Integrand size = 42, antiderivative size = 119 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{3/2} (7 B+12 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 d}+\frac {a^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}+\frac {a B \cos (c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d} \]
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Time = 0.47 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {4157, 4102, 4100, 3859, 209} \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{3/2} (7 B+12 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 d}+\frac {a^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}+\frac {a B \sin (c+d x) \cos (c+d x) \sqrt {a \sec (c+d x)+a}}{2 d} \]
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Rule 209
Rule 3859
Rule 4100
Rule 4102
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos (c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{2} a (5 B+4 C)+\frac {1}{2} a (B+4 C) \sec (c+d x)\right ) \, dx \\ & = \frac {a^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}+\frac {a B \cos (c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {1}{8} (a (7 B+12 C)) \int \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}+\frac {a B \cos (c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac {\left (a^2 (7 B+12 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 )}{4 d} \\ & = \frac {a^{3/2} (7 B+12 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 d}+\frac {a^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}+\frac {a B \cos (c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a \cos (c+d x) \sqrt {a (1+\sec (c+d x))} \left ((7 B+4 C+2 B \cos (c+d x)) \sqrt {1-\sec (c+d x)} \sin (c+d x)+(7 B+12 C) \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)\right )}{4 d (1+\cos (c+d x)) \sqrt {1-\sec (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(103)=206\).
Time = 1.70 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.80
method | result | size |
default | \(\frac {a \left (7 B \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+2 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}+12 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+7 B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+7 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+12 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+4 C \cos \left (d x +c \right ) \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{4 d \left (\cos \left (d x +c \right )+1\right )}\) | \(333\) |
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Time = 0.32 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.69 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {{\left ({\left (7 \, B + 12 \, C\right )} a \cos \left (d x + c\right ) + {\left (7 \, B + 12 \, C\right )} a\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (2 \, B a \cos \left (d x + c\right )^{2} + {\left (7 \, B + 4 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {{\left ({\left (7 \, B + 12 \, C\right )} a \cos \left (d x + c\right ) + {\left (7 \, B + 12 \, C\right )} a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (2 \, B a \cos \left (d x + c\right )^{2} + {\left (7 \, B + 4 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^3\,\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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