Integrand size = 23, antiderivative size = 268 \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {3 b \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (4 a^2+3 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {3 b \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{4 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 a d} \]
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Time = 0.90 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2881, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {\left (4 a^2+3 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {3 b \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{4 a^2 d}+\frac {3 b \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2881
Rule 2884
Rule 2886
Rule 3081
Rule 3134
Rule 3138
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {\int \frac {\left (-\frac {3 b}{2}+a \cos (c+d x)+\frac {1}{2} b \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 a} \\ & = -\frac {3 b \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{4 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {\int \frac {\left (\frac {1}{4} \left (4 a^2+3 b^2\right )+\frac {1}{2} a b \cos (c+d x)+\frac {3}{4} b^2 \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {3 b \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{4 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {\int \frac {\left (-\frac {1}{4} b \left (4 a^2+3 b^2\right )+\frac {1}{4} a b^2 \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 a^2 b}+\frac {(3 b) \int \sqrt {a+b \cos (c+d x)} \, dx}{8 a^2} \\ & = -\frac {3 b \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{4 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {b \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{8 a}+\frac {1}{8} \left (4+\frac {3 b^2}{a^2}\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx+\frac {\left (3 b \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{8 a^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \\ & = \frac {3 b \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {3 b \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{4 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {\left (b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{8 a \sqrt {a+b \cos (c+d x)}}+\frac {\left (\left (4+\frac {3 b^2}{a^2}\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{8 \sqrt {a+b \cos (c+d x)}} \\ & = \frac {3 b \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (4+\frac {3 b^2}{a^2}\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 d \sqrt {a+b \cos (c+d x)}}-\frac {3 b \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{4 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 a d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.37 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.93 \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {\frac {8 a b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2+9 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {6 i b^2 \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{16 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (-\frac {3 b \tan (c+d x)}{4 a^2}+\frac {\sec (c+d x) \tan (c+d x)}{2 a}\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(709\) vs. \(2(329)=658\).
Time = 3.67 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.65
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}^{2}}+\frac {3 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{2 a^{2} \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}-\frac {b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )}{4 a \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, b E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )}{4 a \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {3 b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )}{4 a^{2} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, \Pi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \sqrt {-\frac {2 b}{a -b}}\right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, \Pi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \sqrt {-\frac {2 b}{a -b}}\right ) b^{2}}{4 a^{2} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}\, d}\) | \(710\) |
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Timed out. \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sqrt {a + b \cos {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
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