∫ (a + bArcSin(c x)) dx [373]

Optimal. Leaf size=31 \[ a x+b x \csc ^{-1}\left (\frac {x}{c}\right )+b c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right ) \]

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4916, 5323, 272, 65, 214} \begin {gather*} a x+b c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right )+b x \csc ^{-1}\left (\frac {x}{c}\right ) \end {gather*}

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,

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

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]]

Int[ArcSin[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcCsc[a/c + b*(x^n/c)]^m, x] /;

Int[ArcCsc[(c_.)*(x_)], x_Symbol] :> Simp[x*ArcCsc[c*x], x] + Dist[1/c, Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x], x

\begin {align*} \int \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=a x+b \int \sin ^{-1}\left (\frac {c}{x}\right ) \, dx\\ &=a x+b \int \csc ^{-1}\left (\frac {x}{c}\right ) \, dx\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )+(b c) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}} x} \, dx\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )+\frac {b \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-\frac {c^2}{x^2}}\right )}{c}\\ &=a x+b x \csc ^{-1}\left (\frac {x}{c}\right )+b c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{x^2}}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(31)=62\).
time = 0.06, size = 89, normalized size = 2.87 \begin {gather*} a x+b x \text {ArcSin}\left (\frac {c}{x}\right )+\frac {b c \sqrt {-c^2+x^2} \left (-\log \left (1-\frac {x}{\sqrt {-c^2+x^2}}\right )+\log \left (1+\frac {x}{\sqrt {-c^2+x^2}}\right )\right )}{2 \sqrt {1-\frac {c^2}{x^2}} x} \end {gather*}

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

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method	result	size

default	\(a x -b c \left (-\frac {x \arcsin \left (\frac {c}{x}\right )}{c}-\arctanh \left (\frac {1}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )\right )\)	\(37\)
derivativedivides	\(-c \left (-\frac {a x}{c}+b \left (-\frac {x \arcsin \left (\frac {c}{x}\right )}{c}-\arctanh \left (\frac {1}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )\right )\right )\)	\(42\)

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Maxima [A]
time = 0.47, size = 52, normalized size = 1.68 \begin {gather*} \frac {1}{2} \, {\left (c {\left (\log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right ) - \log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} - 1\right )\right )} + 2 \, x \arcsin \left (\frac {c}{x}\right )\right )} b + a x \end {gather*}

1/2*(c*(log(sqrt(-c^2/x^2 + 1) + 1) - log(sqrt(-c^2/x^2 + 1) - 1)) + 2*x*arcsin(c/x))*b + a*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).
time = 2.68, size = 75, normalized size = 2.42 \begin {gather*} -b c \log \left (x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} - x\right ) + a x + {\left (b x - b\right )} \arcsin \left (\frac {c}{x}\right ) - 2 \, b \arctan \left (\frac {x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} - x}{c}\right ) \end {gather*}

-b*c*log(x*sqrt(-(c^2 - x^2)/x^2) - x) + a*x + (b*x - b)*arcsin(c/x) - 2*b*arctan((x*sqrt(-(c^2 - x^2)/x^2) -

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Sympy [A]
time = 1.09, size = 32, normalized size = 1.03 \begin {gather*} a x + b \left (c \left (\begin {cases} \operatorname {acosh}{\left (\frac {x}{c} \right )} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {x}{c} \right )} & \text {otherwise} \end {cases}\right ) + x \operatorname {asin}{\left (\frac {c}{x} \right )}\right ) \end {gather*}

a*x + b*(c*Piecewise((acosh(x/c), Abs(x**2/c**2) > 1), (-I*asin(x/c), True)) + x*asin(c/x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
time = 0.41, size = 60, normalized size = 1.94 \begin {gather*} a x + \frac {{\left (c^{2} {\left (\log \left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )\right )} + 2 \, c x \arcsin \left (\frac {c}{x}\right )\right )} b}{2 \, c} \end {gather*}

a*x + 1/2*(c^2*(log(sqrt(-c^2/x^2 + 1) + 1) - log(-sqrt(-c^2/x^2 + 1) + 1)) + 2*c*x*arcsin(c/x))*b/c

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Mupad [B]
time = 0.74, size = 32, normalized size = 1.03 \begin {gather*} a\,x+b\,x\,\mathrm {asin}\left (\frac {c}{x}\right )+b\,c\,\mathrm {sign}\left (x\right )\,\ln \left (x+\sqrt {x^2-c^2}\right ) \end {gather*}

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3.4.73 \(\int (a+b \text {ArcSin}(\frac {c}{x})) \, dx\) [373]