Integrand size = 20, antiderivative size = 296 \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {2 b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {198, 197, 5337, 12, 541, 538, 438, 437, 435, 432, 430} \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}-\frac {b c e x^2 \sqrt {c^2 x^2-1}}{3 d^2 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
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Rule 12
Rule 197
Rule 198
Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 538
Rule 541
Rule 5337
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {3 d+2 e x^2}{3 d^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {3 d+2 e x^2}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {d \left (3 c^2 d+2 e\right )+c^2 d e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(2 b c x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}}+\frac {\left (b c^3 x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {\left (b c^3 x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (2 b c 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}{3 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {\left (b c^3 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}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (2 b c 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}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {b c^2 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 )}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {2 b 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 )}{3 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 = 0.42 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x \left (-b c e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )+a \left (c^2 d+e\right ) \left (3 d+2 e x^2\right )+b \left (c^2 d+e\right ) \left (3 d+2 e x^2\right ) \csc ^{-1}(c x)\right )}{3 d^2 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+2 \left (c^2 d+e\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{3 \sqrt {-c^2} d^2 \left (c^2 d+e\right ) \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \,\operatorname {arccsc}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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none
Time = 0.12 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left (2 \, {\left (a c^{3} d^{2} e + a c d e^{2}\right )} x^{3} + 3 \, {\left (a c^{3} d^{3} + a c d^{2} e\right )} x + {\left (2 \, {\left (b c^{3} d^{2} e + b c d e^{2}\right )} x^{3} + 3 \, {\left (b c^{3} d^{3} + b c d^{2} e\right )} x\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (b c^{4} d e^{2} x^{4} + 2 \, b c^{4} d^{2} e x^{2} + b c^{4} d^{3}\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left ({\left (b c^{4} - 3 \, b c^{2}\right )} d e^{2} - 2 \, b e^{3}\right )} x^{4} + {\left (b c^{4} - 3 \, b c^{2}\right )} d^{3} - 2 \, b d^{2} e + 2 \, {\left ({\left (b c^{4} - 3 \, b c^{2}\right )} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3 \, {\left (c^{3} d^{6} + c d^{5} e + {\left (c^{3} d^{4} e^{2} + c d^{3} e^{3}\right )} x^{4} + 2 \, {\left (c^{3} d^{5} e + c d^{4} e^{2}\right )} x^{2}\right )}} \]
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Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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