Optimal. Leaf size=36 \[ \frac {(a+b x) \csc ^{-1}(a+b x)}{b}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5251, 372, 266, 63, 206} \[ \frac {(a+b x) \csc ^{-1}(a+b x)}{b}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rule 372
Rule 5251
Rubi steps
\begin {align*} \int \csc ^{-1}(a+b x) \, dx &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b}+\int \frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}} \, dx\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^2}} x} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{(a+b x)^2}\right )}{2 b}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b}\\ \end {align*}
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Mathematica [B] time = 0.12, size = 114, normalized size = 3.17 \[ \frac {(a+b x) \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (\tanh ^{-1}\left (\frac {a+b x}{\sqrt {a^2+2 a b x+b^2 x^2-1}}\right )-a \tan ^{-1}\left (\sqrt {(a+b x)^2-1}\right )\right )}{b \sqrt {a^2+2 a b x+b^2 x^2-1}}+x \csc ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 75, normalized size = 2.08 \[ \frac {b x \operatorname {arccsc}\left (b x + a\right ) - 2 \, a \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 81, normalized size = 2.25 \[ \frac {1}{2} \, b {\left (\frac {2 \, {\left (b x + a\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{2}} + \frac {\log \left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right )}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 1.39 \[ x \,\mathrm {arccsc}\left (b x +a \right )+\frac {\mathrm {arccsc}\left (b x +a \right ) a}{b}+\frac {\ln \left (b x +a +\left (b x +a \right ) \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 55, normalized size = 1.53 \[ \frac {2 \, {\left (b x + a\right )} \operatorname {arccsc}\left (b x + a\right ) + \log \left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 33, normalized size = 0.92 \[ \frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{{\left (a+b\,x\right )}^2}}}\right )+\mathrm {asin}\left (\frac {1}{a+b\,x}\right )\,\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acsc}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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