Optimal. Leaf size=291 \[ \frac {i b^2 \text {Li}_2\left (-e^{i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4}-\frac {i b^2 \text {Li}_2\left (e^{i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4}-\frac {b^2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \tanh ^{-1}\left (e^{i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e^4}-\frac {b^3 \text {Li}_3\left (-e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Li}_3\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{d e^4} \]
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Rubi [A] time = 0.39, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4805, 12, 4627, 4701, 4709, 4183, 2531, 2282, 6589, 266, 63, 206} \[ \frac {i b^2 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4}-\frac {i b^2 \text {PolyLog}\left (2,e^{i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4}-\frac {b^3 \text {PolyLog}\left (3,-e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {PolyLog}\left (3,e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \tanh ^{-1}\left (e^{i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{d e^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 206
Rule 266
Rule 2282
Rule 2531
Rule 4183
Rule 4627
Rule 4701
Rule 4709
Rule 4805
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{(c e+d e x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{x^3 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{x \sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d e^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d e^4}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \left (a+b \sin ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^2 \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e^4}+\frac {b^2 \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e^4}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,(c+d x)^2\right )}{2 d e^4}\\ &=-\frac {b^2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \left (a+b \sin ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}+\frac {i b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {i b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e^4}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e^4}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-(c+d x)^2}\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \left (a+b \sin ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{d e^4}+\frac {i b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {i b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \left (a+b \sin ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{d e^4}+\frac {i b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {i b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {Li}_3\left (-e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Li}_3\left (e^{i \sin ^{-1}(c+d x)}\right )}{d e^4}\\ \end {align*}
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Mathematica [B] time = 8.02, size = 732, normalized size = 2.52 \[ -\frac {a^3}{3 d e^4 (c+d x)^3}-\frac {a^2 b \sqrt {-c^2-2 c d x-d^2 x^2+1}}{2 d e^4 (c+d x)^2}-\frac {a^2 b \log \left (\sqrt {-c^2-2 c d x-d^2 x^2+1}+1\right )}{2 d e^4}+\frac {a^2 b \log (c+d x)}{2 d e^4}-\frac {a^2 b \sin ^{-1}(c+d x)}{d e^4 (c+d x)^3}+\frac {a b^2 \left (8 i \text {Li}_2\left (-e^{i \sin ^{-1}(c+d x)}\right )-\frac {2 \left (4 i (c+d x)^3 \text {Li}_2\left (e^{i \sin ^{-1}(c+d x)}\right )+4 \sin ^{-1}(c+d x)^2+2 \sin ^{-1}(c+d x) \sin \left (2 \sin ^{-1}(c+d x)\right )-3 (c+d x) \sin ^{-1}(c+d x) \log \left (1-e^{i \sin ^{-1}(c+d x)}\right )+3 (c+d x) \sin ^{-1}(c+d x) \log \left (1+e^{i \sin ^{-1}(c+d x)}\right )+\sin ^{-1}(c+d x) \sin \left (3 \sin ^{-1}(c+d x)\right ) \log \left (1-e^{i \sin ^{-1}(c+d x)}\right )-\sin ^{-1}(c+d x) \sin \left (3 \sin ^{-1}(c+d x)\right ) \log \left (1+e^{i \sin ^{-1}(c+d x)}\right )-2 \cos \left (2 \sin ^{-1}(c+d x)\right )+2\right )}{(c+d x)^3}\right )}{8 d e^4}+\frac {b^3 \left (48 i \sin ^{-1}(c+d x) \text {Li}_2\left (-e^{i \sin ^{-1}(c+d x)}\right )-48 i \sin ^{-1}(c+d x) \text {Li}_2\left (e^{i \sin ^{-1}(c+d x)}\right )-48 \text {Li}_3\left (-e^{i \sin ^{-1}(c+d x)}\right )+48 \text {Li}_3\left (e^{i \sin ^{-1}(c+d x)}\right )-\frac {16 \sin ^{-1}(c+d x)^3 \sin ^4\left (\frac {1}{2} \sin ^{-1}(c+d x)\right )}{(c+d x)^3}+24 \sin ^{-1}(c+d x)^2 \log \left (1-e^{i \sin ^{-1}(c+d x)}\right )-24 \sin ^{-1}(c+d x)^2 \log \left (1+e^{i \sin ^{-1}(c+d x)}\right )-4 \sin ^{-1}(c+d x)^3 \tan \left (\frac {1}{2} \sin ^{-1}(c+d x)\right )-24 \sin ^{-1}(c+d x) \tan \left (\frac {1}{2} \sin ^{-1}(c+d x)\right )-4 \sin ^{-1}(c+d x)^3 \cot \left (\frac {1}{2} \sin ^{-1}(c+d x)\right )-24 \sin ^{-1}(c+d x) \cot \left (\frac {1}{2} \sin ^{-1}(c+d x)\right )-\left ((c+d x) \sin ^{-1}(c+d x)^3 \csc ^4\left (\frac {1}{2} \sin ^{-1}(c+d x)\right )\right )-6 \sin ^{-1}(c+d x)^2 \csc ^2\left (\frac {1}{2} \sin ^{-1}(c+d x)\right )+6 \sin ^{-1}(c+d x)^2 \sec ^2\left (\frac {1}{2} \sin ^{-1}(c+d x)\right )+48 \log \left (\tan \left (\frac {1}{2} \sin ^{-1}(c+d x)\right )\right )\right )}{48 d e^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{2} \arcsin \left (d x + c\right )^{2} + 3 \, a^{2} b \arcsin \left (d x + c\right ) + a^{3}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 716, normalized size = 2.46 \[ -\frac {a^{3}}{3 d \,e^{4} \left (d x +c \right )^{3}}-\frac {b^{3} \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{2 d \,e^{4} \left (d x +c \right )^{2}}-\frac {b^{3} \arcsin \left (d x +c \right )^{3}}{3 d \,e^{4} \left (d x +c \right )^{3}}-\frac {b^{3} \arcsin \left (d x +c \right )}{d \,e^{4} \left (d x +c \right )}-\frac {b^{3} \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{2 d \,e^{4}}-\frac {i a \,b^{2} \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}-\frac {b^{3} \polylog \left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}+\frac {b^{3} \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{2 d \,e^{4}}+\frac {i a \,b^{2} \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}+\frac {b^{3} \polylog \left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}-\frac {2 b^{3} \arctanh \left (i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}-\frac {a \,b^{2} \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{d \,e^{4} \left (d x +c \right )^{2}}-\frac {a \,b^{2} \arcsin \left (d x +c \right )^{2}}{d \,e^{4} \left (d x +c \right )^{3}}-\frac {a \,b^{2}}{d \,e^{4} \left (d x +c \right )}-\frac {a \,b^{2} \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}+\frac {i b^{3} \arcsin \left (d x +c \right ) \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}+\frac {a \,b^{2} \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}-\frac {i b^{3} \arcsin \left (d x +c \right ) \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d \,e^{4}}-\frac {a^{2} b \arcsin \left (d x +c \right )}{d \,e^{4} \left (d x +c \right )^{3}}-\frac {a^{2} b \sqrt {1-\left (d x +c \right )^{2}}}{2 d \,e^{4} \left (d x +c \right )^{2}}-\frac {a^{2} b \arctanh \left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{2 d \,e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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 \[ \frac {\int \frac {a^{3}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a^{2} b \operatorname {asin}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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