Optimal. Leaf size=784 \[ \frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2} \]
[Out]
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Rubi [A] time = 2.35353, antiderivative size = 784, normalized size of antiderivative = 1., number of steps used = 51, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {5240, 4734, 4628, 266, 63, 208, 4668, 4744, 725, 206, 4742, 4520, 2190, 2279, 2391} \[ \frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 e^{5/2}}-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5240
Rule 4734
Rule 4628
Rule 266
Rule 63
Rule 208
Rule 4668
Rule 4744
Rule 725
Rule 206
Rule 4742
Rule 4520
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{x^2 \left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{e^2 x^2}-\frac{d \left (a+b \cos ^{-1}\left (\frac{x}{c}\right )\right )}{e \left (e+d x^2\right )^2}-\frac{d \left (a+b \cos ^{-1}\left (\frac{x}{c}\right )\right )}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{x^2} \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{e}\\ &=\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c e^2}+\frac{d \operatorname{Subst}\left (\int \left (\frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \cos ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d \left (a+b \cos ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{d \left (a+b \cos ^{-1}\left (\frac{x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )}{e}\\ &=\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 e^{5/2}}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c e^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 e^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 e^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac{1}{x}\right )}{2 e^2}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{d \operatorname{Subst}\left (\int \frac{(a+b x) \sin (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}-\frac{d \operatorname{Subst}\left (\int \frac{(a+b x) \sin (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{c^2-c^2 x^2} \, dx,x,\sqrt{1-\frac{1}{c^2 x^2}}\right )}{e^2}-\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}-d x\right ) \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c e^2}+\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}+d x\right ) \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c e^2}-\frac{d^2 \operatorname{Subst}\left (\int \left (-\frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}-\frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{2 e^2}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 e^{5/2}}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \cos ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 e^{5/2}}+\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{d^2+\frac{d e}{c^2}-x^2} \, dx,x,\frac{-d+\frac{\sqrt{-d} \sqrt{e}}{c^2 x}}{\sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 c e^2}-\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{d^2+\frac{d e}{c^2}-x^2} \, dx,x,\frac{d+\frac{\sqrt{-d} \sqrt{e}}{c^2 x}}{\sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 c e^2}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^{5/2}}-\frac{d \operatorname{Subst}\left (\int \frac{(a+b x) \sin (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}-\frac{d \operatorname{Subst}\left (\int \frac{(a+b x) \sin (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^{5/2}}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}+\frac{(i d) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{4 e^{5/2}}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{4 e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{4 e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{4 e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{4 e^{5/2}}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{4 e^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e^2}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 e^2 \sqrt{c^2 d+e}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}\\ \end{align*}
Mathematica [A] time = 2.25693, size = 1331, normalized size = 1.7 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 9.726, size = 1887, normalized size = 2.4 \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 x^{4} \operatorname{arcsec}\left (c x\right ) + a x^{4}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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