Integrand size = 45, antiderivative size = 466 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (48 A b^3-63 a^3 B-56 a b^2 B+a^2 (44 A b+70 b C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^5 d}+\frac {2 \sqrt {a+b} \left (48 A b^3-4 a b^2 (3 A+14 B)+a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^4 d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (6 A b-7 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^2-28 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 a^3 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Time = 1.57 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3134, 3077, 2895, 3073} \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=-\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{35 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{105 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-63 a^3 B+a^2 (44 A b+70 b C)-56 a b^2 B+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{105 a^5 d}+\frac {2 \sqrt {a+b} \cot (c+d x) \left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{105 a^4 d}+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)} \]
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Rule 2895
Rule 3073
Rule 3077
Rule 3134
Rubi steps \begin{align*} \text {integral}& = \frac {2 A \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \int \frac {\frac {1}{2} (-6 A b+7 a B)+\frac {1}{2} a (5 A+7 C) \cos (c+d x)+2 A b \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{7 a} \\ & = \frac {2 A \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (6 A b-7 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} \left (24 A b^2-28 a b B+5 a^2 (5 A+7 C)\right )+\frac {1}{4} a (2 A b+21 a B) \cos (c+d x)-\frac {1}{2} b (6 A b-7 a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{35 a^2} \\ & = \frac {2 A \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (6 A b-7 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^2-28 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 a^3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {\frac {1}{8} \left (-48 A b^3+63 a^3 B+56 a b^2 B-2 a^2 b (22 A+35 C)\right )-\frac {1}{8} a \left (12 A b^2-14 a b B-5 a^2 (5 A+7 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{105 a^3} \\ & = \frac {2 A \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (6 A b-7 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^2-28 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (48 A b^3-63 a^3 B-56 a b^2 B+a^2 (44 A b+70 b C)\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{105 a^3}+\frac {\left (48 A b^3-4 a b^2 (3 A+14 B)+a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{105 a^3} \\ & = -\frac {2 (a-b) \sqrt {a+b} \left (48 A b^3-63 a^3 B-56 a b^2 B+a^2 (44 A b+70 b 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^5 d}+\frac {2 \sqrt {a+b} \left (48 A b^3-4 a b^2 (3 A+14 B)+a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 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^4 d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (6 A b-7 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^2-28 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 a^3 d \cos ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.46 (sec) , antiderivative size = 1468, normalized size of antiderivative = 3.15 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\frac {-\frac {4 a \left (25 a^4 A+32 a^2 A b^2+48 A b^4-49 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 (44 a^3 A b+48 a A b^3-63 a^4 B-56 a^2 b^2 B+70 a^3 b 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 (44 a^2 A b^2+48 A b^4-63 a^3 b B-56 a b^3 B+70 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^4 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {2 \sec ^3(c+d x) (-6 A b \sin (c+d x)+7 a B \sin (c+d x))}{35 a^2}+\frac {2 \sec ^2(c+d x) \left (25 a^2 A \sin (c+d x)+24 A b^2 \sin (c+d x)-28 a b B \sin (c+d x)+35 a^2 C \sin (c+d x)\right )}{105 a^3}+\frac {2 \sec (c+d x) \left (-44 a^2 A b \sin (c+d x)-48 A b^3 \sin (c+d x)+63 a^3 B \sin (c+d x)+56 a b^2 B \sin (c+d x)-70 a^2 b C \sin (c+d x)\right )}{105 a^4}+\frac {2 A \sec ^3(c+d x) \tan (c+d x)}{7 a}\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(5790\) vs. \(2(427)=854\).
Time = 38.61 (sec) , antiderivative size = 5791, normalized size of antiderivative = 12.43
method | result | size |
parts | \(\text {Expression too large to display}\) | \(5791\) |
default | \(\text {Expression too large to display}\) | \(5875\) |
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\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
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