Optimal. Leaf size=271 \[ -\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-i b d^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{32} b^2 c^4 d^2 x^4+\frac {13}{32} b^2 c^2 d^2 x^2+\frac {1}{2} b^2 d^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.41, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4699, 4625, 3717, 2190, 2531, 2282, 6589, 4647, 4641, 30, 4649, 14} \[ -i b d^2 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} b^2 d^2 \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{32} b^2 c^4 d^2 x^4+\frac {13}{32} b^2 c^2 d^2 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 4641
Rule 4647
Rule 4649
Rule 4699
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{2} \left (b c d^2\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{8} \left (3 b c d^2\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (b c d^2\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {1}{8} \left (b^2 c^2 d^2\right ) \int x \left (1-c^2 x^2\right ) \, dx\\ &=-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \operatorname {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{16} \left (3 b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} \left (b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{8} \left (b^2 c^2 d^2\right ) \int \left (x-c^2 x^3\right ) \, dx+\frac {1}{16} \left (3 b^2 c^2 d^2\right ) \int x \, dx+\frac {1}{2} \left (b^2 c^2 d^2\right ) \int x \, dx\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\left (2 i d^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (2 b d^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (i b^2 d^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} \left (b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {11}{32} d^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} b^2 d^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.47, size = 353, normalized size = 1.30 \[ \frac {1}{768} d^2 \left (192 a^2 c^4 x^4-768 a^2 c^2 x^2+768 a^2 \log (c x)+384 a b c^4 x^4 \sin ^{-1}(c x)-624 a b c x \sqrt {1-c^2 x^2}-1536 a b c^2 x^2 \sin ^{-1}(c x)+96 a b c^3 x^3 \sqrt {1-c^2 x^2}-768 i a b \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-768 i a b \sin ^{-1}(c x)^2+624 a b \sin ^{-1}(c x)+1536 a b \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+768 i b^2 \sin ^{-1}(c x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )+384 b^2 \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )+256 i b^2 \sin ^{-1}(c x)^3-288 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )-12 b^2 \sin ^{-1}(c x) \sin \left (4 \sin ^{-1}(c x)\right )+768 b^2 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )-144 b^2 \cos \left (2 \sin ^{-1}(c x)\right )+288 b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )-3 b^2 \cos \left (4 \sin ^{-1}(c x)\right )+24 b^2 \sin ^{-1}(c x)^2 \cos \left (4 \sin ^{-1}(c x)\right )-32 i \pi ^3 b^2\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 2.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.43, size = 560, normalized size = 2.07 \[ d^{2} a^{2} \ln \left (c x \right )-\frac {d^{2} b^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {3 d^{2} b^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{16}+2 d^{2} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d^{2} a b \arcsin \left (c x \right )^{2}+2 d^{2} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {i d^{2} b^{2} \arcsin \left (c x \right )^{3}}{3}+d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 d^{2} b^{2} \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {d^{2} a^{2} c^{4} x^{4}}{4}-d^{2} a^{2} c^{2} x^{2}-\frac {d^{2} a b \sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {3 d^{2} a b \sin \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {d^{2} b^{2} \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{32}-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 d^{2} b^{2} \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{8}+\frac {d^{2} a b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {3 d^{2} a b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, a^{2} c^{4} d^{2} x^{4} - a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} \log \relax (x) + \int \frac {{\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ d^{2} \left (\int \frac {a^{2}}{x}\, dx + \int \left (- 2 a^{2} c^{2} x\right )\, dx + \int a^{2} c^{4} x^{3}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x}\, dx + \int \left (- 2 b^{2} c^{2} x \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{4} x^{3} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 4 a b c^{2} x \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{4} x^{3} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________