Integrand size = 33, antiderivative size = 150 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=-\frac {2 (A b-a B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 B \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b d}-\frac {2 a (A b-a B) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{b^2 (a+b) d}+\frac {2 B \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}} \]
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Time = 0.91 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3033, 3079, 3134, 3138, 2719, 3081, 2720, 2884} \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=-\frac {2 (A b-a B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac {2 a (A b-a B) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{b^2 d (a+b)}+\frac {2 (A b-a B) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {2 B \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b d}+\frac {2 B \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 2884
Rule 3033
Rule 3079
Rule 3081
Rule 3134
Rule 3138
Rubi steps \begin{align*} \text {integral}& = \int \frac {B+A \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (b+a \cos (c+d x))} \, dx \\ & = \frac {2 B \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \int \frac {\frac {3}{2} (A b-a B)+\frac {1}{2} b B \cos (c+d x)+\frac {1}{2} a B \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{3 b} \\ & = \frac {2 B \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {4 \int \frac {\frac {1}{4} \left (-3 a A b+3 a^2 B+b^2 B\right )-\frac {1}{4} b (3 A b-4 a B) \cos (c+d x)-\frac {3}{4} a (A b-a B) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 b^2} \\ & = \frac {2 B \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}-\frac {4 \int \frac {\frac {1}{4} a \left (3 a A b-3 a^2 B-b^2 B\right )-\frac {1}{4} a^2 b B \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 a b^2}-\frac {(A b-a B) \int \sqrt {\cos (c+d x)} \, dx}{b^2} \\ & = -\frac {2 (A b-a B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 B \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {B \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b}-\frac {(a (A b-a B)) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{b^2} \\ & = -\frac {2 (A b-a B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 B \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b d}-\frac {2 a (A b-a B) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{b^2 (a+b) d}+\frac {2 B \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 2.50 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.73 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\frac {\frac {2 b \left (-9 a A b+9 a^2 B+2 b^2 B\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {b \left (-6 A b^2+8 a b B\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{a}+\frac {4 b^2 B \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {12 b (A b-a B) \sin (c+d x)}{\sqrt {\cos (c+d x)}}+\frac {6 (-A b+a B) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a \sqrt {\sin ^2(c+d x)}}}{6 b^3 d} \]
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Time = 15.71 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.93
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (\frac {2 B \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{6 \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {1}{2}\right )^{2}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}\right )}{b}+\frac {2 \left (A b -B a \right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\right )}{b^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+\frac {2 \left (A b -B a \right ) a^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 a}{a -b}, \sqrt {2}\right )}{b^{2} \left (a^{2}-a b \right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(439\) |
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Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{5/2}\,\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )} \,d x \]
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