Optimal. Leaf size=491 \[ \frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e} \]
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Rubi [A] time = 0.74, antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {4733, 4741, 4521, 2190, 2279, 2391} \[ -\frac {i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}-\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}-\frac {i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}-\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}+\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e}-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 4521
Rule 4733
Rule 4741
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{d+e x^2} \, dx &=\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\\ &=-\frac {\int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {e}}+\frac {\int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {e}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt {e}}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt {e}}\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e}-\frac {i \operatorname {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 \sqrt {e}}-\frac {i \operatorname {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 \sqrt {e}}+\frac {i \operatorname {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 \sqrt {e}}+\frac {i \operatorname {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 \sqrt {e}}\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b 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}+\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}+\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}+\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}-\frac {b \operatorname {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}-\frac {b \operatorname {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}-\frac {b \operatorname {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}-\frac {b \operatorname {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}\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b 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}+\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}+\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}+\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}+\frac {(i b) \operatorname {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}+\frac {(i b) \operatorname {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}+\frac {(i b) \operatorname {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}+\frac {(i b) \operatorname {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}\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b 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}+\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}+\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}+\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}-\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}-\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}-\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}-\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}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 399, normalized size = 0.81 \[ -\frac {i \left (i a \log \left (d+e x^2\right )+b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {d c^2+e}}\right )+b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d c^2+e}-c \sqrt {d}}\right )+b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )+b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )+i b \sin ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+i b \sin ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}-c \sqrt {d}}\right )+i b \sin ^{-1}(c x) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+c \sqrt {d}}\right )+i b \sin ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+c \sqrt {d}}\right )+b \sin ^{-1}(c x)^2\right )}{2 e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \arcsin \left (c x\right ) + a x}{e x^{2} + d}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.41, size = 2749, normalized size = 5.60 \[ \text {Expression too large to display} \]
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 \[ b \int \frac {x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{e x^{2} + d}\,{d x} + \frac {a \log \left (e x^{2} + d\right )}{2 \, e} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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