Optimal. Leaf size=31 \[ a x+\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c}+b x \csc ^{-1}(c x) \]
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Rubi [A] time = 0.0205758, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5215, 266, 63, 208} \[ a x+\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c}+b x \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5215
Rule 266
Rule 63
Rule 208
Rubi steps
\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 \operatorname{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) \operatorname{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*}
Mathematica [A] time = 0.0657152, size = 58, normalized size = 1.87 \[ a x+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{\sqrt{c^2 x^2-1}}+b x \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.169, size = 37, normalized size = 1.2 \begin{align*} ax+bx{\rm arccsc} \left (cx\right )+{\frac{b}{c}\ln \left ( cx+cx\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956712, size = 72, normalized size = 2.32 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.46162, size = 154, normalized size = 4.97 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acsc}{\left (c x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09465, size = 63, normalized size = 2.03 \begin{align*}{\left (x \arcsin \left (\frac{1}{c x}\right ) - \frac{c \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{{\left | c \right |}^{2} \mathrm{sgn}\left (x\right )}\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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