Integrand size = 33, antiderivative size = 136 \[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 a^{3/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {a^2 (3 A-8 C) \tan (c+d x)}{3 d \sqrt {a+a \sec (c+d x)}}-\frac {a (3 A-2 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d} \]
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Time = 0.49 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4172, 4002, 4000, 3859, 209, 3877} \[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 a^{3/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-\frac {a^2 (3 A-8 C) \tan (c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {a (3 A-2 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{d} \]
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Rule 209
Rule 3859
Rule 3877
Rule 4000
Rule 4002
Rule 4172
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{d}+\frac {\int (a+a \sec (c+d x))^{3/2} \left (\frac {3 a A}{2}-\frac {1}{2} a (3 A-2 C) \sec (c+d x)\right ) \, dx}{a} \\ & = \frac {A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {a (3 A-2 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d}+\frac {2 \int \sqrt {a+a \sec (c+d x)} \left (\frac {9 a^2 A}{4}-\frac {1}{4} a^2 (3 A-8 C) \sec (c+d x)\right ) \, dx}{3 a} \\ & = \frac {A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {a (3 A-2 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{2} (3 a A) \int \sqrt {a+a \sec (c+d x)} \, dx-\frac {1}{6} (a (3 A-8 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {a^2 (3 A-8 C) \tan (c+d x)}{3 d \sqrt {a+a \sec (c+d x)}}-\frac {a (3 A-2 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d}-\frac {\left (3 a^2 A\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 )}{d} \\ & = \frac {3 a^{3/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {a^2 (3 A-8 C) \tan (c+d x)}{3 d \sqrt {a+a \sec (c+d x)}}-\frac {a (3 A-2 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.77 \[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 \left (9 A \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)+\sqrt {1-\sec (c+d x)} (3 A \sin (c+d x)+2 C (5+\sec (c+d x)) \tan (c+d x))\right )}{3 d \sqrt {1-\sec (c+d x)} \sqrt {a (1+\sec (c+d x))}} \]
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Time = 0.55 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.40
method | result | size |
default | \(\frac {a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (9 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+9 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+3 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+10 C \sin \left (d x +c \right )+2 C \tan \left (d x +c \right )\right )}{3 d \left (\cos \left (d x +c \right )+1\right )}\) | \(191\) |
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Time = 0.31 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.43 \[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {9 \, {\left (A a \cos \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (3 \, A a \cos \left (d x + c\right )^{2} + 10 \, C a \cos \left (d x + c\right ) + 2 \, C a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}}, -\frac {9 \, {\left (A a \cos \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (3 \, A a \cos \left (d x + c\right )^{2} + 10 \, C a \cos \left (d x + c\right ) + 2 \, C a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}}\right ] \]
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Timed out. \[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (120) = 240\).
Time = 0.43 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.91 \[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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\[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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