Integrand size = 23, antiderivative size = 362 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=-\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {b c^2 \left (2 c^2 d-5 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
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Time = 0.33 (sec) , antiderivative size = 362, 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 = {277, 270, 5347, 12, 594, 597, 538, 438, 437, 435, 432, 430} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}-\frac {2 b x \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}} \]
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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 {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{3 d^2 x^4 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {d \left (2 c^2 d-5 e\right )+\left (c^2 d-6 e\right ) e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {d \left (c^2 d-6 e\right ) e-c^2 d \left (2 c^2 d-5 e\right ) e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^3 \left (2 c^2 d-5 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}}-\frac {\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^3 \left (2 c^2 d-5 e\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+e\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}{9 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^3 \left (2 c^2 d-5 e\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}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+e\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}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {b c^2 \left (2 c^2 d-5 e\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 )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\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 )}{9 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 = 6.53 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.69 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=-\frac {\sqrt {d+e x^2} \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+2 c^2 d x^2-5 e x^2\right )+3 a \left (d-2 e x^2\right )+3 b \left (d-2 e x^2\right ) \csc ^{-1}(c x)\right )}{9 d^2 x^3}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (2 c^2 d-5 e\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+2 \left (-c^4 d^2+2 c^2 d e+3 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{9 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \,\operatorname {arccsc}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}d x\]
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none
Time = 0.12 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.55 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\frac {{\left (6 \, a c d e x^{2} - 3 \, a c d^{2} + 3 \, {\left (2 \, b c d e x^{2} - b c d^{2}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b c d^{2} + {\left (2 \, b c^{3} d^{2} - 5 \, b c d e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (2 \, b c^{6} d^{2} - 5 \, b c^{4} d e\right )} x^{3} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (2 \, b c^{6} d^{2} - {\left (5 \, b c^{4} - b c^{2}\right )} d e - 6 \, b e^{2}\right )} x^{3} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{9 \, c d^{3} x^{3}} \]
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\[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \]
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