3.2.76 \(\int \frac {x^3 (a+b \text {csch}^{-1}(c x))}{\sqrt {1-c^4 x^4}} \, dx\) [176]

3.2.76.1 Optimal result

Integrand size = 26, antiderivative size = 130 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {b x \sqrt {1-c^4 x^4}}{2 c^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}-\frac {b x \arctan \left (\frac {\sqrt {1-c^4 x^4}}{\sqrt {-1-c^2 x^2}}\right )}{2 c^3 \sqrt {-c^2 x^2}} \]

output

-1/2*b*x*arctan((-c^4*x^4+1)^(1/2)/(-c^2*x^2-1)^(1/2))/c^3/(-c^2*x^2)^(1/2 
)-1/2*(a+b*arccsch(c*x))*(-c^4*x^4+1)^(1/2)/c^4+1/2*b*x*(-c^4*x^4+1)^(1/2) 
/c^3/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)
 

3.2.76.2 Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {a \sqrt {1-c^4 x^4}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1-c^4 x^4}}{1+c^2 x^2}+b \sqrt {1-c^4 x^4} \text {csch}^{-1}(c x)+b \log \left (x+c^2 x^3\right )-b \log \left (1+c^2 x^2+c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1-c^4 x^4}\right )}{2 c^4} \]

input

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

output

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

3.2.76.3 Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6864, 27, 1896, 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.

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

\(\Big \downarrow \) 6864

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1896

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

input

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

output

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

3.2.76.3.1 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 1896

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_)*((a_) + (c_.)*(x_)^(n2_.))^( 
p_.), x_Symbol] :> Simp[(e^IntPart[q]*((d + e*x^mn)^FracPart[q]/(1 + d*(1/( 
x^mn*e)))^FracPart[q]))/x^(mn*FracPart[q])   Int[x^(m + mn*q)*(1 + d*(1/(x^ 
mn*e)))^q*(a + c*x^n2)^p, x], x] /; FreeQ[{a, c, d, e, m, mn, p, q}, x] && 
EqQ[n2, -2*mn] &&  !IntegerQ[p] &&  !IntegerQ[q] && PosQ[n2]
 

rule 6864

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

3.2.76.4 Maple [F]

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

input

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

output

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

3.2.76.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (110) = 220\).

Time = 0.28 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.04 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {2 \, \sqrt {-c^{4} x^{4} + 1} b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (b c^{2} x^{2} + b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} x^{2} + b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + {\left (b c^{2} x^{2} + b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (a c^{2} x^{2} + a\right )}}{4 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]

input

integrate(x^3*(a+b*arccsch(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 - sqr 
t(-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)
 

3.2.76.6 Sympy [F]

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

input

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

output

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

3.2.76.7 Maxima [F]

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

input

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

output

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

3.2.76.8 Giac [F]

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

input

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

output

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

3.2.76.9 Mupad [F(-1)]

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

input

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

output

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