Integrand size = 23, antiderivative size = 294 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (c^2 d+9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {2 b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{15 e^2 \sqrt {c^2 x^2}}-\frac {b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt {c^2 x^2}} \]
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Time = 0.30 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {272, 45, 5347, 12, 587, 159, 163, 65, 223, 212, 95, 210} \[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {2 b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{15 e^2 \sqrt {c^2 x^2}}-\frac {b x \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{120 c^3 e \sqrt {c^2 x^2}} \]
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 159
Rule 163
Rule 210
Rule 212
Rule 223
Rule 272
Rule 587
Rule 5347
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{15 e^2 x \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{x \sqrt {-1+c^2 x^2}} \, dx}{15 e^2 \sqrt {c^2 x^2}} \\ & = -\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {(b c x) \text {Subst}\left (\int \frac {(d+e x)^{3/2} (-2 d+3 e x)}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{30 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {(b x) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (-4 c^2 d^2+\frac {1}{2} e \left (c^2 d+9 e\right ) x\right )}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{60 c e^2 \sqrt {c^2 x^2}} \\ & = \frac {b \left (c^2 d+9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {(b x) \text {Subst}\left (\int \frac {-4 c^4 d^3-\frac {1}{4} e \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{60 c^3 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b \left (c^2 d+9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {\left (b c d^3 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{15 e^2 \sqrt {c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{240 c^3 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (c^2 d+9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {\left (2 b c d^3 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 )}{15 e^2 \sqrt {c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\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 )}{120 c^5 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (c^2 d+9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {2 b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{15 e^2 \sqrt {c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\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 )}{120 c^5 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (c^2 d+9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {2 b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{15 e^2 \sqrt {c^2 x^2}}-\frac {b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt {c^2 x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.44 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.89 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {16 a \left (d+e x^2\right )^2 \left (-2 d+3 e x^2\right )+\frac {2 b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (9 e+c^2 \left (7 d+6 e x^2\right )\right )}{c^3}+\frac {b \left (16 c^2 d^3 \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+\frac {e \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )}{\sqrt {1-c^2 x^2}}\right )}{c^3 x}+16 b \left (d+e x^2\right )^2 \left (-2 d+3 e x^2\right ) \csc ^{-1}(c x)}{240 e^2 \sqrt {d+e x^2}} \]
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\[\int x^{3} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]
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Time = 1.05 (sec) , antiderivative size = 1379, normalized size of antiderivative = 4.69 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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\[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
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Exception generated. \[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^3\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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