Optimal. Leaf size=579 \[ \frac{i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d} \]
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Rubi [A] time = 0.915763, antiderivative size = 579, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {4733, 4627, 266, 63, 208, 4667, 4741, 4521, 2190, 2279, 2391} \[ \frac{i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4733
Rule 4627
Rule 266
Rule 63
Rule 208
Rule 4667
Rule 4741
Rule 4521
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx &=\int \left (\frac{a+b \sin ^{-1}(c x)}{d x^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \sin ^{-1}(c x)}{x^2} \, dx}{d}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{d+e x^2} \, dx}{d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d x}+\frac{(b c) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{d}-\frac{e \int \left (\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d x}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 (-d)^{3/2}}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c d}-\frac{e \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 (-d)^{3/2}}-\frac{e \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 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d}-\frac{(i e) \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 (-d)^{3/2}}-\frac{(i e) \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 (-d)^{3/2}}-\frac{(i e) \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 (-d)^{3/2}}-\frac{(i e) \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 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d}+\frac{\sqrt{e} \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 (-d)^{3/2}}-\frac{\sqrt{e} \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 (-d)^{3/2}}+\frac{\sqrt{e} \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 (-d)^{3/2}}-\frac{\sqrt{e} \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 (-d)^{3/2}}-\frac{\left (b \sqrt{e}\right ) \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 (-d)^{3/2}}+\frac{\left (b \sqrt{e}\right ) \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 (-d)^{3/2}}-\frac{\left (b \sqrt{e}\right ) \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 (-d)^{3/2}}+\frac{\left (b \sqrt{e}\right ) \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 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d}+\frac{\sqrt{e} \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 (-d)^{3/2}}-\frac{\sqrt{e} \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 (-d)^{3/2}}+\frac{\sqrt{e} \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 (-d)^{3/2}}-\frac{\sqrt{e} \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 (-d)^{3/2}}+\frac{\left (i b \sqrt{e}\right ) \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 (-d)^{3/2}}-\frac{\left (i b \sqrt{e}\right ) \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 (-d)^{3/2}}+\frac{\left (i b \sqrt{e}\right ) \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 (-d)^{3/2}}-\frac{\left (i b \sqrt{e}\right ) \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 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d}+\frac{\sqrt{e} \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 (-d)^{3/2}}-\frac{\sqrt{e} \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 (-d)^{3/2}}+\frac{\sqrt{e} \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 (-d)^{3/2}}-\frac{\sqrt{e} \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 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.377149, size = 455, normalized size = 0.79 \[ \frac{b \sqrt{e} x \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )-b \sqrt{e} x \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )-4 a \sqrt{e} x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )-4 a \sqrt{d}-4 b \sqrt{d} \left (c x \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\sin ^{-1}(c x)\right )}{4 d^{3/2} x} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.511, size = 363, normalized size = 0.6 \begin{align*} -{\frac{ae}{d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{a}{dx}}-{\frac{b\arcsin \left ( cx \right ) }{dx}}-{\frac{be}{8\,c{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{4\,{{\it \_R1}}^{2}{c}^{2}d+{{\it \_R1}}^{2}e-e}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e-2\,{c}^{2}d-e \right ) } \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }}+{\frac{be}{8\,c{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}e-4\,{c}^{2}d-e}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e-2\,{c}^{2}d-e \right ) } \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }}+{\frac{bc}{d}\ln \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1}-1 \right ) }-{\frac{bc}{d}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arcsin \left (c x\right ) + a}{e x^{4} + d x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asin}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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