Integrand size = 35, antiderivative size = 192 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {5 a^{5/2} (5 A+8 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}+\frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \]
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Time = 0.78 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4172, 4102, 4103, 4100, 3859, 209} \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {5 a^{5/2} (5 A+8 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 d}+\frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a \sec (c+d x)+a}}-\frac {a^2 (3 A-8 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d}+\frac {5 a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d} \]
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Rule 209
Rule 3859
Rule 4100
Rule 4102
Rule 4103
Rule 4172
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {5 a A}{2}-\frac {1}{2} a (A-6 C) \sec (c+d x)\right ) \, dx}{3 a} \\ & = \frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (31 A+24 C)-\frac {3}{4} a^2 (3 A-8 C) \sec (c+d x)\right ) \, dx}{6 a} \\ & = -\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (49 A-24 C)+\frac {1}{8} a^3 (13 A+72 C) \sec (c+d x)\right ) \, dx}{3 a} \\ & = \frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{16} \left (5 a^2 (5 A+8 C)\right ) \int \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}-\frac {\left (5 a^3 (5 A+8 C)\right ) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d} \\ & = \frac {5 a^{5/2} (5 A+8 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}+\frac {a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (3 A-8 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {5 a A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.91 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 \left (15 (11 A+24 C) \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+(31 A+144 C+3 (53 A+24 C) \cos (c+d x)+31 A \cos (2 (c+d x))-2 A \cos (3 (c+d x))) \sqrt {1-\sec (c+d x)}+192 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{72 d (1+\cos (c+d x)) \sqrt {1-\sec (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(168)=336\).
Time = 186.58 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {a^{2} \left (8 A \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+75 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+34 A \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+75 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+75 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+120 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+24 C \cos \left (d x +c \right ) \sin \left (d x +c \right )+48 C \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{24 d \left (\cos \left (d x +c \right )+1\right )}\) | \(361\) |
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Time = 0.31 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.98 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {15 \, {\left ({\left (5 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (5 \, A + 8 \, C\right )} a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (8 \, A a^{2} \cos \left (d x + c\right )^{3} + 34 \, A a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (25 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (5 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (5 \, A + 8 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (8 \, A a^{2} \cos \left (d x + c\right )^{3} + 34 \, A a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (25 \, A + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^3\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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