Integrand size = 23, antiderivative size = 274 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {277, 197, 5346, 12, 597, 538, 438, 437, 435, 432, 430} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}+\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}} \]
[In]
[Out]
Rule 12
Rule 197
Rule 277
Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 538
Rule 597
Rule 5346
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d-2 e x^2}{d^2 x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d-2 e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {c^2 x^2}} \\ & = \frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 d e+c^2 d e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^3 \sqrt {c^2 x^2}} \\ & = \frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {\left (b c^3 x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{d^2 \sqrt {c^2 x^2}}+\frac {\left (b c \left (c^2 d+2 e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {c^2 x^2}} \\ & = \frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{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}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (b c \left (c^2 d+2 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}{d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{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}{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+2 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}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \sec ^{-1}(c x)\right )}{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 )}{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+2 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 )}{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 = 4.82 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.77 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-a \left (d+2 e x^2\right )-b \left (d+2 e x^2\right ) \sec ^{-1}(c x)}{d^2 x \sqrt {d+e x^2}}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (c^2 d+2 e\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{\sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
[In]
[Out]
\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
[In]
[Out]
none
Time = 0.12 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.69 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, a c d e x^{2} + a c d^{2} + {\left (2 \, b c d e x^{2} + b c d^{2}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (b c d e x^{2} + b c d^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (b c^{4} d e x^{3} + b c^{4} d^{2} x\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left (b c^{4} d e + 2 \, b e^{2}\right )} x^{3} + {\left (b c^{4} d^{2} + 2 \, b d e\right )} x\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{c d^{3} e x^{3} + c d^{4} x} \]
[In]
[Out]
\[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
[In]
[Out]