Integrand size = 41, antiderivative size = 217 \[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=-\frac {6 b^2 B \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^3 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^3 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}} \]
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Time = 0.42 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {16, 3100, 2827, 2716, 2721, 2719, 2720} \[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {2 b^3 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 b^3 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {6 b^2 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}} \]
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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^7 \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx \\ & = \frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {1}{7} \left (2 b^4\right ) \int \frac {\frac {7 b^2 B}{2}+\frac {1}{2} b^2 (5 A+7 C) \cos (c+d x)}{(b \cos (c+d x))^{7/2}} \, dx \\ & = \frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\left (b^6 B\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx+\frac {1}{7} \left (b^5 (5 A+7 C)\right ) \int \frac {1}{(b \cos (c+d x))^{5/2}} \, dx \\ & = \frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {1}{5} \left (3 b^4 B\right ) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx+\frac {1}{21} \left (b^3 (5 A+7 C)\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx \\ & = \frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^3 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {1}{5} \left (3 b^2 B\right ) \int \sqrt {b \cos (c+d x)} \, dx+\frac {\left (b^3 (5 A+7 C) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}} \\ & = \frac {2 b^3 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^3 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {\left (3 b^2 B \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}} \\ & = -\frac {6 b^2 B \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^3 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^3 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 2.37 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62 \[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {(b \cos (c+d x))^{5/2} \sec ^6(c+d x) \left (-504 B \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 (5 A+7 C) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 (110 A+70 C+273 B \cos (c+d x)+10 (5 A+7 C) \cos (2 (c+d x))+63 B \cos (3 (c+d x))) \sin (c+d x)\right )}{420 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(727\) vs. \(2(241)=482\).
Time = 3137.82 (sec) , antiderivative size = 728, normalized size of antiderivative = 3.35
method | result | size |
default | \(\text {Expression too large to display}\) | \(728\) |
parts | \(\text {Expression too large to display}\) | \(1008\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.12 \[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {-5 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 63 i \, \sqrt {2} B b^{\frac {5}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 63 i \, \sqrt {2} B b^{\frac {5}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (63 \, B b^{2} \cos \left (d x + c\right )^{3} + 5 \, {\left (5 \, A + 7 \, C\right )} b^{2} \cos \left (d x + c\right )^{2} + 21 \, B b^{2} \cos \left (d x + c\right ) + 15 \, A b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]
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\[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \sec \left (d x + c\right )^{7} \,d x } \]
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\[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \sec \left (d x + c\right )^{7} \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^7} \,d x \]
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