Integrand size = 42, antiderivative size = 209 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (163 B+200 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{64 d}+\frac {a^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (95 B+104 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (11 B+8 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a B \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d} \]
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Time = 0.85 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4157, 4102, 4100, 3890, 3859, 209} \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (163 B+200 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{64 d}+\frac {a^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (95 B+104 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (11 B+8 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{24 d}+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d} \]
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Rule 209
Rule 3859
Rule 3890
Rule 4100
Rule 4102
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (11 B+8 C)+\frac {1}{2} a (3 B+8 C) \sec (c+d x)\right ) \, dx \\ & = \frac {a^2 (11 B+8 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a B \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (95 B+104 C)+\frac {3}{4} a^2 (17 B+24 C) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (95 B+104 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (11 B+8 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a B \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {1}{64} \left (a^2 (163 B+200 C)\right ) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (95 B+104 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (11 B+8 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a B \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {1}{128} \left (a^2 (163 B+200 C)\right ) \int \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (95 B+104 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (11 B+8 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a B \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac {\left (a^3 (163 B+200 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 )}{64 d} \\ & = \frac {a^{5/2} (163 B+200 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{64 d}+\frac {a^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (95 B+104 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (11 B+8 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a B \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.91 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.75 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 \left (6075 B \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+6600 C \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+2079 B \sqrt {1-\sec (c+d x)}+1240 C \sqrt {1-\sec (c+d x)}+7641 B \cos (c+d x) \sqrt {1-\sec (c+d x)}+6360 C \cos (c+d x) \sqrt {1-\sec (c+d x)}+2097 B \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+1240 C \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+522 B \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}-80 C \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+18 B \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+7680 C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}+4608 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},5,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{2880 d (1+\cos (c+d x)) \sqrt {1-\sec (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(402\) vs. \(2(185)=370\).
Time = 0.40 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.93
\[\frac {a^{2} \left (48 B \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+184 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}+64 C \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+489 B \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+326 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}+600 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+272 C \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+489 B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+489 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+600 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+600 C \cos \left (d x +c \right ) \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{192 d \left (\cos \left (d x +c \right )+1\right )}\]
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Time = 0.32 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.01 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {3 \, {\left ({\left (163 \, B + 200 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (163 \, B + 200 \, C\right )} a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (48 \, B a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (23 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (163 \, B + 136 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (163 \, B + 200 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{384 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {3 \, {\left ({\left (163 \, B + 200 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (163 \, B + 200 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (48 \, B a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (23 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (163 \, B + 136 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (163 \, B + 200 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{192 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
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Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^5\,\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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