Integrand size = 23, antiderivative size = 252 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {b \left (9 c^2 d-e\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {c^2 x^2}} \]
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Time = 0.78 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {272, 45, 5347, 12, 1629, 163, 65, 223, 212, 95, 210} \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {b x \left (9 c^2 d-e\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 163
Rule 210
Rule 212
Rule 223
Rule 272
Rule 1629
Rule 5347
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 e^3 \sqrt {c^2 x^2}} \\ & = -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b c x) \text {Subst}\left (\int \frac {-8 d^2-4 d e x+e^2 x^2}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e^3 \sqrt {c^2 x^2}} \\ & = \frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {(b x) \text {Subst}\left (\int \frac {-8 c^2 d^2 e-\frac {1}{2} \left (9 c^2 d-e\right ) e^2 x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 c e^4 \sqrt {c^2 x^2}} \\ & = \frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {\left (4 b c d^2 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {\left (b \left (9 c^2 d-e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{12 c e^2 \sqrt {c^2 x^2}} \\ & = \frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {\left (8 b c d^2 x\right ) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {\left (b \left (9 c^2 d-e\right ) x\right ) \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 )}{6 c^3 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {\left (b \left (9 c^2 d-e\right ) x\right ) \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 )}{6 c^3 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {b \left (9 c^2 d-e\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {c^2 x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.53 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {16 b d^2 \sqrt {1+\frac {d}{e x^2}} \left (-1+c^2 x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+b e \left (-9 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x^4 \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )+2 x \left (-1+c^2 x^2\right ) \left (b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-2 a c \left (8 d^2+4 d e x^2-e^2 x^4\right )-2 b c \left (8 d^2+4 d e x^2-e^2 x^4\right ) \csc ^{-1}(c x)\right )}{12 c e^3 x \left (-1+c^2 x^2\right ) \sqrt {d+e x^2}} \]
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\[\int \frac {x^{5} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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Time = 0.52 (sec) , antiderivative size = 1480, normalized size of antiderivative = 5.87 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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