Optimal. Leaf size=272 \[ \frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {i \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 i a^2 \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {2 i a^2 \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \]
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Rubi [A]
time = 0.16, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5367, 4512,
4275, 4268, 2317, 2438, 4269, 3556, 4270} \begin {gather*} \frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}-\frac {2 i a^2 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 i a^2 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {i \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {x}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 3556
Rule 4268
Rule 4269
Rule 4270
Rule 4275
Rule 4512
Rule 5367
Rubi steps
\begin {align*} \int x^2 \csc ^{-1}(a+b x)^2 \, dx &=-\frac {\text {Subst}\left (\int x^2 \cot (x) \csc (x) (-a+\csc (x))^2 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int x (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int \left (-a^3 x+3 a^2 x \csc (x)-3 a x \csc ^2(x)+x \csc ^3(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int x \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}+\frac {(2 a) \text {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}+\frac {(2 a) \text {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}+\frac {\text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}-\frac {\text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}-\frac {\left (2 i a^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\left (2 i a^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {2 i a^2 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 i a^2 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {i \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 i a^2 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {2 i a^2 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 3.14, size = 314, normalized size = 1.15 \begin {gather*} -\frac {-2 \left (2-12 a \csc ^{-1}(a+b x)+\left (1+6 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+2 \csc ^{-1}(a+b x) \left (-1+3 a \csc ^{-1}(a+b x)\right ) \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {\csc ^{-1}(a+b x)^2 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}-48 a \log \left (\frac {1}{a+b x}\right )+8 \left (1+6 a^2\right ) \left (\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-\text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )\right )\right )+2 \csc ^{-1}(a+b x) \left (1+3 a \csc ^{-1}(a+b x)\right ) \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-8 (a+b x)^3 \csc ^{-1}(a+b x)^2 \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \left (2+12 a \csc ^{-1}(a+b x)+\left (1+6 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{24 b^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 10.44, size = 500, normalized size = 1.84
method | result | size |
derivativedivides | \(\frac {\mathrm {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-\mathrm {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+\frac {\mathrm {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-2 \,\mathrm {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+\frac {\mathrm {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}}{3}-\frac {i \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+\frac {b x}{3}+\frac {a}{3}+\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+\frac {i \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-2 i a^{2} \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 a \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 a \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right )-4 a \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 a^{2} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 i \mathrm {arccsc}\left (b x +a \right ) a -2 a^{2} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 i a^{2} \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}\) | \(500\) |
default | \(\frac {\mathrm {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-\mathrm {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+\frac {\mathrm {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-2 \,\mathrm {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+\frac {\mathrm {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}}{3}-\frac {i \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+\frac {b x}{3}+\frac {a}{3}+\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+\frac {i \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-2 i a^{2} \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 a \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 a \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right )-4 a \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 a^{2} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 i \mathrm {arccsc}\left (b x +a \right ) a -2 a^{2} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 i a^{2} \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}\) | \(500\) |
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} \operatorname {acsc}^{2}{\left (a + b x \right )}\, 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\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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