Optimal. Leaf size=513 \[ \frac {x \left (c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.98, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {4777, 4775, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181, 4641} \[ -\frac {i b^2 \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {x \left (c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4181
Rule 4641
Rule 4651
Rule 4657
Rule 4675
Rule 4677
Rule 4763
Rule 4775
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {\left (c^2 f^2+g^2+2 c^2 f g x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (c^2 f^2+g^2+2 c^2 f g x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \int \left (\frac {c^2 f^2 \left (1+\frac {g^2}{c^2 f^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {2 c^2 f g x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 f g \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}+\frac {\left (\left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 i b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 2.26, size = 259, normalized size = 0.50 \[ \frac {\sqrt {1-c^2 x^2} \left (-3 (c f+g)^2 \left (-\tan \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)+4 i b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )\right )+4 b^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )\right )\right )+3 (g-c f)^2 \left (-\cot \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)-4 i b \log \left (1+i e^{-i \sin ^{-1}(c x)}\right )\right )+4 b^2 \text {Li}_2\left (-i e^{-i \sin ^{-1}(c x)}\right )\right )\right )-\frac {2 g^2 \left (a+b \sin ^{-1}(c x)\right )^3}{b}\right )}{6 c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} g^{2} x^{2} + 2 \, a^{2} f g x + a^{2} f^{2} + {\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g^{2} x^{2} + 2 \, a b f g x + a b f^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.91, size = 1861, normalized size = 3.63 \[ \text {result too large to display} \]
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 \[ a^{2} g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} d^{\frac {3}{2}}}\right )} + \frac {2 \, a b f^{2} x \arcsin \left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a^{2} f^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {a b f^{2} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{c d^{\frac {3}{2}}} - \sqrt {d} \int \frac {{\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b g^{2} x^{2} + 2 \, a b f g x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c^{2} d^{2} x^{2} - d^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} + \frac {2 \, a^{2} f g}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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