Optimal. Leaf size=483 \[ -\frac{i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{8 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{16 b c^3 \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.804111, antiderivative size = 483, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {4701, 4653, 4675, 3719, 2190, 2279, 2391, 4679, 4419, 4183, 264} \[ -\frac{i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{8 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{16 b c^3 \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4653
Rule 4675
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 4679
Rule 4419
Rule 4183
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{1}{3} \left (4 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (2 b c \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{1}{3} \left (8 c^4\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2}} \, dx}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (16 b c^5 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (16 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (16 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{8 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (32 i b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (8 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{8 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{16 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (4 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (16 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{8 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{16 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{5 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{8 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{16 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.852669, size = 462, normalized size = 0.96 \[ \frac{-3 i b^2 c^3 x^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )-5 i b^2 c^3 x^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+8 a^2 c^4 x^4-4 a^2 c^2 x^2-a^2-a b c x \sqrt{1-c^2 x^2}+10 a b c^3 x^3 \sqrt{1-c^2 x^2} \log (c x)+3 a b c^3 x^3 \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )+16 a b c^4 x^4 \sin ^{-1}(c x)-8 a b c^2 x^2 \sin ^{-1}(c x)-2 a b \sin ^{-1}(c x)+b^2 c^4 x^4-b^2 c^2 x^2+8 b^2 c^4 x^4 \sin ^{-1}(c x)^2-8 i b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2-4 b^2 c^2 x^2 \sin ^{-1}(c x)^2-b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+10 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+6 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )-b^2 \sin ^{-1}(c x)^2}{3 d x^3 \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.372, size = 2845, normalized size = 5.9 \begin{align*} \text{output too large to display} \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{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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