Integrand size = 33, antiderivative size = 262 \[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {2 \left (6 a^2 A b-3 A b^3-8 a^3 B+5 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (6 a A b-8 a^2 B-b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 (A b-a B) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 B \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d} \]
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Time = 0.56 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3067, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 a^2 (A b-a B) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^2 B+6 a A b-b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-8 a^3 B+6 a^2 A b+5 a b^2 B-3 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 B \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b^2 d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 3067
Rule 3102
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 (A b-a B) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \int \frac {\frac {1}{2} a b (A b-a B)+\frac {1}{2} \left (2 a^2-b^2\right ) (A b-a B) \cos (c+d x)+\frac {1}{2} b \left (a^2-b^2\right ) B \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{b^2 \left (a^2-b^2\right )} \\ & = -\frac {2 a^2 (A b-a B) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 B \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {4 \int \frac {\frac {1}{4} b^2 \left (3 a A b-2 a^2 B-b^2 B\right )+\frac {1}{4} b \left (6 a^2 A b-3 A b^3-8 a^3 B+5 a b^2 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )} \\ & = -\frac {2 a^2 (A b-a B) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 B \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}-\frac {\left (6 a A b-8 a^2 B-b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3}+\frac {\left (6 a^2 A b-3 A b^3-8 a^3 B+5 a b^2 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3 b^3 \left (a^2-b^2\right )} \\ & = -\frac {2 a^2 (A b-a B) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 B \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {\left (\left (6 a^2 A b-3 A b^3-8 a^3 B+5 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}{3 b^3 \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (6 a A b-8 a^2 B-b^2 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}{3 b^3 \sqrt {a+b \cos (c+d x)}} \\ & = \frac {2 \left (6 a^2 A b-3 A b^3-8 a^3 B+5 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 )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (6 a A b-8 a^2 B-b^2 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 )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 (A b-a B) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 B \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d} \\ \end{align*}
Time = 1.47 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {2 \left (\frac {\sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (\left (6 a^2 A b-3 A b^3-8 a^3 B+5 a b^2 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+(a-b) \left (-6 a A b+8 a^2 B+b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{a-b}+b \left (\frac {a \left (3 a A b-4 a^2 B+b^2 B\right )}{-a^2+b^2}+b B \cos (c+d x)\right ) \sin (c+d x)\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1274\) vs. \(2(302)=604\).
Time = 14.82 (sec) , antiderivative size = 1275, normalized size of antiderivative = 4.87
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1275\) |
default | \(\text {Expression too large to display}\) | \(1336\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 789, normalized size of antiderivative = 3.01 \[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {6 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - B a b^{4} + {\left (B a^{2} b^{3} - B b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) - {\left (\sqrt {2} {\left (16 i \, B a^{4} b - 12 i \, A a^{3} b^{2} - 16 i \, B a^{2} b^{3} + 15 i \, A a b^{4} - 3 i \, B b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (16 i \, B a^{5} - 12 i \, A a^{4} b - 16 i \, B a^{3} b^{2} + 15 i \, A a^{2} b^{3} - 3 i \, B a b^{4}\right )}\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 ) - {\left (\sqrt {2} {\left (-16 i \, B a^{4} b + 12 i \, A a^{3} b^{2} + 16 i \, B a^{2} b^{3} - 15 i \, A a b^{4} + 3 i \, B b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-16 i \, B a^{5} + 12 i \, A a^{4} b + 16 i \, B a^{3} b^{2} - 15 i \, A a^{2} b^{3} + 3 i \, B a b^{4}\right )}\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 \, {\left (\sqrt {2} {\left (-8 i \, B a^{3} b^{2} + 6 i \, A a^{2} b^{3} + 5 i \, B a b^{4} - 3 i \, A b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-8 i \, B a^{4} b + 6 i \, A a^{3} b^{2} + 5 i \, B a^{2} b^{3} - 3 i \, A a b^{4}\right )}\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 \, {\left (\sqrt {2} {\left (8 i \, B a^{3} b^{2} - 6 i \, A a^{2} b^{3} - 5 i \, B a b^{4} + 3 i \, A b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (8 i \, B a^{4} b - 6 i \, A a^{3} b^{2} - 5 i \, B a^{2} b^{3} + 3 i \, A a b^{4}\right )}\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 )}{9 \, {\left ({\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{4} - a b^{6}\right )} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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