Optimal. Leaf size=803 \[ -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {c^2 d+e}}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {c^2 d+e}}-\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.27, antiderivative size = 803, normalized size of antiderivative = 1.00, 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 =
{5349, 4817, 4723, 272, 65, 214, 4757, 4827, 739, 212, 4825, 4615, 2221, 2317, 2438}
\begin {gather*} \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\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 {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}-\frac {3 i b \sqrt {-d} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 212
Rule 214
Rule 272
Rule 739
Rule 2221
Rule 2317
Rule 2438
Rule 4615
Rule 4723
Rule 4757
Rule 4817
Rule 4825
Rule 4827
Rule 5349
Rubi steps
\begin {align*} \int \frac {x^4 \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^2 \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^2}-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )^2}-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{x^2} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \text {Subst}\left (\int \frac {a+b \sin ^{-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 \csc ^{-1}(c x)\right )}{e^2}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c e^2}+\frac {d \text {Subst}\left (\int \left (\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \sin ^{-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 \text {Subst}\left (\int \left (-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \sin ^{-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 \csc ^{-1}(c x)\right )}{e^2}+\frac {d \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 )}{2 e^{5/2}}+\frac {d \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 )}{2 e^{5/2}}-\frac {b \text {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 \text {Subst}\left (\int \frac {a+b \sin ^{-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 \text {Subst}\left (\int \frac {a+b \sin ^{-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 \text {Subst}\left (\int \frac {a+b \sin ^{-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 \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {d \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 )}{2 e^{5/2}}+\frac {d \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 )}{2 e^{5/2}}+\frac {(b c) \text {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) \text {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) \text {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 \text {Subst}\left (\int \left (-\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \sin ^{-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 \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {d \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 )}{4 e^{5/2}}+\frac {d \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 )}{4 e^{5/2}}+\frac {d \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 )}{2 e^{5/2}}+\frac {d \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 )}{2 e^{5/2}}+\frac {d \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 )}{2 e^{5/2}}+\frac {d \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 )}{2 e^{5/2}}-\frac {(b d) \text {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) \text {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 \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-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 \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 )}{2 e^{5/2}}+\frac {\sqrt {-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 )}{2 e^{5/2}}-\frac {\sqrt {-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 )}{2 e^{5/2}}+\frac {\sqrt {-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 )}{2 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}+\frac {d \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 )}{4 e^{5/2}}+\frac {d \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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-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 \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 )}{2 e^{5/2}}+\frac {\sqrt {-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 )}{2 e^{5/2}}-\frac {\sqrt {-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 )}{2 e^{5/2}}+\frac {\sqrt {-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 )}{2 e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 )}{2 e^{5/2}}+\frac {d \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 )}{4 e^{5/2}}+\frac {d \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 )}{4 e^{5/2}}+\frac {d \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 )}{4 e^{5/2}}+\frac {d \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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-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 \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 )}{4 e^{5/2}}+\frac {3 \sqrt {-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 )}{4 e^{5/2}}-\frac {3 \sqrt {-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 )}{4 e^{5/2}}+\frac {3 \sqrt {-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 )}{4 e^{5/2}}-\frac {i b \sqrt {-d} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-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 \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 )}{4 e^{5/2}}+\frac {3 \sqrt {-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 )}{4 e^{5/2}}-\frac {3 \sqrt {-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 )}{4 e^{5/2}}+\frac {3 \sqrt {-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 )}{4 e^{5/2}}-\frac {i b \sqrt {-d} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-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 \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 )}{4 e^{5/2}}+\frac {3 \sqrt {-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 )}{4 e^{5/2}}-\frac {3 \sqrt {-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 )}{4 e^{5/2}}+\frac {3 \sqrt {-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 )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-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 {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}\\ \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. \(1634\) vs. \(2(803)=1606\).
time = 6.04, size = 1634, normalized size = 2.03 \begin {gather*} \frac {a x}{e^2}+\frac {a d x}{2 e^2 \left (d+e x^2\right )}-\frac {3 a \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+b \left (-\frac {d \left (-\frac {\csc ^{-1}(c x)}{-i \sqrt {d} \sqrt {e}+e x}+\frac {i \left (\frac {\text {ArcSin}\left (\frac {1}{c x}\right )}{\sqrt {e}}-\frac {\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}}\right )}{\sqrt {d}}\right )}{4 e^2}-\frac {d \left (-\frac {\csc ^{-1}(c x)}{i \sqrt {d} \sqrt {e}+e x}-\frac {i \left (\frac {\text {ArcSin}\left (\frac {1}{c x}\right )}{\sqrt {e}}-\frac {\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}}\right )}{\sqrt {d}}\right )}{4 e^2}+\frac {3 \sqrt {d} \left (\pi ^2-4 \pi \csc ^{-1}(c x)+8 \csc ^{-1}(c x)^2-32 \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 )+4 i \pi \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-8 i \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 )+16 i \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 )+4 i \pi \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-8 i \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 )-16 i \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 i \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-4 i \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )+8 \text {PolyLog}\left (2,\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+8 \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 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )}{32 e^{5/2}}-\frac {3 \sqrt {d} \left (\pi ^2-4 \pi \csc ^{-1}(c x)+8 \csc ^{-1}(c x)^2-32 \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 )+4 i \pi \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-8 i \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 )+16 i \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 )+4 i \pi \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-8 i \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 )-16 i \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 i \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-4 i \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )+8 \text {PolyLog}\left (2,\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+8 \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 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )}{32 e^{5/2}}+\frac {\frac {1}{2} \csc ^{-1}(c x) \cot \left (\frac {1}{2} \csc ^{-1}(c x)\right )+\log \left (\cos \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-\log \left (\sin \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )+\frac {1}{2} \csc ^{-1}(c x) \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )}{c e^2}\right ) \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 = 12.79, size = 1888, normalized size = 2.35 \[\text {Expression too large to display}\]
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^{4} \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^4\,\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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