3.2.88 \(\int \frac {x^3 (a+b \text {sech}^{-1}(c x))}{\sqrt {1-c^4 x^4}} \, dx\) [188]

3.2.88.1 Optimal result

Integrand size = 26, antiderivative size = 159 \[ \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}+\frac {b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{2 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x} \]

output

1/2*b*arctanh((c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^5/x/(-1+1/c/x)^(1/2) 
/(1+1/c/x)^(1/2)-1/2*b*(-c^2*x^2+1)^(1/2)*(c^2*x^2+1)^(1/2)/c^5/x/(-1+1/c/ 
x)^(1/2)/(1+1/c/x)^(1/2)-1/2*(a+b*arcsech(c*x))*(-c^4*x^4+1)^(1/2)/c^4
 

3.2.88.2 Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {a \sqrt {1-c^4 x^4}+\frac {b \sqrt {1-c^4 x^4}}{\sqrt {\frac {1-c x}{1+c x}} (1+c x)}+b \sqrt {1-c^4 x^4} \text {sech}^{-1}(c x)-b \log (x (1-c x))+b \log \left (1-c x-\sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^4 x^4}\right )}{2 c^4} \]

input

Integrate[(x^3*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]
 

output

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

3.2.88.3 Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.71, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6863, 27, 1388, 243, 60, 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.

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

\(\Big \downarrow \) 6863

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {1-c^4 x^4}}{x \sqrt {1-c^2 x^2}}dx}{2 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}\)

\(\Big \downarrow \) 1388

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {c^2 x^2+1}}{x}dx}{2 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {c^2 x^2+1}}{x^2}dx^2}{4 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \left (\int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2+2 \sqrt {c^2 x^2+1}\right )}{4 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c^2}+2 \sqrt {c^2 x^2+1}\right )}{4 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^4}-\frac {b \sqrt {1-c^2 x^2} \left (2 \sqrt {c^2 x^2+1}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )}{4 c^5 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}\)

input

Int[(x^3*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]
 

output

-1/2*(Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/c^4 - (b*Sqrt[1 - c^2*x^2]*( 
2*Sqrt[1 + c^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]))/(4*c^5*Sqrt[-1 + 1/(c 
*x)]*Sqrt[1 + 1/(c*x)]*x)
 

3.2.88.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

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]
 

rule 221

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]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 6863

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHid 
e[u, x]}, Simp[(a + b*ArcSech[c*x])   v, x] + Simp[b*(Sqrt[1 - c^2*x^2]/(c* 
x*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]))   Int[SimplifyIntegrand[v/(x*Sqrt[ 
1 - c^2*x^2]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c} 
, x]
 

3.2.88.4 Maple [F]

\[\int \frac {x^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}d x\]

input

int(x^3*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x)
 

output

int(x^3*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x)
 

3.2.88.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (135) = 270\).

Time = 0.27 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.75 \[ \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {2 \, \sqrt {-c^{4} x^{4} + 1} b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (b c^{2} x^{2} - b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} x^{2} - b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) + {\left (b c^{2} x^{2} - b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) - 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (a c^{2} x^{2} - a\right )}}{4 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]

input

integrate(x^3*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="fricas")
 

output

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

3.2.88.6 Sympy [F]

\[ \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \]

input

integrate(x**3*(a+b*asech(c*x))/(-c**4*x**4+1)**(1/2),x)
 

output

Integral(x**3*(a + b*asech(c*x))/sqrt(-(c*x - 1)*(c*x + 1)*(c**2*x**2 + 1) 
), x)
 

3.2.88.7 Maxima [F]

\[ \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]

input

integrate(x^3*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="maxima")
 

output

1/2*b*((c^4*x^4 - 1)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/(sqrt(c^2*x^2 + 
 1)*sqrt(c*x + 1)*sqrt(-c*x + 1)*c^4) - 2*integrate(1/2*(2*c^2*x^5*log(c) 
+ 4*c^2*x^5*log(sqrt(x)) + (4*c^2*x^5*log(sqrt(x)) + (c^2*x^2*(2*log(c) + 
1) + 1)*x^3)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)))/((c^2*x^2*e^(log(c* 
x + 1) + log(-c*x + 1)) + c^2*x^2*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)) 
)*sqrt(c^2*x^2 + 1)), x)) - 1/2*sqrt(-c^4*x^4 + 1)*a/c^4
 

3.2.88.8 Giac [F]

\[ \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]

input

integrate(x^3*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="giac")
 

output

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

3.2.88.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]

input

int((x^3*(a + b*acosh(1/(c*x))))/(1 - c^4*x^4)^(1/2),x)
 

output

int((x^3*(a + b*acosh(1/(c*x))))/(1 - c^4*x^4)^(1/2), x)