Optimal. Leaf size=309 \[ -\frac{2 b^2 c \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)} \]
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Rubi [A] time = 0.528895, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {4743, 4773, 3323, 2264, 2190, 2279, 2391} \[ -\frac{2 b^2 c \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 4743
Rule 4773
Rule 3323
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{(2 b c) \int \frac{a+b \sin ^{-1}(c x)}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{a+b x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{(4 b c) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac{(4 i b c) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c d-2 \sqrt{c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{c^2 d^2-e^2}}+\frac{(4 i b c) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c d+2 \sqrt{c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{c^2 d^2-e^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{\left (2 i b^2 c\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e e^{i x}}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (2 i b^2 c\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e e^{i x}}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e \sqrt{c^2 d^2-e^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i e x}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i e x}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e \sqrt{c^2 d^2-e^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}\\ \end{align*}
Mathematica [A] time = 0.338029, size = 231, normalized size = 0.75 \[ \frac{-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac{2 b c \left (-b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )\right )\right )}{\sqrt{c^2 d^2-e^2}}}{e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.393, size = 646, normalized size = 2.1 \begin{align*} -{\frac{c{a}^{2}}{ \left ( ecx+dc \right ) e}}-{\frac{c{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{ \left ( ecx+dc \right ) e}}+2\,{\frac{c{b}^{2}\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}\arcsin \left ( cx \right ) }{e \left ({c}^{2}{d}^{2}-{e}^{2} \right ) }\ln \left ({\frac{idc+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) e+\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}}{idc+\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}}} \right ) }-2\,{\frac{c{b}^{2}\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}\arcsin \left ( cx \right ) }{e \left ({c}^{2}{d}^{2}-{e}^{2} \right ) }\ln \left ({\frac{idc+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) e-\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}}{idc-\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}}} \right ) }+{\frac{2\,ic{b}^{2}}{e \left ({c}^{2}{d}^{2}-{e}^{2} \right ) }\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}{\it dilog} \left ({ \left ( idc+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) e-\sqrt{-{c}^{2}{d}^{2}+{e}^{2}} \right ) \left ( idc-\sqrt{-{c}^{2}{d}^{2}+{e}^{2}} \right ) ^{-1}} \right ) }-{\frac{2\,ic{b}^{2}}{e \left ({c}^{2}{d}^{2}-{e}^{2} \right ) }\sqrt{-{c}^{2}{d}^{2}+{e}^{2}}{\it dilog} \left ({ \left ( idc+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) e+\sqrt{-{c}^{2}{d}^{2}+{e}^{2}} \right ) \left ( idc+\sqrt{-{c}^{2}{d}^{2}+{e}^{2}} \right ) ^{-1}} \right ) }-2\,{\frac{cab\arcsin \left ( cx \right ) }{ \left ( ecx+dc \right ) e}}-2\,{\frac{cab}{{e}^{2}}\ln \left ({ \left ( -2\,{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }+2\,\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}\sqrt{- \left ( cx+{\frac{dc}{e}} \right ) ^{2}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}} \right ) \left ( cx+{\frac{dc}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}} \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^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}{e^{2} x^{2} + 2 \, d e x + d^{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{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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