∫ (a + b csc−1(cx)) dx [47]

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

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5323, 272, 65, 214} \begin {gather*} a x+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}+b x \csc ^{-1}(c x) \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[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 \csc ^{-1}(c x)\right ) \, dx &=a x+b \int \csc ^{-1}(c x) \, dx\\ &=a x+b x \csc ^{-1}(c x)+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x} \, dx}{c}\\ &=a x+b x \csc ^{-1}(c x)-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=a x+b x \csc ^{-1}(c x)+(b c) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )\\ &=a x+b x \csc ^{-1}(c x)+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 58, normalized size = 1.87 \begin {gather*} a x+b x \csc ^{-1}(c x)+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}} \end {gather*}

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

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method	result	size

default	\(a x +b x \,\mathrm {arccsc}\left (c x \right )+\frac {b \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}\)	\(37\)
derivativedivides	\(\frac {a c x +c x b \,\mathrm {arccsc}\left (c x \right )+\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) b}{c}\)	\(40\)

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Maxima [A]
time = 0.26, size = 53, normalized size = 1.71 \begin {gather*} a x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b}{2 \, c} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
time = 0.43, size = 64, normalized size = 2.06 \begin {gather*} \frac {a c x - 2 \, b c \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c x - b c\right )} \operatorname {arccsc}\left (c x\right ) - b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c} \end {gather*}

(a*c*x - 2*b*c*arctan(-c*x + sqrt(c^2*x^2 - 1)) + (b*c*x - b*c)*arccsc(c*x) - b*log(-c*x + sqrt(c^2*x^2 - 1)))

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Sympy [A]
time = 1.78, size = 32, normalized size = 1.03 \begin {gather*} a x + b \left (x \operatorname {acsc}{\left (c x \right )} + \frac {\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}}{c}\right ) \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).
time = 0.42, size = 62, normalized size = 2.00 \begin {gather*} \frac {1}{2} \, b c {\left (\frac {2 \, x \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}}\right )} + a x \end {gather*}

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

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Mupad [B]
time = 0.00, size = 33, normalized size = 1.06 \begin {gather*} a\,x+b\,x\,\mathrm {asin}\left (\frac {1}{c\,x}\right )+\frac {b\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{c^2\,x^2}}}\right )}{c} \end {gather*}

a*x + b*x*asin(1/(c*x)) + (b*atanh(1/(1 - 1/(c^2*x^2))^(1/2)))/c

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3.1.47 \(\int (a+b \csc ^{-1}(c x)) \, dx\) [47]