### 3.60 $$\int \frac{e^{\text{csch}^{-1}(c x)} x^5}{1+c^2 x^2} \, dx$$

Optimal. Leaf size=92 $\frac{x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{4 c^2}+\frac{x^3}{3 c^3}-\frac{3 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{8 c^4}+\frac{3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{8 c^6}-\frac{x}{c^5}+\frac{\tan ^{-1}(c x)}{c^6}$

[Out]

-(x/c^5) - (3*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(8*c^4) + x^3/(3*c^3) + (Sqrt[1 + 1/(c^2*x^2)]*x^4)/(4*c^2) + ArcTan[
c*x]/c^6 + (3*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(8*c^6)

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Rubi [A]  time = 0.108045, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {6342, 266, 51, 63, 208, 302, 203} $\frac{x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{4 c^2}+\frac{x^3}{3 c^3}-\frac{3 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{8 c^4}+\frac{3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{8 c^6}-\frac{x}{c^5}+\frac{\tan ^{-1}(c x)}{c^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(E^ArcCsch[c*x]*x^5)/(1 + c^2*x^2),x]

[Out]

-(x/c^5) - (3*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(8*c^4) + x^3/(3*c^3) + (Sqrt[1 + 1/(c^2*x^2)]*x^4)/(4*c^2) + ArcTan[
c*x]/c^6 + (3*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(8*c^6)

Rule 6342

Int[(E^ArcCsch[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d^2/(a*c^2), Int[(d*x)
^(m - 2)/Sqrt[1 + 1/(c^2*x^2)], x], x] + Dist[d/c, Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d,
m}, x] && EqQ[b - a*c^2, 0]

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)} x^5}{1+c^2 x^2} \, dx &=\frac{\int \frac{x^3}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{c^2}+\frac{\int \frac{x^4}{1+c^2 x^2} \, dx}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c^2}+\frac{\int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{c}\\ &=-\frac{x}{c^5}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\int \frac{1}{1+c^2 x^2} \, dx}{c^5}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{8 c^4}\\ &=-\frac{x}{c^5}-\frac{3 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{8 c^4}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\tan ^{-1}(c x)}{c^6}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{16 c^6}\\ &=-\frac{x}{c^5}-\frac{3 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{8 c^4}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\tan ^{-1}(c x)}{c^6}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{8 c^4}\\ &=-\frac{x}{c^5}-\frac{3 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{8 c^4}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\tan ^{-1}(c x)}{c^6}+\frac{3 \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{8 c^6}\\ \end{align*}

Mathematica [A]  time = 0.207719, size = 85, normalized size = 0.92 $\frac{c x \left (6 c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}+8 c^2 x^2-9 c x \sqrt{\frac{1}{c^2 x^2}+1}-24\right )+9 \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+24 \tan ^{-1}(c x)}{24 c^6}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcCsch[c*x]*x^5)/(1 + c^2*x^2),x]

[Out]

(c*x*(-24 - 9*c*Sqrt[1 + 1/(c^2*x^2)]*x + 8*c^2*x^2 + 6*c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3) + 24*ArcTan[c*x] + 9*Lo
g[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(24*c^6)

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Maple [B]  time = 0.201, size = 165, normalized size = 1.8 \begin{align*}{\frac{x}{8\,{c}^{6}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}} \left ( 2\,x \left ({\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}} \right ) ^{3/2}{c}^{4}-5\,x\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}{c}^{2}-5\,\ln \left ( x+\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}} \right ) +8\,\ln \left ( x+\sqrt{-{\frac{ \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }{{c}^{4}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}+{\frac{{x}^{3}}{3\,{c}^{3}}}-{\frac{x}{{c}^{5}}}+{\frac{\arctan \left ( cx \right ) }{{c}^{6}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1/c/x+(1+1/c^2/x^2)^(1/2))*x^5/(c^2*x^2+1),x)

[Out]

1/8*((c^2*x^2+1)/c^2/x^2)^(1/2)*x*(2*x*((c^2*x^2+1)/c^2)^(3/2)*c^4-5*x*((c^2*x^2+1)/c^2)^(1/2)*c^2-5*ln(x+((c^
2*x^2+1)/c^2)^(1/2))+8*ln(x+(-(x*c^2+(-c^2)^(1/2))*(-x*c^2+(-c^2)^(1/2))/c^4)^(1/2)))/((c^2*x^2+1)/c^2)^(1/2)/
c^6+1/3*x^3/c^3-x/c^5+arctan(c*x)/c^6

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Maxima [B]  time = 1.53694, size = 216, normalized size = 2.35 \begin{align*} \frac{c^{2} x^{3} - 3 \, x}{3 \, c^{5}} + \frac{\frac{2 \,{\left (3 \, \left (\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}\right )^{\frac{3}{2}} - 5 \, \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )}}{\frac{2 \,{\left (c^{2} x^{2} + 1\right )}}{c^{2} x^{2}} - \frac{{\left (c^{2} x^{2} + 1\right )}^{2}}{c^{4} x^{4}} - 1} + 3 \, \log \left (\sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1\right ) - 3 \, \log \left (\sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 1\right )}{16 \, c^{6}} + \frac{\arctan \left (c x\right )}{c^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/c/x+(1+1/c^2/x^2)^(1/2))*x^5/(c^2*x^2+1),x, algorithm="maxima")

[Out]

1/3*(c^2*x^3 - 3*x)/c^5 + 1/16*(2*(3*((c^2*x^2 + 1)/(c^2*x^2))^(3/2) - 5*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/(2*(c^
2*x^2 + 1)/(c^2*x^2) - (c^2*x^2 + 1)^2/(c^4*x^4) - 1) + 3*log(sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1) - 3*log(sqrt(
(c^2*x^2 + 1)/(c^2*x^2)) - 1))/c^6 + arctan(c*x)/c^6

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Fricas [A]  time = 3.02309, size = 209, normalized size = 2.27 \begin{align*} \frac{8 \, c^{3} x^{3} - 24 \, c x + 3 \,{\left (2 \, c^{4} x^{4} - 3 \, c^{2} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 24 \, \arctan \left (c x\right ) - 9 \, \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right )}{24 \, c^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/c/x+(1+1/c^2/x^2)^(1/2))*x^5/(c^2*x^2+1),x, algorithm="fricas")

[Out]

1/24*(8*c^3*x^3 - 24*c*x + 3*(2*c^4*x^4 - 3*c^2*x^2)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 24*arctan(c*x) - 9*log(c*
x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x))/c^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac{c x^{5} \sqrt{1 + \frac{1}{c^{2} x^{2}}}}{c^{2} x^{2} + 1}\, dx}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/c/x+(1+1/c**2/x**2)**(1/2))*x**5/(c**2*x**2+1),x)

[Out]

(Integral(x**4/(c**2*x**2 + 1), x) + Integral(c*x**5*sqrt(1 + 1/(c**2*x**2))/(c**2*x**2 + 1), x))/c

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Giac [A]  time = 1.20418, size = 120, normalized size = 1.3 \begin{align*} \frac{1}{8} \, \sqrt{c^{2} x^{2} + 1} x{\left (\frac{2 \, x^{2}{\left | c \right |} \mathrm{sgn}\left (x\right )}{c^{4}} - \frac{3 \,{\left | c \right |} \mathrm{sgn}\left (x\right )}{c^{6}}\right )} - \frac{3 \, \log \left (-x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1}\right ) \mathrm{sgn}\left (x\right )}{8 \, c^{6}} + \frac{\arctan \left (c x\right )}{c^{6}} + \frac{c^{6} x^{3} - 3 \, c^{4} x}{3 \, c^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/c/x+(1+1/c^2/x^2)^(1/2))*x^5/(c^2*x^2+1),x, algorithm="giac")

[Out]

1/8*sqrt(c^2*x^2 + 1)*x*(2*x^2*abs(c)*sgn(x)/c^4 - 3*abs(c)*sgn(x)/c^6) - 3/8*log(-x*abs(c) + sqrt(c^2*x^2 + 1
))*sgn(x)/c^6 + arctan(c*x)/c^6 + 1/3*(c^6*x^3 - 3*c^4*x)/c^9