Integrand size = 39, antiderivative size = 185 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=-\frac {2 (3 A+5 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {2 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac {2 (3 A+5 C) \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}} \]
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Time = 0.38 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {16, 3100, 2827, 2716, 2721, 2720, 2719} \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=-\frac {2 (3 A+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {2 (3 A+5 C) \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}}+\frac {2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}} \]
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Rule 16
Rule 2716
Rule 2719
Rule 2720
Rule 2721
Rule 2827
Rule 3100
Rubi steps \begin{align*} \text {integral}& = b \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx \\ & = \frac {2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 \int \frac {\frac {5 b^2 B}{2}+\frac {1}{2} b^2 (3 A+5 C) \cos (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx}{5 b^2} \\ & = \frac {2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+B \int \frac {1}{(b \cos (c+d x))^{5/2}} \, dx+\frac {(3 A+5 C) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx}{5 b} \\ & = \frac {2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac {2 (3 A+5 C) \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}}+\frac {B \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx}{3 b^2}-\frac {(3 A+5 C) \int \sqrt {b \cos (c+d x)} \, dx}{5 b^3} \\ & = \frac {2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac {2 (3 A+5 C) \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}}+\frac {\left (B \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b^2 \sqrt {b \cos (c+d x)}}-\frac {\left ((3 A+5 C) \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 b^3 \sqrt {\cos (c+d x)}} \\ & = -\frac {2 (3 A+5 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {2 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac {2 (3 A+5 C) \sin (c+d x)}{5 b^2 d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.64 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (-3 (3 A+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+9 A \sin (c+d x)+15 C \sin (c+d x)+5 B \tan (c+d x)+3 A \sec (c+d x) \tan (c+d x)\right )}{15 b^2 d \sqrt {b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(807\) vs. \(2(213)=426\).
Time = 19.29 (sec) , antiderivative size = 808, normalized size of antiderivative = 4.37
method | result | size |
default | \(\text {Expression too large to display}\) | \(808\) |
parts | \(\text {Expression too large to display}\) | \(808\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.21 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\frac {-5 i \, \sqrt {2} B \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} B \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 \, \sqrt {2} {\left (3 i \, A + 5 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 \, \sqrt {2} {\left (-3 i \, A - 5 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, {\left (3 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 5 \, B \cos \left (d x + c\right ) + 3 \, A\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, b^{3} d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{\cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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