Integrand size = 40, antiderivative size = 205 \[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {i \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {i b \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \]
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Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {6813, 4721, 3798, 2221, 2611, 2320, 6724} \[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}+\frac {i \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{3 b c}-\frac {\log \left (1-e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 c} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4721
Rule 6724
Rule 6813
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^2}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = -\frac {\text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}+\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {i b \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = \frac {i \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {i b \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \\ & = \frac {i \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {i b \left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \\ \end{align*}
\[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (226 ) = 452\).
Time = 1.30 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.20
method | result | size |
default | \(-\frac {a^{2} \ln \left (c x -1\right )}{2 c}+\frac {a^{2} \ln \left (c x +1\right )}{2 c}-b^{2} \left (-\frac {i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}+1\right )}{c}-\frac {2 i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, -\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, \frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}\right )-2 a b \left (-\frac {i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}+1\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}-\frac {i \operatorname {polylog}\left (2, \frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}\right )\) | \(657\) |
parts | \(-\frac {a^{2} \ln \left (c x -1\right )}{2 c}+\frac {a^{2} \ln \left (c x +1\right )}{2 c}-b^{2} \left (-\frac {i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}+1\right )}{c}-\frac {2 i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, -\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 \operatorname {polylog}\left (3, \frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}\right )-2 a b \left (-\frac {i \arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}+1\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\arcsin \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}-\frac {i \operatorname {polylog}\left (2, \frac {i \sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1-\frac {-c x +1}{c x +1}}\right )}{c}\right )\) | \(657\) |
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\[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {asin}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \]
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