Integrand size = 33, antiderivative size = 147 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=-\frac {2 (7 A+9 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)}}+\frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {2 (7 A+9 C) \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {16, 3091, 2716, 2721, 2719} \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {2 (7 A+9 C) \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 b d \sqrt {\cos (c+d x)}} \]
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Rule 16
Rule 2716
Rule 2719
Rule 2721
Rule 3091
Rubi steps \begin{align*} \text {integral}& = b^5 \int \frac {A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{11/2}} \, dx \\ & = \frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {1}{9} \left (b^3 (7 A+9 C)\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx \\ & = \frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {1}{15} (b (7 A+9 C)) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx \\ & = \frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {2 (7 A+9 C) \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {(7 A+9 C) \int \sqrt {b \cos (c+d x)} \, dx}{15 b} \\ & = \frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {2 (7 A+9 C) \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {\left ((7 A+9 C) \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 b \sqrt {\cos (c+d x)}} \\ & = -\frac {2 (7 A+9 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)}}+\frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {2 (7 A+9 C) \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.66 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {-6 (7 A+9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6 (7 A+9 C) \sin (c+d x)+2 \sec (c+d x) \left (7 A+9 C+5 A \sec ^2(c+d x)\right ) \tan (c+d x)}{45 d \sqrt {b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(730\) vs. \(2(155)=310\).
Time = 16.57 (sec) , antiderivative size = 731, normalized size of antiderivative = 4.97
method | result | size |
default | \(\text {Expression too large to display}\) | \(731\) |
parts | \(\text {Expression too large to display}\) | \(782\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.09 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=-\frac {3 \, \sqrt {2} {\left (7 i \, A + 9 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-7 i \, A - 9 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} + 5 \, A\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{45 \, b d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{5}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{5}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^5\,\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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