+1)*(a+b*arcsech(c*x))*((-c*x+1)/(c*x+1))^(1/2)/c^4-1/6*b*x^2*(c*x+1)*(a+ 
b*arcsech(c*x))*((-c*x+1)/(c*x+1))^(1/2)/c^2
 

3.1.33.2 Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.71 \[ \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=-\frac {b^2 c^2 x^2-3 a^2 c^4 x^4+4 a b \sqrt {\frac {1-c x}{1+c x}}+4 a b c x \sqrt {\frac {1-c x}{1+c x}}+2 a b c^2 x^2 \sqrt {\frac {1-c x}{1+c x}}+2 a b c^3 x^3 \sqrt {\frac {1-c x}{1+c x}}+2 b \left (-3 a c^4 x^4+b \sqrt {\frac {1-c x}{1+c x}} \left (2+2 c x+c^2 x^2+c^3 x^3\right )\right ) \text {sech}^{-1}(c x)-3 b^2 c^4 x^4 \text {sech}^{-1}(c x)^2+4 b^2 \log (x)}{12 c^4} \]

input

Integrate[x^3*(a + b*ArcSech[c*x])^2,x]
 

output

-1/12*(b^2*c^2*x^2 - 3*a^2*c^4*x^4 + 4*a*b*Sqrt[(1 - c*x)/(1 + c*x)] + 4*a 
*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)] + 2*a*b*c^2*x^2*Sqrt[(1 - c*x)/(1 + c*x)] 
 + 2*a*b*c^3*x^3*Sqrt[(1 - c*x)/(1 + c*x)] + 2*b*(-3*a*c^4*x^4 + b*Sqrt[(1 
 - c*x)/(1 + c*x)]*(2 + 2*c*x + c^2*x^2 + c^3*x^3))*ArcSech[c*x] - 3*b^2*c 
^4*x^4*ArcSech[c*x]^2 + 4*b^2*Log[x])/c^4
 

3.1.33.3 Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6839, 5974, 3042, 4673, 3042, 4672, 26, 3042, 26, 3956}

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.

\(\displaystyle \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx\)

\(\Big \downarrow \) 6839

\(\displaystyle -\frac {\int c^4 x^4 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)}{c^4}\)

\(\Big \downarrow \) 5974

\(\displaystyle -\frac {\frac {1}{2} b \int c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {1}{2} b \int \left (a+b \text {sech}^{-1}(c x)\right ) \csc \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )^4d\text {sech}^{-1}(c x)}{c^4}\)

\(\Big \downarrow \) 4673

\(\displaystyle -\frac {\frac {1}{2} b \left (\frac {2}{3} \int c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)+\frac {1}{3} c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} b c^2 x^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \int \left (a+b \text {sech}^{-1}(c x)\right ) \csc \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )^2d\text {sech}^{-1}(c x)+\frac {1}{3} c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-i b \int -i \sqrt {\frac {1-c x}{c x+1}} (c x+1)d\text {sech}^{-1}(c x)\right )+\frac {1}{3} c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-b \int \sqrt {\frac {1-c x}{c x+1}} (c x+1)d\text {sech}^{-1}(c x)\right )+\frac {1}{3} c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} b c^2 x^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-b \int -i \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )+\frac {1}{3} c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+i b \int \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )+\frac {1}{3} c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {\frac {1}{2} b \left (\frac {1}{3} c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-b \log \left (\frac {1}{c x}\right )\right )+\frac {1}{6} b c^2 x^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^4}\)

input

Int[x^3*(a + b*ArcSech[c*x])^2,x]
 

output

-((-1/4*(c^4*x^4*(a + b*ArcSech[c*x])^2) + (b*((b*c^2*x^2)/6 + (c^2*x^2*Sq 
rt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/3 + (2*(Sqrt[(1 - 
c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]) - b*Log[1/(c*x)]))/3))/2)/c 
^4)
 

3.1.33.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 5974

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + 
(b_.)*(x_)]^(p_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)) 
, x] + Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; 
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

rule 6839

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])
 

3.1.33.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(223\) vs. \(2(110)=220\).

Time = 0.72 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.81

method	result	size
parts	\(\frac {x^{4} a^{2}}{4}+\frac {b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3}+\frac {\operatorname {arcsech}\left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} x^{3}}{6}-\frac {\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{3}-\frac {c^{2} x^{2}}{12}+\frac {\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{3}\right )}{c^{4}}+\frac {2 a b \left (\frac {c^{4} x^{4} \operatorname {arcsech}\left (c x \right )}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (c^{2} x^{2}+2\right )}{12}\right )}{c^{4}}\)	\(224\)
derivativedivides	\(\frac {\frac {a^{2} c^{4} x^{4}}{4}+b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3}+\frac {\operatorname {arcsech}\left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} x^{3}}{6}-\frac {\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{3}-\frac {c^{2} x^{2}}{12}+\frac {\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{3}\right )+2 a b \left (\frac {c^{4} x^{4} \operatorname {arcsech}\left (c x \right )}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (c^{2} x^{2}+2\right )}{12}\right )}{c^{4}}\)	\(225\)
default	\(\frac {\frac {a^{2} c^{4} x^{4}}{4}+b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3}+\frac {\operatorname {arcsech}\left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} x^{3}}{6}-\frac {\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{3}-\frac {c^{2} x^{2}}{12}+\frac {\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{3}\right )+2 a b \left (\frac {c^{4} x^{4} \operatorname {arcsech}\left (c x \right )}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (c^{2} x^{2}+2\right )}{12}\right )}{c^{4}}\)	\(225\)

input

int(x^3*(a+b*arcsech(c*x))^2,x,method=_RETURNVERBOSE)
 

output

1/4*x^4*a^2+b^2/c^4*(-1/3*arcsech(c*x)+1/4*arcsech(c*x)^2*c^4*x^4-1/6*arcs 
ech(c*x)*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c^3*x^3-1/3*arcsech(c*x) 
*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c*x-1/12*c^2*x^2+1/3*ln(1+(1/c/x 
+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2))+2*a*b/c^4*(1/4*c^4*x^4*arcsech(c*x) 
-1/12*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(c^2*x^2+2))
 

3.1.33.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (110) = 220\).

Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.97 \[ \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\frac {3 \, b^{2} c^{4} x^{4} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, a^{2} c^{4} x^{4} - 6 \, a b c^{4} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - b^{2} c^{2} x^{2} - 4 \, b^{2} \log \left (x\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{4} - {\left (b^{2} c^{3} x^{3} + 2 \, b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, {\left (a b c^{3} x^{3} + 2 \, a b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{12 \, c^{4}} \]

input

integrate(x^3*(a+b*arcsech(c*x))^2,x, algorithm="fricas")
 

output

1/12*(3*b^2*c^4*x^4*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^2 
+ 3*a^2*c^4*x^4 - 6*a*b*c^4*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x 
) - b^2*c^2*x^2 - 4*b^2*log(x) + 2*(3*a*b*c^4*x^4 - 3*a*b*c^4 - (b^2*c^3*x 
^3 + 2*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*(a*b*c^3*x^3 + 2*a*b*c*x)*sqrt(-(c^2*x^2 - 1 
)/(c^2*x^2)))/c^4
 

3.1.33.6 Sympy [F]

\[ \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\int x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}\, dx \]

input

integrate(x**3*(a+b*asech(c*x))**2,x)
 

output

Integral(x**3*(a + b*asech(c*x))**2, x)
 

3.1.33.7 Maxima [F]

\[ \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]

input

integrate(x^3*(a+b*arcsech(c*x))^2,x, algorithm="maxima")
 

output

1/4*a^2*x^4 + 1/6*(3*x^4*arcsech(c*x) + (c^2*x^3*(1/(c^2*x^2) - 1)^(3/2) - 
 3*x*sqrt(1/(c^2*x^2) - 1))/c^3)*a*b + b^2*integrate(x^3*log(sqrt(1/(c*x) 
+ 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2, x)
 

3.1.33.8 Giac [F]

\[ \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]

input

integrate(x^3*(a+b*arcsech(c*x))^2,x, algorithm="giac")
 

output

integrate((b*arcsech(c*x) + a)^2*x^3, x)
 

3.1.33.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]

input

int(x^3*(a + b*acosh(1/(c*x)))^2,x)
 

output

int(x^3*(a + b*acosh(1/(c*x)))^2, x)