Integrand size = 21, antiderivative size = 132 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b c \sqrt {d} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e \sqrt {c^2 x^2}}+\frac {b x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c^2 x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5345, 457, 132, 65, 223, 212, 12, 95, 210} \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b c \sqrt {d} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{e \sqrt {c^2 x^2}}+\frac {b x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c^2 x^2}} \]
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Rule 12
Rule 65
Rule 95
Rule 132
Rule 210
Rule 212
Rule 223
Rule 457
Rule 5345
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {(b c x) \int \frac {\sqrt {d+e x^2}}{x \sqrt {-1+c^2 x^2}} \, dx}{e \sqrt {c^2 x^2}} \\ & = \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {(b c x) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 e \sqrt {c^2 x^2}} \\ & = \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 \sqrt {c^2 x^2}}+\frac {(b c x) \text {Subst}\left (\int \frac {d}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e \sqrt {c^2 x^2}} \\ & = \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {(b x) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}+\frac {e x^2}{c^2}}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {c^2 x^2}}+\frac {(b c d x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e \sqrt {c^2 x^2}} \\ & = \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {(b x) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{c \sqrt {c^2 x^2}}+\frac {(b c d x) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{e \sqrt {c^2 x^2}} \\ & = \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b c \sqrt {d} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e \sqrt {c^2 x^2}}+\frac {b x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c^2 x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.81 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},1,\frac {3}{2},\frac {e-c^2 e x^2}{c^2 d+e},1-c^2 x^2\right )}{\sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}}}+b \csc ^{-1}(c x)\right )}{e} \]
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\[\int \frac {x \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
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none
Time = 0.36 (sec) , antiderivative size = 870, normalized size of antiderivative = 6.59 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\left [\frac {b c \sqrt {-d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, \sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {-d} + 8 \, d^{2}}{x^{4}}\right ) + b \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} + 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {e} + e^{2}\right ) + 4 \, \sqrt {e x^{2} + d} {\left (b c \operatorname {arccsc}\left (c x\right ) + a c\right )}}{4 \, c e}, -\frac {2 \, b c \sqrt {d} \arctan \left (-\frac {\sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {d}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - b \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} + 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {e} + e^{2}\right ) - 4 \, \sqrt {e x^{2} + d} {\left (b c \operatorname {arccsc}\left (c x\right ) + a c\right )}}{4 \, c e}, \frac {b c \sqrt {-d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, \sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {-d} + 8 \, d^{2}}{x^{4}}\right ) - 2 \, b \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {-e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) + 4 \, \sqrt {e x^{2} + d} {\left (b c \operatorname {arccsc}\left (c x\right ) + a c\right )}}{4 \, c e}, -\frac {b c \sqrt {d} \arctan \left (-\frac {\sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {d}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) + b \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {-e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) - 2 \, \sqrt {e x^{2} + d} {\left (b c \operatorname {arccsc}\left (c x\right ) + a c\right )}}{2 \, c e}\right ] \]
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x^{2} + d}} \,d x } \]
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x^{2} + d}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
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