Optimal. Leaf size=628 \[ \frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {b d \text {ArcTan}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}+\frac {i b d \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {i b d \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{e^3} \]
[Out]
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Rubi [A]
time = 1.14, antiderivative size = 628, normalized size of antiderivative = 1.00, number of
steps used = 31, number of rules used = 14, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used
= {5349, 4817, 4723, 270, 4721, 3798, 2221, 2317, 2438, 4813, 385, 211, 4825, 4615}
\begin {gather*} -\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}+\frac {2 d \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {b d \text {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 \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-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 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 270
Rule 385
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4615
Rule 4721
Rule 4723
Rule 4813
Rule 4817
Rule 4825
Rule 5349
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\text {Subst}\left (\int \frac {a+b \sin ^{-1}\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 \sin ^{-1}\left (\frac {x}{c}\right )}{e^2 x^3}-\frac {2 d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{e^3 x}+\frac {d^2 x \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )^2}+\frac {2 d^2 x \left (a+b \sin ^{-1}\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 \sin ^{-1}\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 \sin ^{-1}\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 \sin ^{-1}\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 \sin ^{-1}\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 \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}+\frac {(2 d) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{e^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \sin ^{-1}\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 \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {i d \left (a+b \csc ^{-1}(c x)\right )^2}{b e^3}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {a+b \sin ^{-1}\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 \sin ^{-1}\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,\csc ^{-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 \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {i d \left (a+b \csc ^{-1}(c x)\right )^2}{b e^3}-\frac {b d \tan ^{-1}\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 \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{e^3}+\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{e^3}-\frac {(2 b d) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-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 \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {b d \tan ^{-1}\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 \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}+\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{e^3}+\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{e^3}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{e^3}-\frac {(-d)^{3/2} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{e^3}+\frac {(i b d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-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 \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {b d \tan ^{-1}\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 \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}-\frac {i b d \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{e^3}+\frac {(b d) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{e^3}+\frac {(b d) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{e^3}+\frac {(b d) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{e^3}+\frac {(b d) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-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 \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {b d \tan ^{-1}\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 \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}-\frac {i b d \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{e^3}-\frac {(i b d) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{e^3}-\frac {(i b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{e^3}-\frac {(i b d) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{e^3}-\frac {(i b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-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 \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {b d \tan ^{-1}\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 \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}+\frac {i b d \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {i b d \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{e^3}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1480\) vs. \(2(628)=1256\).
time = 4.09, size = 1480, normalized size = 2.36 \begin {gather*} -\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 (i d \pi ^2-\frac {2 e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}-4 i d \pi \csc ^{-1}(c x)-2 e x^2 \csc ^{-1}(c x)+\frac {d^{3/2} \csc ^{-1}(c x)}{\sqrt {d}-i \sqrt {e} x}+\frac {d^{3/2} \csc ^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}+8 i d \csc ^{-1}(c x)^2-2 d \text {ArcSin}\left (\frac {1}{c x}\right )-16 i d \text {ArcSin}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )-16 i d \text {ArcSin}\left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )-2 d \pi \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-8 d \text {ArcSin}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 d \pi \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \csc ^{-1}(c x) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-8 d \text {ArcSin}\left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 d \pi \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \csc ^{-1}(c x) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+8 d \text {ArcSin}\left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 d \pi \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 d \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+8 d \text {ArcSin}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-8 d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+2 d \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )+2 d \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{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 \text {PolyLog}\left (2,\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 i d \text {PolyLog}\left (2,\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 i d \text {PolyLog}\left (2,-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 i d \text {PolyLog}\left (2,\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 i d \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )}{4 e^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 2.61, size = 764, normalized size = 1.22
method | result | size |
derivativedivides | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{6} d \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{e^{3}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{8} d \,\mathrm {arccsc}\left (c x \right ) x^{2}}{e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{8} \mathrm {arccsc}\left (c x \right ) x^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d x}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {i b \,c^{6} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {i b \,c^{6} x^{2}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {i b \,c^{6} \sqrt {e \left (c^{2} d +e \right )}\, \arctanh \left (\frac {2 d \,c^{2} \left (\frac {i}{c x}+\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}{2 e^{3} \left (c^{2} d +e \right )}+\frac {i b \,c^{6} d \left (\munderset {\textit {\_R1} =\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 \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {i}{c x}-\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 b \,c^{6} d \dilog \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {2 b \,c^{6} d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}-\frac {2 i b \,c^{6} d \dilog \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i b \,c^{8} d^{2} \left (\munderset {\textit {\_R1} =\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 \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {i}{c x}-\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}}}{c^{6}}\) | \(764\) |
default | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{6} d \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{e^{3}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{8} d \,\mathrm {arccsc}\left (c x \right ) x^{2}}{e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{8} \mathrm {arccsc}\left (c x \right ) x^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d x}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {i b \,c^{6} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {i b \,c^{6} x^{2}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {i b \,c^{6} \sqrt {e \left (c^{2} d +e \right )}\, \arctanh \left (\frac {2 d \,c^{2} \left (\frac {i}{c x}+\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}{2 e^{3} \left (c^{2} d +e \right )}+\frac {i b \,c^{6} d \left (\munderset {\textit {\_R1} =\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 \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {i}{c x}-\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 b \,c^{6} d \dilog \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {2 b \,c^{6} d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}-\frac {2 i b \,c^{6} d \dilog \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i b \,c^{8} d^{2} \left (\munderset {\textit {\_R1} =\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 \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {i}{c x}-\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}}}{c^{6}}\) | \(764\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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