Optimal. Leaf size=257 \[ \frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e}-\frac {i b \text {PolyLog}\left (2,\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {i b \text {PolyLog}\left (2,\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {i b \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 e} \]
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Rubi [A]
time = 0.29, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5333, 2598}
\begin {gather*} \frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e-\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (\sqrt {e^2-c^2 d^2}+e\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {i b \text {Li}_2\left (\frac {i \left (e-\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {i b \text {Li}_2\left (\frac {i \left (e+\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2598
Rule 5333
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{d+e x} \, dx &=\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e}+\frac {b \int \frac {\log \left (1-\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c e}+\frac {b \int \frac {\log \left (1-\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c e}-\frac {b \int \frac {\log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c e}\\ &=\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e}-\frac {i b \text {Li}_2\left (\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {i b \text {Li}_2\left (\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 411, normalized size = 1.60 \begin {gather*} \frac {a \log (d+e x)}{e}+\frac {b \left (i \left (\pi -2 \csc ^{-1}(c x)\right )^2+32 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {(c d-e) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {-c^2 d^2+e^2}}\right )-4 \left (\pi -2 \csc ^{-1}(c x)+4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )-4 \left (\pi -2 \csc ^{-1}(c x)-4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )-8 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+4 \left (\pi -2 \csc ^{-1}(c x)\right ) \log \left (e+\frac {d}{x}\right )+8 \csc ^{-1}(c x) \log \left (e+\frac {d}{x}\right )+8 i \left (\text {PolyLog}\left (2,\frac {i \left (-e+\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )+\text {PolyLog}\left (2,-\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )\right )+4 i \left (\csc ^{-1}(c x)^2+\text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )\right )}{8 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 892 vs. \(2 (316 ) = 632\).
time = 1.07, size = 893, normalized size = 3.47
method | result | size |
derivativedivides | \(\frac {\frac {a c \ln \left (e c x +c d \right )}{e}-\frac {i b c \dilog \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {i b c e \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i b c e \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {b c e \,\mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {b c e \,\mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i b \,c^{3} d^{2} \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {b \,c^{3} d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {b \,c^{3} d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i b \,c^{3} d^{2} \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {i b c \dilog \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {b c \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}}{c}\) | \(893\) |
default | \(\frac {\frac {a c \ln \left (e c x +c d \right )}{e}-\frac {i b c \dilog \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {i b c e \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i b c e \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {b c e \,\mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {b c e \,\mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i b \,c^{3} d^{2} \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {b \,c^{3} d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {b \,c^{3} d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i b \,c^{3} d^{2} \dilog \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {i b c \dilog \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {b c \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}}{c}\) | \(893\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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