Integrand size = 23, antiderivative size = 453 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
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Time = 0.41 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {277, 270, 5347, 12, 594, 597, 538, 438, 437, 435, 432, 430} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}-\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 270
Rule 277
Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 538
Rule 594
Rule 597
Rule 5347
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{15 d^2 x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{15 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (d \left (12 c^2 d-e\right )+\left (3 c^2 d-10 e\right ) e x^2\right )}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {-d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right )-2 e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right ) x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {-2 d e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right )+c^2 d e \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2}}-\frac {\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.69 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {\sqrt {d+e x^2} \left (15 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (-31 e^2 x^4+d e x^2 \left (8+19 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )+15 b \left (3 d^2+d e x^2-2 e^2 x^4\right ) \csc ^{-1}(c x)\right )}{225 d^2 x^5}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+\left (-24 c^6 d^3-31 c^4 d^2 e+23 c^2 d e^2+30 e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{225 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
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\[\int \frac {\left (a +b \,\operatorname {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{6}}d x\]
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none
Time = 0.11 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\frac {{\left (30 \, a c d e^{2} x^{4} - 15 \, a c d^{2} e x^{2} - 45 \, a c d^{3} + 15 \, {\left (2 \, b c d e^{2} x^{4} - b c d^{2} e x^{2} - 3 \, b c d^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (9 \, b c d^{3} + {\left (24 \, b c^{5} d^{3} + 19 \, b c^{3} d^{2} e - 31 \, b c d e^{2}\right )} x^{4} + 4 \, {\left (3 \, b c^{3} d^{3} + 2 \, b c d^{2} e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (24 \, b c^{8} d^{3} + 19 \, b c^{6} d^{2} e - 31 \, b c^{4} d e^{2}\right )} x^{5} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (24 \, b c^{8} d^{3} + {\left (19 \, b c^{6} + 12 \, b c^{4}\right )} d^{2} e - {\left (31 \, b c^{4} - 8 \, b c^{2}\right )} d e^{2} - 30 \, b e^{3}\right )} x^{5} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{225 \, c d^{3} x^{5}} \]
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\[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]
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