Integrand size = 20, antiderivative size = 109 \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {197, 5336, 12, 432, 430} \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
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Rule 12
Rule 197
Rule 430
Rule 432
Rule 5336
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {1}{d \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d \sqrt {c^2 x^2}} \\ & = \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (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}{d \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (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}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {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 )}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ \end{align*}
Time = 1.71 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d \left (-c+c^3 x^2\right ) \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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none
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {{\left (b e x^{2} + b d\right )} \sqrt {-d} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) + {\left (b c d x \operatorname {arcsec}\left (c x\right ) + a c d x\right )} \sqrt {e x^{2} + d}}{c d^{2} e x^{2} + c d^{3}} \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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