Integrand size = 33, antiderivative size = 320 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \left (56 a^2 A b+63 A b^3-48 a^3 B-44 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (56 a^3 A b+49 a A b^3-48 a^4 B-32 a^2 b^2 B-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (28 a A b-24 a^2 B-25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^3 d}+\frac {2 (7 A b-6 a B) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d} \]
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Time = 0.70 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3069, 3128, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b^3 d}+\frac {2 \left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{35 b^2 d}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 3069
Rule 3102
Rule 3128
Rubi steps \begin{align*} \text {integral}& = \frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d}+\frac {2 \int \frac {\cos (c+d x) \left (2 a B+\frac {5}{2} b B \cos (c+d x)+\frac {1}{2} (7 A b-6 a B) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx}{7 b} \\ & = \frac {2 (7 A b-6 a B) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d}+\frac {4 \int \frac {\frac {1}{2} a (7 A b-6 a B)+\frac {1}{4} b (21 A b+2 a B) \cos (c+d x)-\frac {1}{4} \left (28 a A b-24 a^2 B-25 b^2 B\right ) \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{35 b^2} \\ & = -\frac {2 \left (28 a A b-24 a^2 B-25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^3 d}+\frac {2 (7 A b-6 a B) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d}+\frac {8 \int \frac {\frac {1}{8} b \left (14 a A b-12 a^2 B+25 b^2 B\right )+\frac {1}{8} \left (56 a^2 A b+63 A b^3-48 a^3 B-44 a b^2 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^3} \\ & = -\frac {2 \left (28 a A b-24 a^2 B-25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^3 d}+\frac {2 (7 A b-6 a B) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d}+\frac {\left (56 a^2 A b+63 A b^3-48 a^3 B-44 a b^2 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{105 b^4}-\frac {\left (56 a^3 A b+49 a A b^3-48 a^4 B-32 a^2 b^2 B-25 b^4 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^4} \\ & = -\frac {2 \left (28 a A b-24 a^2 B-25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^3 d}+\frac {2 (7 A b-6 a B) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d}+\frac {\left (\left (56 a^2 A b+63 A b^3-48 a^3 B-44 a b^2 B\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{105 b^4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (56 a^3 A b+49 a A b^3-48 a^4 B-32 a^2 b^2 B-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{105 b^4 \sqrt {a+b \cos (c+d x)}} \\ & = \frac {2 \left (56 a^2 A b+63 A b^3-48 a^3 B-44 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (56 a^3 A b+49 a A b^3-48 a^4 B-32 a^2 b^2 B-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (28 a A b-24 a^2 B-25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^3 d}+\frac {2 (7 A b-6 a B) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (14 a A b-12 a^2 B+25 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-56 a^2 A b-63 A b^3+48 a^3 B+44 a b^2 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+2 b (a+b \cos (c+d x)) \left (-56 a A b+48 a^2 B+65 b^2 B+6 b (7 A b-6 a B) \cos (c+d x)+15 b^2 B \cos (2 (c+d x))\right ) \sin (c+d x)}{210 b^4 d \sqrt {a+b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1304\) vs. \(2(354)=708\).
Time = 13.14 (sec) , antiderivative size = 1305, normalized size of antiderivative = 4.08
method | result | size |
default | \(\text {Expression too large to display}\) | \(1305\) |
parts | \(\text {Expression too large to display}\) | \(1494\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.76 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {\sqrt {2} {\left (-96 i \, B a^{4} + 112 i \, A a^{3} b - 52 i \, B a^{2} b^{2} + 84 i \, A a b^{3} - 75 i \, B b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (96 i \, B a^{4} - 112 i \, A a^{3} b + 52 i \, B a^{2} b^{2} - 84 i \, A a b^{3} + 75 i \, B b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (48 i \, B a^{3} b - 56 i \, A a^{2} b^{2} + 44 i \, B a b^{3} - 63 i \, A b^{4}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (-48 i \, B a^{3} b + 56 i \, A a^{2} b^{2} - 44 i \, B a b^{3} + 63 i \, A b^{4}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (15 \, B b^{4} \cos \left (d x + c\right )^{2} + 24 \, B a^{2} b^{2} - 28 \, A a b^{3} + 25 \, B b^{4} - 3 \, {\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, b^{5} d} \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
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