Integrand size = 27, antiderivative size = 297 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}} \]
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Time = 0.20 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {4785, 4783, 4803, 4268, 2317, 2438, 8, 14} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=\frac {3 c^2 d \sqrt {d-c^2 d x^2} \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}} \]
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Rule 8
Rule 14
Rule 2317
Rule 2438
Rule 4268
Rule 4783
Rule 4785
Rule 4803
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}-\frac {1}{2} \left (3 c^2 d\right ) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x} \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {1-c^2 x^2}{x^2} \, dx}{2 \sqrt {1-c^2 x^2}} \\ & = -\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-c^2+\frac {1}{x^2}\right ) \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt {1-c^2 x^2}} \\ & = -\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{2 \sqrt {1-c^2 x^2}} \\ & = -\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (3 b c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{2 \sqrt {1-c^2 x^2}} \\ & = -\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (3 i b c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {\left (3 i b c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}} \\ & = -\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 1.58 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=-\frac {a d \left (1+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{2 x^2}-\frac {3}{2} a c^2 d^{3/2} \log (x)+\frac {3}{2} a c^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b c^2 d \sqrt {d-c^2 d x^2} \left (c x-\sqrt {1-c^2 x^2} \arcsin (c x)-\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {b c^2 d^2 \sqrt {1-c^2 x^2} \left (-2 \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+4 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-4 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+4 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+\arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{8 \sqrt {d-c^2 d x^2}} \]
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Time = 0.17 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.47
method | result | size |
default | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}-\frac {3 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {d \left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2} d}{2 c^{2} x^{2}-2}\right )\) | \(437\) |
parts | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}-\frac {3 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {d \left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2} d}{2 c^{2} x^{2}-2}\right )\) | \(437\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{3}}\, dx \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^3} \,d x \]
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