Optimal. Leaf size=220 \[ \frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5331, 4495,
4271, 3855, 4268, 2611, 2320, 6724} \begin {gather*} -\frac {i b^2 \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {i b^2 \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}+\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b^3 \text {Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 3855
Rule 4268
Rule 4271
Rule 4495
Rule 5331
Rule 6724
Rubi steps
\begin {align*} \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int (a+b x)^3 \cot (x) \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^3}-\frac {b^3 \text {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}+\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {\left (i b^3\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(580\) vs. \(2(220)=440\).
time = 7.31, size = 580, normalized size = 2.64 \begin {gather*} \frac {a^3 x^3}{3}+\frac {a^2 b x^2 \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+a^2 b x^3 \csc ^{-1}(c x)+\frac {a^2 b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{2 c^3}+\frac {a b^2 \left (-8 i \text {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+2 c^3 x^3 \left (2+4 \csc ^{-1}(c x)^2-2 \cos \left (2 \csc ^{-1}(c x)\right )-\frac {3 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )}{c x}+\frac {3 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )}{c x}+\frac {4 i \text {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3 x^3}+2 \csc ^{-1}(c x) \sin \left (2 \csc ^{-1}(c x)\right )+\csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )-\csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )\right )\right )}{8 c^3}+\frac {b^3 \left (24 \csc ^{-1}(c x) \cot \left (\frac {1}{2} \csc ^{-1}(c x)\right )+4 \csc ^{-1}(c x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(c x)\right )+6 \csc ^{-1}(c x)^2 \csc ^2\left (\frac {1}{2} \csc ^{-1}(c x)\right )+\frac {\csc ^{-1}(c x)^3 \csc ^4\left (\frac {1}{2} \csc ^{-1}(c x)\right )}{c x}-24 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+24 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )-48 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-48 i \csc ^{-1}(c x) \text {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+48 i \csc ^{-1}(c x) \text {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )+48 \text {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-48 \text {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )-6 \csc ^{-1}(c x)^2 \sec ^2\left (\frac {1}{2} \csc ^{-1}(c x)\right )+16 c^3 x^3 \csc ^{-1}(c x)^3 \sin ^4\left (\frac {1}{2} \csc ^{-1}(c x)\right )+24 \csc ^{-1}(c x) \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )+4 \csc ^{-1}(c x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )}{48 c^3} \end {gather*}
Warning: Unable to verify antiderivative.
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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. 602 vs. \(2 (270 ) = 540\).
time = 1.00, size = 603, normalized size = 2.74
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} x^{3} a^{3}}{3}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{3} c^{3} x^{3}}{3}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}{2}+b^{3} \mathrm {arccsc}\left (c x \right ) c x -\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i a \,b^{2} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+b^{3} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 b^{3} \arctanh \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} c^{3} x^{3}+a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}-a \,b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+a \,b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i a \,b^{2} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+a \,b^{2} c x +a^{2} b \,c^{3} x^{3} \mathrm {arccsc}\left (c x \right )+\frac {a^{2} b \left (c^{2} x^{2}-1\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {a^{2} b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{3}}\) | \(603\) |
default | \(\frac {\frac {c^{3} x^{3} a^{3}}{3}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{3} c^{3} x^{3}}{3}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}{2}+b^{3} \mathrm {arccsc}\left (c x \right ) c x -\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i a \,b^{2} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+b^{3} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 b^{3} \arctanh \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} c^{3} x^{3}+a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}-a \,b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+a \,b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i a \,b^{2} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+a \,b^{2} c x +a^{2} b \,c^{3} x^{3} \mathrm {arccsc}\left (c x \right )+\frac {a^{2} b \left (c^{2} x^{2}-1\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {a^{2} b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{3}}\) | \(603\) |
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 x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 x^2\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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