Integrand size = 21, antiderivative size = 608 \[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}+\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {i b d \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{e^3} \]
[Out]
Time = 0.91 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5348, 4818, 4724, 270, 4722, 3800, 2221, 2317, 2438, 4814, 385, 211, 4826, 4616} \[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}-\frac {d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}+\frac {2 d \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{e^3}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}+\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{e^3}-\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c e^2}-\frac {i b d \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{e^3} \]
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Rule 211
Rule 270
Rule 385
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4616
Rule 4722
Rule 4724
Rule 4814
Rule 4818
Rule 4826
Rule 5348
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \arccos \left (\frac {x}{c}\right )}{x^3 \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {a+b \arccos \left (\frac {x}{c}\right )}{e^2 x^3}-\frac {2 d \left (a+b \arccos \left (\frac {x}{c}\right )\right )}{e^3 x}+\frac {d^2 x \left (a+b \arccos \left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )^2}+\frac {2 d^2 x \left (a+b \arccos \left (\frac {x}{c}\right )\right )}{e^3 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {(2 d) \text {Subst}\left (\int \frac {a+b \arccos \left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \arccos \left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\text {Subst}\left (\int \frac {a+b \arccos \left (\frac {x}{c}\right )}{x^3} \, dx,x,\frac {1}{x}\right )}{e^2}-\frac {d^2 \text {Subst}\left (\int \frac {x \left (a+b \arccos \left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{e^2} \\ & = \frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}-\frac {(2 d) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{e^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \arccos \left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \arccos \left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {b \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e^2}+\frac {(b d) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c e^2} \\ & = -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}-\frac {i d \left (a+b \sec ^{-1}(c x)\right )^2}{b e^3}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {a+b \arccos \left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {a+b \arccos \left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {(4 i d) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(c x)\right )}{e^3}+\frac {(b d) \text {Subst}\left (\int \frac {1}{e-\left (-d-\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 c e^2} \\ & = -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}-\frac {i d \left (a+b \sec ^{-1}(c x)\right )^2}{b e^3}+\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}+\frac {2 d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^3}+\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {(a+b x) \sin (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{e^3}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {(a+b x) \sin (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{e^3}-\frac {(2 b d) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{e^3} \\ & = -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}+\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}+\frac {2 d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^3}-\frac {\left (i (-d)^{3/2}\right ) \text {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 )}{e^3}-\frac {\left (i (-d)^{3/2}\right ) \text {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 )}{e^3}+\frac {\left (i (-d)^{3/2}\right ) \text {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 )}{e^3}+\frac {\left (i (-d)^{3/2}\right ) \text {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 )}{e^3}+\frac {(i b d) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sec ^{-1}(c x)}\right )}{e^3} \\ & = -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}+\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}-\frac {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 )}{e^3}-\frac {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 )}{e^3}-\frac {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 )}{e^3}-\frac {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 )}{e^3}+\frac {2 d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^3}-\frac {i b d \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{e^3}+\frac {(b d) \text {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 )}{e^3}+\frac {(b d) \text {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 )}{e^3}+\frac {(b d) \text {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 )}{e^3}+\frac {(b d) \text {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 )}{e^3} \\ & = -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}+\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}-\frac {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 )}{e^3}-\frac {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 )}{e^3}-\frac {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 )}{e^3}-\frac {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 )}{e^3}+\frac {2 d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^3}-\frac {i b d \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{e^3}-\frac {(i b d) \text {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 )}{e^3}-\frac {(i b d) \text {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 )}{e^3}-\frac {(i b d) \text {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 )}{e^3}-\frac {(i b d) \text {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 )}{e^3} \\ & = -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e^2}+\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}-\frac {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 )}{e^3}-\frac {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 )}{e^3}-\frac {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 )}{e^3}-\frac {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 )}{e^3}+\frac {2 d \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {i b d \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{e^3} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1255\) vs. \(2(608)=1216\).
Time = 3.18 (sec) , antiderivative size = 1255, normalized size of antiderivative = 2.06 \[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {-2 a e x^2+\frac {2 a d^2}{d+e x^2}+4 a d \log \left (d+e x^2\right )+b \left (\frac {2 e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}-2 e x^2 \sec ^{-1}(c x)+\frac {d^{3/2} \sec ^{-1}(c x)}{\sqrt {d}-i \sqrt {e} x}+\frac {d^{3/2} \sec ^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}+2 d \arcsin \left (\frac {1}{c x}\right )+16 i d \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \tan \left (\frac {1}{2} \sec ^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )+16 i d \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \tan \left (\frac {1}{2} \sec ^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )+4 d \sec ^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+8 d \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \sec ^{-1}(c x) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+8 d \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \sec ^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-8 d \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \sec ^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-8 d \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-8 d \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-\frac {d \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (\sqrt {e}+c \left (i c \sqrt {d}-\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}-\frac {d \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (-\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}-4 i d \operatorname {PolyLog}\left (2,-\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-4 i d \operatorname {PolyLog}\left (2,\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-4 i d \operatorname {PolyLog}\left (2,-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-4 i d \operatorname {PolyLog}\left (2,\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+4 i d \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )\right )}{4 e^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 9.59 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.15
method | result | size |
parts | \(\frac {a \,x^{2}}{2 e^{2}}-\frac {a \,d^{2}}{2 e^{3} \left (e \,x^{2}+d \right )}-\frac {a d \ln \left (e \,x^{2}+d \right )}{e^{3}}+\frac {b \left (\frac {c^{4} \left (2 c^{4} d \,\operatorname {arcsec}\left (c x \right ) x^{2}+\operatorname {arcsec}\left (c x \right ) e \,c^{4} x^{4}-\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x -\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-i c^{2} d -i e \,c^{2} x^{2}\right )}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) e^{2}}-\frac {i \sqrt {e \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 c^{2} d {\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}+2 c^{2} d +4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right ) d \,c^{6}}{2 \left (c^{2} d +e \right ) e^{3}}-\frac {2 i d \,c^{6} \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}+\frac {i d^{2} c^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}+\frac {i d \,c^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}-\frac {2 i d \,c^{6} \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}+\frac {2 d \,c^{6} \operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}+\frac {2 d \,c^{6} \operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}\right )}{c^{6}}\) | \(701\) |
derivativedivides | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a \,c^{6} d \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{e^{3}}+b \,c^{4} \left (\frac {2 c^{4} d \,\operatorname {arcsec}\left (c x \right ) x^{2}+\operatorname {arcsec}\left (c x \right ) e \,c^{4} x^{4}-\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x -\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-i c^{2} d -i e \,c^{2} x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) e^{2}}+\frac {i c^{4} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}+\frac {i c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}-\frac {2 i c^{2} d \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}-\frac {2 i c^{2} d \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}-\frac {i \sqrt {e \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 c^{2} d {\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}+2 c^{2} d +4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right ) d \,c^{2}}{2 e^{3} \left (c^{2} d +e \right )}+\frac {2 c^{2} d \,\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \,\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}\right )}{c^{6}}\) | \(725\) |
default | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a \,c^{6} d \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{e^{3}}+b \,c^{4} \left (\frac {2 c^{4} d \,\operatorname {arcsec}\left (c x \right ) x^{2}+\operatorname {arcsec}\left (c x \right ) e \,c^{4} x^{4}-\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x -\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-i c^{2} d -i e \,c^{2} x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) e^{2}}+\frac {i c^{4} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}+\frac {i c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}-\frac {2 i c^{2} d \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}-\frac {2 i c^{2} d \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}-\frac {i \sqrt {e \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 c^{2} d {\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}+2 c^{2} d +4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right ) d \,c^{2}}{2 e^{3} \left (c^{2} d +e \right )}+\frac {2 c^{2} d \,\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \,\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{3}}\right )}{c^{6}}\) | \(725\) |
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\[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^5 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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