Optimal. Leaf size=176 \[ \frac {10}{3} b c^3 d \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac {5}{3} i b^2 c^3 d \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+\frac {5}{3} i b^2 c^3 d \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-\frac {b^2 c^2 d}{3 x} \]
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Rubi [A] time = 0.38, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4695, 4627, 4709, 4183, 2279, 2391, 4693, 30} \[ -\frac {5}{3} i b^2 c^3 d \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+\frac {5}{3} i b^2 c^3 d \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}+\frac {10}{3} b c^3 d \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {b^2 c^2 d}{3 x} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2279
Rule 2391
Rule 4183
Rule 4627
Rule 4693
Rule 4695
Rule 4709
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} (2 b c d) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx-\frac {1}{3} \left (2 c^2 d\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d\right ) \int \frac {1}{x^2} \, dx-\frac {1}{3} \left (b c^3 d\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx-\frac {1}{3} \left (4 b c^3 d\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac {1}{3} \left (b c^3 d\right ) \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{3} \left (4 b c^3 d\right ) \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {10}{3} b c^3 d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\frac {1}{3} \left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{3} \left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{3} \left (4 b^2 c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{3} \left (4 b^2 c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {10}{3} b c^3 d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac {1}{3} \left (i b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\frac {1}{3} \left (i b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\frac {1}{3} \left (4 i b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\frac {1}{3} \left (4 i b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {2 c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {10}{3} b c^3 d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac {5}{3} i b^2 c^3 d \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+\frac {5}{3} i b^2 c^3 d \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.77, size = 266, normalized size = 1.51 \[ \frac {d \left (3 a^2 c^2 x^2-a^2-a b c x \sqrt {1-c^2 x^2}+6 a b c^2 x^2 \sin ^{-1}(c x)+5 a b c^3 x^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-2 a b \sin ^{-1}(c x)-5 i b^2 c^3 x^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+5 i b^2 c^3 x^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-5 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+5 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-b^2 c^2 x^2+3 b^2 c^2 x^2 \sin ^{-1}(c x)^2-b^2 c x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)-b^2 \sin ^{-1}(c x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 291, normalized size = 1.65 \[ -\frac {d \,a^{2}}{3 x^{3}}+\frac {c^{2} d \,a^{2}}{x}+\frac {c^{2} d \,b^{2} \arcsin \left (c x \right )^{2}}{x}-\frac {c d \,b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{3 x^{2}}-\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{3 x^{3}}-\frac {b^{2} c^{2} d}{3 x}+\frac {5 c^{3} d \,b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {5 i b^{2} c^{3} d \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {5 c^{3} d \,b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {5 i b^{2} c^{3} d \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {2 d a b \arcsin \left (c x \right )}{3 x^{3}}+\frac {2 c^{2} d a b \arcsin \left (c x \right )}{x}+\frac {5 c^{3} d a b \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{3}-\frac {c d a b \sqrt {-c^{2} x^{2}+1}}{3 x^{2}} \]
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 \[ 2 \, {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} a b c^{2} d - \frac {1}{3} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac {2 \, \arcsin \left (c x\right )}{x^{3}}\right )} a b d + \frac {a^{2} c^{2} d}{x} - \frac {a^{2} d}{3 \, x^{3}} + \frac {2 \, x^{3} \int \frac {{\left (3 \, b^{2} c^{3} d x^{2} - b^{2} c d\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{2} x^{5} - x^{3}}\,{d x} + {\left (3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right )}{x^4} \,d x \]
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 \[ - d \left (\int \left (- \frac {a^{2}}{x^{4}}\right )\, dx + \int \frac {a^{2} c^{2}}{x^{2}}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \frac {b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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