Optimal. Leaf size=795 \[ -\frac{3 \sqrt{e} \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{d c^2+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{d c^2+e}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}} \]
[Out]
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Rubi [A] time = 1.99793, antiderivative size = 795, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 14, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4733, 4627, 266, 63, 208, 4667, 4743, 725, 206, 4741, 4521, 2190, 2279, 2391} \[ -\frac{3 \sqrt{e} \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{d c^2+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{d c^2+e}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4733
Rule 4627
Rule 266
Rule 63
Rule 208
Rule 4667
Rule 4743
Rule 725
Rule 206
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 )^2} \, dx &=\int \left (\frac{a+b \sin ^{-1}(c x)}{d^2 x^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d \left (d+e x^2\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \sin ^{-1}(c x)}{x^2} \, dx}{d^2}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{d+e x^2} \, dx}{d^2}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{(b c) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{d^2}-\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^2}-\frac{e \int \left (-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d^2}+\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 (-d)^{5/2}}+\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 (-d)^{5/2}}+\frac{e^2 \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d^2}+\frac{e^2 \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d^2}+\frac{e^2 \int \frac{a+b \sin ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d^2}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\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^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)^{5/2}}+\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)^{5/2}}-\frac{(b c e) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 d^2}+\frac{(b c e) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 d^2}+\frac{e^2 \int \left (-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d^2}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^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)^{5/2}}+\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)^{5/2}}+\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)^{5/2}}+\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)^{5/2}}+\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{5/2}}+\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{5/2}}+\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{-e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 d^2}-\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 d^2}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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 )}{4 (-d)^{5/2}}+\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 )}{4 (-d)^{5/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}+\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 )}{4 (-d)^{5/2}}+\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 )}{4 (-d)^{5/2}}+\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 )}{4 (-d)^{5/2}}+\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 )}{4 (-d)^{5/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{3 \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 )}{4 (-d)^{5/2}}+\frac{3 \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 )}{4 (-d)^{5/2}}-\frac{3 \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 )}{4 (-d)^{5/2}}+\frac{3 \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 )}{4 (-d)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}+\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 )}{4 (-d)^{5/2}}-\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 )}{4 (-d)^{5/2}}+\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 )}{4 (-d)^{5/2}}-\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 )}{4 (-d)^{5/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{3 \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 )}{4 (-d)^{5/2}}+\frac{3 \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 )}{4 (-d)^{5/2}}-\frac{3 \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 )}{4 (-d)^{5/2}}+\frac{3 \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 )}{4 (-d)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}-\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 )}{4 (-d)^{5/2}}+\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 )}{4 (-d)^{5/2}}-\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 )}{4 (-d)^{5/2}}+\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 )}{4 (-d)^{5/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d^2 \sqrt{c^2 d+e}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{3 \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 )}{4 (-d)^{5/2}}+\frac{3 \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 )}{4 (-d)^{5/2}}-\frac{3 \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 )}{4 (-d)^{5/2}}+\frac{3 \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 )}{4 (-d)^{5/2}}-\frac{3 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 )}{4 (-d)^{5/2}}+\frac{3 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 )}{4 (-d)^{5/2}}-\frac{3 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 )}{4 (-d)^{5/2}}+\frac{3 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 )}{4 (-d)^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.47593, size = 672, normalized size = 0.85 \[ \frac{b \left (3 \sqrt{e} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )-3 \sqrt{e} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )-2 i \sqrt{d} \sqrt{e} \left (\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left (\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}\right )+2 \sqrt{d} \sqrt{e} \left (-\frac{c \tanh ^{-1}\left (\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right )-\frac{8 \sqrt{d} \left (c x \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\sin ^{-1}(c x)\right )}{x}\right )-\frac{4 a \sqrt{d} e x}{d+e x^2}-12 a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )-\frac{8 a \sqrt{d}}{x}}{8 d^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.112, size = 1839, normalized size = 2.3 \begin{align*} \text{result 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{b \arcsin \left (c x\right ) + a}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, 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(-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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