Optimal. Leaf size=574 \[ \frac {d (a+b \text {ArcSin}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \text {ArcSin}(c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2} \]
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Rubi [A]
time = 0.70, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4817, 4813,
385, 211, 4825, 4617, 2221, 2317, 2438} \begin {gather*} \frac {(a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {d (a+b \text {ArcSin}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \text {ArcSin}(c x))^2}{2 b e^2}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {b c \sqrt {d} \text {ArcTan}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 385
Rule 2221
Rule 2317
Rule 2438
Rule 4617
Rule 4813
Rule 4817
Rule 4825
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {d x \left (a+b \sin ^{-1}(c x)\right )}{e \left (d+e x^2\right )^2}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=\frac {d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {(b c d) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e^2}+\frac {\int \left (-\frac {a+b \sin ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e}\\ &=\frac {d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {(b c d) \text {Subst}\left (\int \frac {1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-c^2 x^2}}\right )}{2 e^2}-\frac {\int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{3/2}}+\frac {\int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{3/2}}\\ &=\frac {d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {b c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{3/2}}\\ &=\frac {d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}-\frac {i \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}-\sqrt {c^2 d+e}-\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{3/2}}-\frac {i \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}+\sqrt {c^2 d+e}-\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {i \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}-\sqrt {c^2 d+e}+\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {i \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}+\sqrt {c^2 d+e}+\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{3/2}}\\ &=\frac {d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}\\ &=\frac {d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^2}\\ &=\frac {d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 593, normalized size = 1.03 \begin {gather*} \frac {\frac {2 a d}{d+e x^2}+2 a \log \left (d+e x^2\right )+b \left (\sqrt {d} \left (\frac {\text {ArcSin}(c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \text {ArcTan}\left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-i \sqrt {d} \left (-\frac {\text {ArcSin}(c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-i \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-i \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.87, size = 2962, normalized size = 5.16
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2962\) |
default | \(\text {Expression too large to display}\) | \(2962\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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