Integrand size = 35, antiderivative size = 151 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{3/2} (7 A+8 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 d}+\frac {a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}-\frac {a (A-4 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d} \]
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Time = 0.70 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4172, 4103, 4100, 3859, 209} \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{3/2} (7 A+8 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 d}+\frac {a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {a (A-4 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d} \]
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Rule 209
Rule 3859
Rule 4100
Rule 4103
Rule 4172
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {3 a A}{2}-\frac {1}{2} a (A-4 C) \sec (c+d x)\right ) \, dx}{2 a} \\ & = -\frac {a (A-4 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}+\frac {\int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (5 A-8 C)+\frac {1}{4} a^2 (A+8 C) \sec (c+d x)\right ) \, dx}{a} \\ & = \frac {a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}-\frac {a (A-4 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}+\frac {1}{8} (a (7 A+8 C)) \int \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}-\frac {a (A-4 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d}-\frac {\left (a^2 (7 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 )}{4 d} \\ & = \frac {a^{3/2} (7 A+8 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 d}+\frac {a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}}-\frac {a (A-4 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.72 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sec (c+d x))} \left (\sqrt {2} (7 A+8 C) \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cos (c+d x)}+2 (A+8 C+7 A \cos (c+d x)+A \cos (2 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(326\) vs. \(2(131)=262\).
Time = 0.56 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.17
method | result | size |
default | \(\frac {a \left (7 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+2 A \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+8 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+7 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+7 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+8 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+8 C \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{4 d \left (\cos \left (d x +c \right )+1\right )}\) | \(327\) |
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Time = 0.33 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.11 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {{\left ({\left (7 \, A + 8 \, C\right )} a \cos \left (d x + c\right ) + {\left (7 \, A + 8 \, C\right )} a\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (2 \, A a \cos \left (d x + c\right )^{2} + 7 \, A a \cos \left (d x + c\right ) + 8 \, C a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {{\left ({\left (7 \, A + 8 \, C\right )} a \cos \left (d x + c\right ) + {\left (7 \, A + 8 \, C\right )} a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (2 \, A a \cos \left (d x + c\right )^{2} + 7 \, A a \cos \left (d x + c\right ) + 8 \, C a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
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Timed out. \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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