Optimal. Leaf size=210 \[ -i \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-i \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\frac {1}{2} i \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]
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Rubi [A] time = 0.30, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5259, 4552, 4529, 3717, 2190, 2279, 2391, 4519} \[ -i \text {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-i \text {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\frac {1}{2} i \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4519
Rule 4529
Rule 4552
Rule 5259
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}(a+b x)}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {x \cot (x) \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {x \cot (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\left (a \operatorname {Subst}\left (\int \frac {x \cos (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\right )-\operatorname {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \operatorname {Subst}\left (\int \frac {e^{i x} x}{1-\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \operatorname {Subst}\left (\int \frac {e^{i x} x}{1+\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-\operatorname {Subst}\left (\int \log \left (1+\frac {i a e^{i x}}{1-\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )-\operatorname {Subst}\left (\int \log \left (1+\frac {i a e^{i x}}{1+\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+\operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )+i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{1-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )+i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )\\ &=\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-i \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-i \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {1}{2} i \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.44, size = 375, normalized size = 1.79 \[ \frac {1}{8} \left (8 i \left (\text {Li}_2\left (-\frac {i \left (\sqrt {1-a^2}-1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )+\text {Li}_2\left (\frac {i \left (\sqrt {1-a^2}+1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )\right )-4 \log \left (1+\frac {i \left (\sqrt {1-a^2}-1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right ) \left (-2 \csc ^{-1}(a+b x)+4 \sin ^{-1}\left (\frac {\sqrt {\frac {a-1}{a}}}{\sqrt {2}}\right )+\pi \right )-4 \log \left (1-\frac {i \left (\sqrt {1-a^2}+1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right ) \left (-2 \csc ^{-1}(a+b x)-4 \sin ^{-1}\left (\frac {\sqrt {\frac {a-1}{a}}}{\sqrt {2}}\right )+\pi \right )-32 i \sin ^{-1}\left (\frac {\sqrt {\frac {a-1}{a}}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {(a+1) \cot \left (\frac {1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt {1-a^2}}\right )+4 i \left (\csc ^{-1}(a+b x)^2+\text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )\right )+i \left (\pi -2 \csc ^{-1}(a+b x)\right )^2+4 \log \left (\frac {b x}{a+b x}\right ) \left (\pi -2 \csc ^{-1}(a+b x)\right )-8 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )+8 \log \left (\frac {b x}{a+b x}\right ) \csc ^{-1}(a+b x)\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccsc}\left (b x + a\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccsc}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.13, size = 607, normalized size = 2.89 \[ \frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right ) a^{2}}{a^{2}-1}+\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right ) a^{2}}{a^{2}-1}-\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {i \dilog \left (\frac {-a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right ) a^{2}}{a^{2}-1}-\frac {i \dilog \left (\frac {a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right ) a^{2}}{a^{2}-1}+\frac {i \dilog \left (\frac {-a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}+\frac {i \dilog \left (\frac {a \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-i \dilog \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+i \dilog \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccsc}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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