Integrand size = 25, antiderivative size = 225 \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {2 \left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) (5 A b+3 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 (5 A b+3 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
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Time = 0.40 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=-\frac {2 \left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (3 a B+5 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{15 d}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2}{5} \int \sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} (5 a A+3 b B)+\frac {1}{2} (5 A b+3 a B) \cos (c+d x)\right ) \, dx \\ & = \frac {2 (5 A b+3 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {4}{15} \int \frac {\frac {1}{4} \left (15 a^2 A+5 A b^2+12 a b B\right )+\frac {1}{4} \left (20 a A b+3 a^2 B+9 b^2 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 (5 A b+3 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac {\left (\left (a^2-b^2\right ) (5 A b+3 a B)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 b}+\frac {\left (20 a A b+3 a^2 B+9 b^2 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{15 b} \\ & = \frac {2 (5 A b+3 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\left (\left (20 a A b+3 a^2 B+9 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}{15 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) (5 A b+3 a B) \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}{15 b \sqrt {a+b \cos (c+d x)}} \\ & = \frac {2 \left (20 a A b+3 a^2 B+9 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 )}{15 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) (5 A b+3 a B) \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 )}{15 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 (5 A b+3 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.90 \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {2 \left (b \left (15 a^2 A+5 A b^2+12 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )+b (a+b \cos (c+d x)) (5 A b+6 a B+3 b B \cos (c+d x)) \sin (c+d x)\right )}{15 b d \sqrt {a+b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(992\) vs. \(2(263)=526\).
Time = 13.12 (sec) , antiderivative size = 993, normalized size of antiderivative = 4.41
method | result | size |
default | \(\text {Expression too large to display}\) | \(993\) |
parts | \(\text {Expression too large to display}\) | \(1115\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.19 \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {\sqrt {2} {\left (6 i \, B a^{3} - 5 i \, A a^{2} b - 18 i \, B a b^{2} - 15 i \, A b^{3}\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 (-6 i \, B a^{3} + 5 i \, A a^{2} b + 18 i \, B a b^{2} + 15 i \, A b^{3}\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 (-3 i \, B a^{2} b - 20 i \, A a b^{2} - 9 i \, B b^{3}\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 (3 i \, B a^{2} b + 20 i \, A a b^{2} + 9 i \, B b^{3}\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 (3 \, B b^{3} \cos \left (d x + c\right ) + 6 \, B a b^{2} + 5 \, A b^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45 \, b^{2} d} \]
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\[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\int \left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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