Integrand size = 23, antiderivative size = 169 \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=-\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,e^{2 i \arcsin (c+d x)}\right )}{4 d e} \]
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Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4889, 12, 4721, 3798, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{2 d e}-\frac {3 i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2}{2 d e}-\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {\log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3}{d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,e^{2 i \arcsin (c+d x)}\right )}{4 d e} \]
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4721
Rule 4889
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{x} \, dx,x,c+d x\right )}{d e} \\ & = \frac {\text {Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\arcsin (c+d x)\right )}{d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}-\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1-e^{2 i x}} \, dx,x,\arcsin (c+d x)\right )}{d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )}{2 d e}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{2 i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{2 d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {\left (3 i b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i \arcsin (c+d x)}\right )}{4 d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,e^{2 i \arcsin (c+d x)}\right )}{4 d e} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=-\frac {i \left (8 a b^2 \pi ^3+b^3 \pi ^4+96 a^2 b \arcsin (c+d x)^2-64 a b^2 \arcsin (c+d x)^3-16 b^3 \arcsin (c+d x)^4+192 i a b^2 \arcsin (c+d x)^2 \log \left (1-e^{-2 i \arcsin (c+d x)}\right )+64 i b^3 \arcsin (c+d x)^3 \log \left (1-e^{-2 i \arcsin (c+d x)}\right )+192 i a^2 b \arcsin (c+d x) \log \left (1-e^{2 i \arcsin (c+d x)}\right )+64 i a^3 \log (c+d x)-96 b^2 \arcsin (c+d x) (2 a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c+d x)}\right )+96 a^2 b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )+96 i a b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c+d x)}\right )+96 i b^3 \arcsin (c+d x) \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c+d x)}\right )+48 b^3 \operatorname {PolyLog}\left (4,e^{-2 i \arcsin (c+d x)}\right )\right )}{64 d e} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (206 ) = 412\).
Time = 0.71 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.86
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{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 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\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 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{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 )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(652\) |
default | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{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 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\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 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{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 )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(652\) |
parts | \(\frac {a^{3} \ln \left (d x +c \right )}{e d}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{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 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\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 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{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 )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}\) | \(660\) |
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\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\frac {\int \frac {a^{3}}{c + d x}\, dx + \int \frac {b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a^{2} b \operatorname {asin}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
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\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{c\,e+d\,e\,x} \,d x \]
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