Integrand size = 45, antiderivative size = 333 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {(19 A-15 B+11 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{2 \sqrt {2} a^{3/2} d}-\frac {(1201 A-1029 B+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}} \]
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Time = 1.35 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4306, 3120, 3063, 12, 2861, 211} \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {(19 A-15 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(11 A-7 B+7 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{14 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B+C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac {(67 A-63 B+35 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{70 a d \sqrt {a \cos (c+d x)+a}}+\frac {(397 A-273 B+245 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{210 a d \sqrt {a \cos (c+d x)+a}}-\frac {(1201 A-1029 B+665 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{210 a d \sqrt {a \cos (c+d x)+a}} \]
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Rule 12
Rule 211
Rule 2861
Rule 3063
Rule 3120
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx \\ & = -\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a (11 A-7 B+7 C)-2 a (2 A-2 B+C) \cos (c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{4} a^2 (67 A-63 B+35 C)+\frac {3}{2} a^2 (11 A-7 B+7 C) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{7 a^3} \\ & = -\frac {(67 A-63 B+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a^3 (397 A-273 B+245 C)-\frac {1}{2} a^3 (67 A-63 B+35 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{35 a^4} \\ & = \frac {(397 A-273 B+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a^4 (1201 A-1029 B+665 C)+\frac {1}{8} a^4 (397 A-273 B+245 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{105 a^5} \\ & = -\frac {(1201 A-1029 B+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {105 a^5 (19 A-15 B+11 C)}{32 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{105 a^6} \\ & = -\frac {(1201 A-1029 B+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left ((19 A-15 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{4 a} \\ & = -\frac {(1201 A-1029 B+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}-\frac {\left ((19 A-15 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{2 d} \\ & = \frac {(19 A-15 B+11 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{2 \sqrt {2} a^{3/2} d}-\frac {(1201 A-1029 B+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 8.65 (sec) , antiderivative size = 3137, normalized size of antiderivative = 9.42 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(694\) vs. \(2(286)=572\).
Time = 2.05 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.09
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {9}{2}}\left (d x +c \right )\right ) \left (1995 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-1575 \left (\cos ^{6}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, B \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+1155 C \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3990 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1201 A \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-3150 \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, B \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )-1029 \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right ) B \sqrt {2}+2310 C \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+665 C \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+1995 A \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+804 A \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-1575 B \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-756 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) B \sqrt {2}+1155 C \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+420 C \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-196 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+84 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) B \sqrt {2}-140 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+36 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-84 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) B \sqrt {2}-60 A \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}\right )}{420 a^{2} d \left (1+\cos \left (d x +c \right )\right )^{2}}\) | \(695\) |
parts | \(\text {Expression too large to display}\) | \(762\) |
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Time = 0.33 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.77 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=-\frac {105 \, \sqrt {2} {\left ({\left (19 \, A - 15 \, B + 11 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (19 \, A - 15 \, B + 11 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (19 \, A - 15 \, B + 11 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left ({\left (1201 \, A - 1029 \, B + 665 \, C\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (67 \, A - 63 \, B + 35 \, C\right )} \cos \left (d x + c\right )^{3} - 28 \, {\left (7 \, A - 3 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 12 \, {\left (3 \, A - 7 \, B\right )} \cos \left (d x + c\right ) - 60 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{420 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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