Optimal. Leaf size=452 \[ -\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b c \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.62, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {4701, 4655, 4653, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4705, 4679, 4419, 4183} \[ -\frac {5 i b^2 c \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b c \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4183
Rule 4419
Rule 4653
Rule 4655
Rule 4675
Rule 4677
Rule 4679
Rule 4701
Rule 4705
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\left (4 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (8 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 x}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (32 i b c \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 )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b c \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 )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 c \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 )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 c \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 )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b c \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 )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 i b^2 c \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 )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b c \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 )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 2.91, size = 352, normalized size = 0.78 \[ -\frac {c \left (\frac {a^2 \left (8 c^4 x^4-12 c^2 x^2+3\right )}{c x}+a b \sqrt {1-c^2 x^2} \left (6 \left (c^2 x^2-1\right ) \log (c x)+5 \left (c^2 x^2-1\right ) \log \left (1-c^2 x^2\right )+1\right )+\frac {2 a b \left (8 c^4 x^4-12 c^2 x^2+3\right ) \sin ^{-1}(c x)}{c x}-b^2 \left (1-c^2 x^2\right )^{3/2} \left (\frac {c x}{\sqrt {1-c^2 x^2}}-\frac {3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{c x}+\frac {5 c x \sin ^{-1}(c x)^2}{\sqrt {1-c^2 x^2}}+\frac {c x \sin ^{-1}(c x)^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {\sin ^{-1}(c x)}{c^2 x^2-1}-5 i \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )-3 i \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-8 i \sin ^{-1}(c x)^2+6 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+10 \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )\right )\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{8} - 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} - d^{3} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.49, size = 3777, normalized size = 8.36 \[ \text {output too large to display} \]
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 \[ \frac {1}{3} \, a^{2} {\left (\frac {8 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {4 \, c^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {3}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}\right )} - \sqrt {d} \int \frac {{\left (b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{6} d^{3} x^{8} - 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} - d^{3} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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