3.140 \(\int \frac{x (a+b \text{csch}^{-1}(c x))}{\sqrt{d+e x^2}} \, dx\)

Optimal. Leaf size=135 \[ \frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{\sqrt{e} \sqrt{-c^2 x^2}}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{e \sqrt{-c^2 x^2}} \]

[Out]

(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e + (b*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(Sqr
t[e]*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(e*Sqrt[-(c^2*x^
2)])

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Rubi [A]  time = 0.153675, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {6300, 446, 105, 63, 217, 203, 93, 204} \[ \frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{\sqrt{e} \sqrt{-c^2 x^2}}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{e \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]

[Out]

(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e + (b*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(Sqr
t[e]*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(e*Sqrt[-(c^2*x^
2)])

Rule 6300

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcCsch[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[-(c^2*x^2)]), Int[(d + e*x^2)^(p
+ 1)/(x*Sqrt[-1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{\sqrt{d+e x^2}} \, dx &=\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}-\frac{(b c x) \int \frac{\sqrt{d+e x^2}}{x \sqrt{-1-c^2 x^2}} \, dx}{e \sqrt{-c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{2 e \sqrt{-c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 \sqrt{-c^2 x^2}}-\frac{(b c d x) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 e \sqrt{-c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{(b x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{e}{c^2}-\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1-c^2 x^2}\right )}{c \sqrt{-c^2 x^2}}-\frac{(b c d x) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1-c^2 x^2}}\right )}{e \sqrt{-c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{e \sqrt{-c^2 x^2}}+\frac{(b x) \operatorname{Subst}\left (\int \frac{1}{1+\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1-c^2 x^2}}{\sqrt{d+e x^2}}\right )}{c \sqrt{-c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{\sqrt{e} \sqrt{-c^2 x^2}}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{e \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.238083, size = 223, normalized size = 1.65 \[ \frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{e}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^3 \sqrt{d} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right )-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right )\right )}{c^2 e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]

[Out]

(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*Sqr
t[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + c^3*Sqr
t[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(c^2*e*Sqrt[1 + c^2*x^2]*Sqrt[d +
e*x^2])

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Maple [F]  time = 0.46, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{\rm arccsch} \left (cx\right ) \right ){\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x)

[Out]

int(x*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b{\left (\frac{\sqrt{e x^{2} + d} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{e} + \int \frac{c^{2} e x^{3} + c^{2} d x}{{\left (c^{2} e x^{2} + e\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{e x^{2} + d} +{\left (c^{2} e x^{2} + e\right )} \sqrt{e x^{2} + d}}\,{d x} - \int \frac{{\left (e \log \left (c\right ) + e\right )} c^{2} x^{3} +{\left (c^{2} d + e \log \left (c\right )\right )} x +{\left (c^{2} e x^{3} + e x\right )} \log \left (x\right )}{{\left (c^{2} e x^{2} + e\right )} \sqrt{e x^{2} + d}}\,{d x}\right )} + \frac{\sqrt{e x^{2} + d} a}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.93202, size = 2398, normalized size = 17.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(e*x^2 + d)*b*c*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + b*c*sqrt(d)*log(((c^4*d^2 + 6
*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt((c^
2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) + 4*sqrt(e*x^2 + d)*a*c + b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 + 6*c^2*d*
e + 8*(c^4*d*e + c^2*e^2)*x^2 + 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt((c^2*x^2 + 1)
/(c^2*x^2)) + e^2))/(c*e), 1/4*(4*sqrt(e*x^2 + d)*b*c*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + b*c
*sqrt(d)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e
*x^2 + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) + 4*sqrt(e*x^2 + d)*a*c - 2*b*sqrt(-e)*arctan(1/
2*(2*c^2*e*x^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*e^2*x^4 + (c^2*d*e
+ e^2)*x^2 + d*e)))/(c*e), 1/4*(2*b*c*sqrt(-d)*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(
-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d*e*x^4 + (c^2*d^2 + d*e)*x^2 + d^2)) + 4*sqrt(e*x^2 + d)*b*c*log((c*x*
sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*sqrt(e*x^2 + d)*a*c + b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 + 6*
c^2*d*e + 8*(c^4*d*e + c^2*e^2)*x^2 + 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt((c^2*x^
2 + 1)/(c^2*x^2)) + e^2))/(c*e), 1/2*(b*c*sqrt(-d)*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sq
rt(-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d*e*x^4 + (c^2*d^2 + d*e)*x^2 + d^2)) + 2*sqrt(e*x^2 + d)*b*c*log((c
*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 2*sqrt(e*x^2 + d)*a*c - b*sqrt(-e)*arctan(1/2*(2*c^2*e*x^3 + (c
^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*e))
)/(c*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \operatorname{acsch}{\left (c x \right )}\right )}{\sqrt{d + e x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acsch(c*x))/(e*x**2+d)**(1/2),x)

[Out]

Integral(x*(a + b*acsch(c*x))/sqrt(d + e*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x}{\sqrt{e x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x/sqrt(e*x^2 + d), x)