Integrand size = 45, antiderivative size = 580 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^2 d}-\frac {2 \sqrt {a+b} \left (15 b^3 (A-7 B)-a^3 (25 A-63 B+35 C)+a^2 b (145 A-119 B+245 C)-a b^2 (135 A-161 B+315 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a d}-\frac {2 b^2 \sqrt {a+b} C \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{d}+\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Time = 1.95 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3126, 3132, 2888, 3077, 2895, 3073} \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{105 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {a+b} \cot (c+d x) \left (-\left (a^3 (25 A-63 B+35 C)\right )+a^2 b (145 A-119 B+245 C)-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{105 a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{105 a^2 d}+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 b^2 C \sqrt {a+b} \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d} \]
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Rule 2888
Rule 2895
Rule 3073
Rule 3077
Rule 3126
Rule 3132
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} (5 A b+7 a B)+\frac {1}{2} (5 a A+7 b B+7 a C) \cos (c+d x)+\frac {7}{2} b C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4}{35} \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right )+\frac {1}{4} \left (40 a A b+21 a^2 B+35 b^2 B+70 a b C\right ) \cos (c+d x)+\frac {35}{4} b^2 C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8}{105} \int \frac {\frac {1}{8} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right )+\frac {1}{8} \left (119 a^2 b B+105 b^3 B+45 a b^2 (3 A+7 C)+5 a^3 (5 A+7 C)\right ) \cos (c+d x)+\frac {105}{8} b^3 C \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8}{105} \int \frac {\frac {1}{8} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right )+\frac {1}{8} \left (119 a^2 b B+105 b^3 B+45 a b^2 (3 A+7 C)+5 a^3 (5 A+7 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx+\left (b^3 C\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx \\ & = -\frac {2 b^2 \sqrt {a+b} C \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{d}+\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{105} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx+\frac {1}{105} \left (-15 b^3 (A-7 B)+a^3 (25 A-63 B+35 C)-a^2 b (145 A-119 B+245 C)+a b^2 (135 A-161 B+315 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 (a-b) \sqrt {a+b} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^2 d}-\frac {2 \sqrt {a+b} \left (15 b^3 (A-7 B)-a^3 (25 A-63 B+35 C)+a^2 b (145 A-119 B+245 C)-a b^2 (135 A-161 B+315 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a d}-\frac {2 b^2 \sqrt {a+b} C \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{d}+\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.84 (sec) , antiderivative size = 1472, normalized size of antiderivative = 2.54 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {-\frac {4 a \left (25 a^4 A-10 a^2 A b^2-15 A b^4+56 a^3 b B-56 a b^3 B+35 a^4 C+70 a^2 b^2 C\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-4 a \left (-145 a^3 A b-15 a A b^3-63 a^4 B-161 a^2 b^2 B-245 a^3 b C+105 a b^3 C\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )+2 \left (-145 a^2 A b^2-15 A b^4-63 a^3 b B-161 a b^3 B-245 a^2 b^2 C\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right )|-\frac {2 a}{-a-b}\right ) \sec (c+d x)}{b \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {\frac {(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )}{b}+\frac {\sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt {\cos (c+d x)}}\right )}{105 a d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {2}{35} \sec ^3(c+d x) \left (15 a A b \sin (c+d x)+7 a^2 B \sin (c+d x)\right )+\frac {2}{105} \sec ^2(c+d x) \left (25 a^2 A \sin (c+d x)+45 A b^2 \sin (c+d x)+77 a b B \sin (c+d x)+35 a^2 C \sin (c+d x)\right )+\frac {2 \sec (c+d x) \left (145 a^2 A b \sin (c+d x)+15 A b^3 \sin (c+d x)+63 a^3 B \sin (c+d x)+161 a b^2 B \sin (c+d x)+245 a^2 b C \sin (c+d x)\right )}{105 a}+\frac {2}{7} a^2 A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(6987\) vs. \(2(534)=1068\).
Time = 39.27 (sec) , antiderivative size = 6988, normalized size of antiderivative = 12.05
method | result | size |
default | \(\text {Expression too large to display}\) | \(6988\) |
parts | \(\text {Expression too large to display}\) | \(7085\) |
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^{9/2}} \,d x \]
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