Optimal. Leaf size=821 \[ -\frac{\text{PolyLog}\left (3,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}+\frac{\text{PolyLog}\left (3,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}-\frac{\text{PolyLog}\left (3,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}+\frac{\text{PolyLog}\left (3,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}+\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}-\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}+\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}-\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.34202, antiderivative size = 821, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {4667, 4741, 4521, 2190, 2531, 2282, 6589} \[ -\frac{\text{PolyLog}\left (3,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}+\frac{\text{PolyLog}\left (3,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}-\frac{\text{PolyLog}\left (3,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}+\frac{\text{PolyLog}\left (3,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b^2}{\sqrt{-d} \sqrt{e}}+\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}-\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}+\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}-\frac{i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) b}{\sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4667
Rule 4741
Rule 4521
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d+e x^2} \, dx &=\int \left (\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2 \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2 \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)^2}{i c \sqrt{-d}-\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{i \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)^2}{i c \sqrt{-d}+\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{i \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)^2}{i c \sqrt{-d}-\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{i \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)^2}{i c \sqrt{-d}+\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int (a+b x) \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 )}{\sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int (a+b x) \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 )}{\sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int (a+b x) \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 )}{\sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int (a+b x) \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 )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{-i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.776722, size = 1101, normalized size = 1.34 \[ \frac{2 \sqrt{-d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) a^2-2 b \sqrt{d} \sin ^{-1}(c x) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right ) a+2 b \sqrt{d} \sin ^{-1}(c x) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-i c \sqrt{-d}}+1\right ) a+2 b \sqrt{d} \sin ^{-1}(c x) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) a-2 b \sqrt{d} \sin ^{-1}(c x) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right ) a+2 i b \sqrt{d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) a-2 i b \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) a-b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right )+b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-i c \sqrt{-d}}+1\right )+b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )-b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right )-2 i b \sqrt{d} \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )+2 i b \sqrt{d} \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-i c \sqrt{-d}}\right )+2 i b^2 \sqrt{d} \sin ^{-1}(c x) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )-2 i b^2 \sqrt{d} \sin ^{-1}(c x) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )+2 b^2 \sqrt{d} \text{PolyLog}\left (3,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )-2 b^2 \sqrt{d} \text{PolyLog}\left (3,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-i c \sqrt{-d}}\right )-2 b^2 \sqrt{d} \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )+2 b^2 \sqrt{d} \text{PolyLog}\left (3,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{2 \sqrt{-d^2} \sqrt{e}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.711, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{e{x}^{2}+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{d + e x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]