Integrand size = 23, antiderivative size = 182 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {163 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{64 d}+\frac {163 a^3 \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {17 a^3 \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d} \]
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Time = 0.41 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3898, 4100, 3890, 3859, 209} \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {163 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{64 d}+\frac {163 a^3 \sin (c+d x)}{64 d \sqrt {a \sec (c+d x)+a}}+\frac {17 a^3 \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt {a \sec (c+d x)+a}}+\frac {163 a^3 \sin (c+d x) \cos (c+d x)}{96 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d} \]
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Rule 209
Rule 3859
Rule 3890
Rule 3898
Rule 4100
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {1}{4} a \int \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {17 a}{2}+\frac {13}{2} a \sec (c+d x)\right ) \, dx \\ & = \frac {17 a^3 \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {1}{48} \left (163 a^2\right ) \int \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {163 a^3 \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {17 a^3 \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {1}{64} \left (163 a^2\right ) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {163 a^3 \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {17 a^3 \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {1}{128} \left (163 a^2\right ) \int \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {163 a^3 \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {17 a^3 \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d}-\frac {\left (163 a^3\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 {163 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{64 d}+\frac {163 a^3 \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}+\frac {17 a^3 \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.82 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {a^2 \left (675 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+(231+849 \cos (c+d x)+233 \cos (2 (c+d x))+58 \cos (3 (c+d x))+2 \cos (4 (c+d x))) \sqrt {1-\sec (c+d x)}+512 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},5,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{320 d (1+\cos (c+d x)) \sqrt {1-\sec (c+d x)}} \]
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Time = 0.94 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.21
\[\frac {a^{2} \left (48 \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+184 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+489 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+326 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+489 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+489 \cos \left (d x +c \right ) \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{192 d \left (\cos \left (d x +c \right )+1\right )}\]
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none
Time = 0.30 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.90 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\left [\frac {489 \, {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (48 \, a^{2} \cos \left (d x + c\right )^{4} + 184 \, a^{2} \cos \left (d x + c\right )^{3} + 326 \, a^{2} \cos \left (d x + c\right )^{2} + 489 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{384 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {489 \, {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (48 \, a^{2} \cos \left (d x + c\right )^{4} + 184 \, a^{2} \cos \left (d x + c\right )^{3} + 326 \, a^{2} \cos \left (d x + c\right )^{2} + 489 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{192 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
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Timed out. \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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