Integrand size = 6, antiderivative size = 36 \[ \int \csc ^{-1}(a+b x) \, dx=\frac {(a+b x) \csc ^{-1}(a+b x)}{b}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b} \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5359, 379, 272, 65, 212} \[ \int \csc ^{-1}(a+b x) \, dx=\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b}+\frac {(a+b x) \csc ^{-1}(a+b x)}{b} \]
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Rule 65
Rule 212
Rule 272
Rule 379
Rule 5359
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 {\text {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 {\text {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 {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 469, normalized size of antiderivative = 13.03 \[ \int \csc ^{-1}(a+b x) \, dx=x \csc ^{-1}(a+b x)-\frac {(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (\sqrt [4]{-1} \left (-i+\sqrt {-1+a^2}\right ) \sqrt {2 i-i a^2+2 \sqrt {-1+a^2}} \arctan \left (\frac {(-1)^{3/4} \sqrt {2 i-i a^2+2 \sqrt {-1+a^2}} b x}{a \sqrt {-1+a^2}-a \sqrt {-1+a^2+2 a b x+b^2 x^2}}\right )+(-1)^{3/4} \left (i+\sqrt {-1+a^2}\right ) \sqrt {-2 i+i a^2+2 \sqrt {-1+a^2}} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-2 i+i a^2+2 \sqrt {-1+a^2}} b x}{a \sqrt {-1+a^2}-a \sqrt {-1+a^2+2 a b x+b^2 x^2}}\right )+a \left (a \arctan \left (\frac {\sqrt {-1+a^2} b^2 x^2}{a^4+a^3 b x+b^2 x^2-a^2 \left (1+\sqrt {-1+a^2} \sqrt {-1+a^2+2 a b x+b^2 x^2}\right )}\right )-\log \left (\sqrt {-1+a^2}-b x-\sqrt {-1+a^2+2 a b x+b^2 x^2}\right )+\log \left (b^2 \left (\sqrt {-1+a^2}+b x-\sqrt {-1+a^2+2 a b x+b^2 x^2}\right )\right )\right )\right )}{a b \sqrt {-1+a^2+2 a b x+b^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )+\ln \left (b x +a +\left (b x +a \right ) \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}\) | \(43\) |
default | \(\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )+\ln \left (b x +a +\left (b x +a \right ) \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}\) | \(43\) |
parts | \(x \,\operatorname {arccsc}\left (b x +a \right )+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) \sqrt {b^{2}}+\ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b \right )}{b \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\) | \(143\) |
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.08 \[ \int \csc ^{-1}(a+b x) \, dx=\frac {b x \operatorname {arccsc}\left (b x + a\right ) - 2 \, a \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{b} \]
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\[ \int \csc ^{-1}(a+b x) \, dx=\int \operatorname {acsc}{\left (a + b x \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.53 \[ \int \csc ^{-1}(a+b x) \, dx=\frac {2 \, {\left (b x + a\right )} \operatorname {arccsc}\left (b x + a\right ) + \log \left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (34) = 68\).
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.25 \[ \int \csc ^{-1}(a+b x) \, dx=\frac {1}{2} \, b {\left (\frac {2 \, {\left (b x + a\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{2}} + \frac {\log \left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 1\right )}{b^{2}}\right )} \]
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Time = 1.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \csc ^{-1}(a+b x) \, dx=\frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{{\left (a+b\,x\right )}^2}}}\right )+\mathrm {asin}\left (\frac {1}{a+b\,x}\right )\,\left (a+b\,x\right )}{b} \]
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