3.2.40 \(\int \frac {x (a+b \text {csch}^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [140]

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

[Out]

b*c*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2-1)^(1/2))*d^(1/2)/e/(-c^2*x^2)^(1/2)+b*x*arctan(e^(1/2)*(-c^2*x
^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/e^(1/2)/(-c^2*x^2)^(1/2)+(a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/e

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Rubi [A]
time = 0.10, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6435, 457, 132, 65, 223, 209, 12, 95, 210} \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b x \text {ArcTan}\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 \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{e \sqrt {-c^2 x^2}} \end {gather*}

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 12

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

Rule 65

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 95

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 132

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

Rule 209

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

Rule 210

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

Rule 223

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 457

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 6435

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]

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) \text {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) \text {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) \text {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) \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.54, size = 137, normalized size = 1.01 \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e}-\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (c \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )-\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{e \sqrt {1+c^2 x^2}} \end {gather*}

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*(c*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[1 + c^2
*x^2])/Sqrt[d + e*x^2]] - Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + e*x^2])]))/(e*Sqrt[1 + c^2*x
^2])

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (112) = 224\).
time = 0.52, size = 885, normalized size = 6.56 \begin {gather*} \left [\frac {4 \, \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} b c \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + b c \sqrt {d} \log \left (\frac {c^{4} d^{2} x^{4} + 8 \, c^{2} d^{2} x^{2} + x^{4} \cosh \left (1\right )^{2} + x^{4} \sinh \left (1\right )^{2} - 4 \, {\left (c^{3} d x^{3} + c x^{3} \cosh \left (1\right ) + c x^{3} \sinh \left (1\right ) + 2 \, c d x\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 8 \, d^{2} + 2 \, {\left (3 \, c^{2} d x^{4} + 4 \, d x^{2}\right )} \cosh \left (1\right ) + 2 \, {\left (3 \, c^{2} d x^{4} + x^{4} \cosh \left (1\right ) + 4 \, d x^{2}\right )} \sinh \left (1\right )}{x^{4}}\right ) + 4 \, \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} a c + b \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} \log \left (c^{4} d^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \sinh \left (1\right )^{2} + 4 \, {\left (c^{4} d x + {\left (2 \, c^{4} x^{3} + c^{2} x\right )} \cosh \left (1\right ) + {\left (2 \, c^{4} x^{3} + c^{2} x\right )} \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}{4 \, {\left (c \cosh \left (1\right ) + c \sinh \left (1\right )\right )}}, \frac {2 \, b c \sqrt {-d} \arctan \left (\frac {{\left (c^{3} d x^{3} + c x^{3} \cosh \left (1\right ) + c x^{3} \sinh \left (1\right ) + 2 \, c d x\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {-d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d^{2} x^{2} + d^{2} + {\left (c^{2} d x^{4} + d x^{2}\right )} \cosh \left (1\right ) + {\left (c^{2} d x^{4} + d x^{2}\right )} \sinh \left (1\right )\right )}}\right ) + 4 \, \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} b c \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 4 \, \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} a c + b \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} \log \left (c^{4} d^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \sinh \left (1\right )^{2} + 4 \, {\left (c^{4} d x + {\left (2 \, c^{4} x^{3} + c^{2} x\right )} \cosh \left (1\right ) + {\left (2 \, c^{4} x^{3} + c^{2} x\right )} \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}{4 \, {\left (c \cosh \left (1\right ) + c \sinh \left (1\right )\right )}}\right ] \end {gather*}

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(x^2*cosh(1) + x^2*sinh(1) + 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*x^4 + 8*c^2*d^2*x^2 + x^4*cosh(1)^2 + x^4*sinh(1)^2 - 4*(c^3*d*x^3 + c*x^3*cosh(1) + c*x^3*sin
h(1) + 2*c*d*x)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2 + 2*(3*c^2*d
*x^4 + 4*d*x^2)*cosh(1) + 2*(3*c^2*d*x^4 + x^4*cosh(1) + 4*d*x^2)*sinh(1))/x^4) + 4*sqrt(x^2*cosh(1) + x^2*sin
h(1) + d)*a*c + b*sqrt(cosh(1) + sinh(1))*log(c^4*d^2 + (8*c^4*x^4 + 8*c^2*x^2 + 1)*cosh(1)^2 + (8*c^4*x^4 + 8
*c^2*x^2 + 1)*sinh(1)^2 + 4*(c^4*d*x + (2*c^4*x^3 + c^2*x)*cosh(1) + (2*c^4*x^3 + c^2*x)*sinh(1))*sqrt(x^2*cos
h(1) + x^2*sinh(1) + d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))*sqrt(cosh(1) + sinh(1)) + 2*(4*c^4*d*x^2 + 3*c^2*d)*cosh
(1) + 2*(4*c^4*d*x^2 + 3*c^2*d + (8*c^4*x^4 + 8*c^2*x^2 + 1)*cosh(1))*sinh(1)))/(c*cosh(1) + c*sinh(1)), 1/4*(
2*b*c*sqrt(-d)*arctan(1/2*(c^3*d*x^3 + c*x^3*cosh(1) + c*x^3*sinh(1) + 2*c*d*x)*sqrt(x^2*cosh(1) + x^2*sinh(1)
+ d)*sqrt(-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d^2*x^2 + d^2 + (c^2*d*x^4 + d*x^2)*cosh(1) + (c^2*d*x^4 + d
*x^2)*sinh(1))) + 4*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*b*c*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))
+ 4*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*a*c + b*sqrt(cosh(1) + sinh(1))*log(c^4*d^2 + (8*c^4*x^4 + 8*c^2*x^2
+ 1)*cosh(1)^2 + (8*c^4*x^4 + 8*c^2*x^2 + 1)*sinh(1)^2 + 4*(c^4*d*x + (2*c^4*x^3 + c^2*x)*cosh(1) + (2*c^4*x^3
+ c^2*x)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))*sqrt(cosh(1) + sinh(1)) +
2*(4*c^4*d*x^2 + 3*c^2*d)*cosh(1) + 2*(4*c^4*d*x^2 + 3*c^2*d + (8*c^4*x^4 + 8*c^2*x^2 + 1)*cosh(1))*sinh(1)))
/(c*cosh(1) + c*sinh(1))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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