Integrand size = 41, antiderivative size = 209 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=-\frac {6 B \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}} \]
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Time = 0.31 (sec) , antiderivative size = 209, 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} \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 b d \sqrt {\cos (c+d x)}} \]
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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^4 \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^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {1}{7} (2 b) \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^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\left (b^3 B\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx+\frac {1}{7} \left (b^2 (5 A+7 C)\right ) \int \frac {1}{(b \cos (c+d x))^{5/2}} \, dx \\ & = \frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {1}{5} (3 b B) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx+\frac {1}{21} (5 A+7 C) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx \\ & = \frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {(3 B) \int \sqrt {b \cos (c+d x)} \, dx}{5 b}+\frac {\left ((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 (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^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {\left (3 B \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 b \sqrt {\cos (c+d x)}} \\ & = -\frac {6 B \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)}}+\frac {2 (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^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.64 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {2 \left (-63 B \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+63 B \sin (c+d x)+25 A \tan (c+d x)+35 C \tan (c+d x)+21 B \sec (c+d x) \tan (c+d x)+15 A \sec ^2(c+d x) \tan (c+d x)\right )}{105 d \sqrt {b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(726\) vs. \(2(233)=466\).
Time = 20.22 (sec) , antiderivative size = 727, normalized size of antiderivative = 3.48
method | result | size |
default | \(\text {Expression too large to display}\) | \(727\) |
parts | \(\text {Expression too large to display}\) | \(874\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=-\frac {5 \, \sqrt {2} {\left (5 i \, A + 7 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-5 i \, A - 7 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 63 i \, \sqrt {2} B \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 63 i \, \sqrt {2} B \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (63 \, B \cos \left (d x + c\right )^{3} + 5 \, {\left (5 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{2} + 21 \, B \cos \left (d x + c\right ) + 15 \, A\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, b d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^4\,\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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