Integrand size = 23, antiderivative size = 112 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {2 b E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac {2 \left (a^2+3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^3 d}-\frac {2 b^3 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^3 (a+b) d}+\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d} \]
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Time = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4349, 3938, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {2 b^3 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^3 d (a+b)}-\frac {2 b E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac {2 \left (a^2+3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 a d} \]
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Rule 2719
Rule 2720
Rule 2884
Rule 3856
Rule 3872
Rule 3934
Rule 3938
Rule 4191
Rule 4349
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx \\ & = \frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3 b}{2}+\frac {1}{2} a \sec (c+d x)+\frac {1}{2} b \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 a} \\ & = \frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3 a b}{2}-\left (-\frac {a^2}{2}-\frac {3 b^2}{2}\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 a^3}-\frac {\left (b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{a^3} \\ & = \frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d}-\frac {b^3 \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a^3}-\frac {\left (b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{a^2}+\frac {\left (\left (a^2+3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{3 a^3} \\ & = -\frac {2 b^3 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^3 (a+b) d}+\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d}-\frac {b \int \sqrt {\cos (c+d x)} \, dx}{a^2}+\frac {\left (a^2+3 b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a^3} \\ & = -\frac {2 b E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac {2 \left (a^2+3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^3 d}-\frac {2 b^3 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^3 (a+b) d}+\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d} \\ \end{align*}
Time = 1.83 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {6 b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+4 \sqrt {\cos (c+d x)} \sin (c+d x)-\frac {6 \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a^2 \sqrt {\sin ^2(c+d x)}}}{6 a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(551\) vs. \(2(184)=368\).
Time = 7.42 (sec) , antiderivative size = 552, normalized size of antiderivative = 4.93
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (4 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b -2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+2 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a^{2} b +3 a \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, b^{3}+3 a^{2} b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a \,b^{2}+3 b^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 a}{a -b}, \sqrt {2}\right )\right )}{3 a^{3} \left (a -b \right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(552\) |
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Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\cos ^{\frac {3}{2}}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {3}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {3}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{3/2}}{a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]
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