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

Optimal. Leaf size=130 \[ -\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}+\frac {b x \sqrt {1-c^4 x^4}}{2 c^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}-\frac {b x \tan ^{-1}\left (\frac {\sqrt {1-c^4 x^4}}{\sqrt {-c^2 x^2-1}}\right )}{2 c^3 \sqrt {-c^2 x^2}} \]

[Out]

-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)

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 133, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {261, 6310, 12, 1572, 1252, 848, 50, 63, 208} \[ -\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}-\frac {b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{2 c^5 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {b \sqrt {c^2 x^2+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 c^5 x \sqrt {\frac {1}{c^2 x^2}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 50

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] + Dist[(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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 1572

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Dist[(e^IntPart[
q]*(d + e*x^mn)^FracPart[q])/(x^(mn*FracPart[q])*(1 + d/(x^mn*e))^FracPart[q]), Int[x^(m + mn*q)*(1 + d/(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] &&  !In
tegerQ[q] && PosQ[n2]

Rule 6310

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

Rubi steps

\begin {align*} \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}+\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}-\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}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {1-c^4 x^4}}{x \sqrt {1+c^2 x^2}} \, dx}{2 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-c^4 x^2}}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-c^2 x}}{x} \, dx,x,x^2\right )}{4 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{2 c^7 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^4}+\frac {b \sqrt {1+c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 c^5 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.34, size = 141, normalized size = 1.08 \[ -\frac {a \sqrt {1-c^4 x^4}+b \sqrt {1-c^4 x^4} \text {csch}^{-1}(c x)+b \log \left (c^2 x^3+x\right )+\frac {b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {1-c^4 x^4}}{c^2 x^2+1}-b \log \left (c^2 x^2+c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {1-c^4 x^4}+1\right )}{2 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-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

________________________________________________________________________________________

fricas [B]  time = 0.68, size = 265, normalized size = 2.04 \[ -\frac {2 \, \sqrt {-c^{4} x^{4} + 1} b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (b c^{2} x^{2} + b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} x^{2} + b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + {\left (b c^{2} x^{2} + b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {-c^{4} x^{4} + 1} {\left (a c^{2} x^{2} + a\right )}}{4 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 - sqrt(-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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{\sqrt {-c^{4} x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b {\left (\frac {{\left (c^{4} x^{4} - 1\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {-c x + 1} c^{4}} - 2 \, \int {\left (x^{3} \log \relax (c) + x^{3} \log \relax (x)\right )} e^{\left (-\frac {1}{2} \, \log \left (c^{2} x^{2} + 1\right ) - \frac {1}{2} \, \log \left (c x + 1\right ) - \frac {1}{2} \, \log \left (-c x + 1\right )\right )}\,{d x} - 2 \, \int \frac {c^{2} x^{3} - x}{2 \, {\left (\sqrt {c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {-c x + 1} c^{2} + \sqrt {c x + 1} \sqrt {-c x + 1} c^{2}\right )}}\,{d x}\right )} - \frac {\sqrt {-c^{4} x^{4} + 1} a}{2 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*integ
rate((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*integr
ate(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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________