Optimal. Leaf size=331 \[ -\frac {2 \left (2 a^3 A b-6 a A b^3-8 a^4 B+15 a^2 b^2 B-3 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (2 a^2 A b-3 A b^3-8 a^3 B+9 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 A b-6 A b^3-5 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}} \]
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Rubi [A]
time = 0.36, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3067, 3100,
2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 a^2 (A b-a B) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (-5 a^3 B+2 a^2 A b+9 a b^2 B-6 A b^3\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-8 a^3 B+2 a^2 A b+9 a b^2 B-3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 B+2 a^3 A b+15 a^2 b^2 B-6 a A b^3-3 b^4 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 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 3067
Rule 3100
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx &=-\frac {2 a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \int \frac {\frac {3}{2} a b (A b-a B)+\frac {1}{2} \left (2 a^2-3 b^2\right ) (A b-a B) \cos (c+d x)+\frac {3}{2} b \left (a^2-b^2\right ) B \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {2 a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 A b-6 A b^3-5 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 \int \frac {-\frac {1}{4} b^2 \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right )+\frac {1}{4} b \left (2 a^3 A b-6 a A b^3-8 a^4 B+15 a^2 b^2 B-3 b^4 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {2 a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 A b-6 A b^3-5 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (2 a^2 A b-3 A b^3-8 a^3 B+9 a b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )}-\frac {\left (2 a^3 A b-6 a A b^3-8 a^4 B+15 a^2 b^2 B-3 b^4 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {2 a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 A b-6 A b^3-5 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (\left (2 a^3 A b-6 a A b^3-8 a^4 B+15 a^2 b^2 B-3 b^4 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 )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (2 a^2 A b-3 A b^3-8 a^3 B+9 a 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 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\\ &=-\frac {2 \left (2 a^3 A b-6 a A b^3-8 a^4 B+15 a^2 b^2 B-3 b^4 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 )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (2 a^2 A b-3 A b^3-8 a^3 B+9 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 (A b-a B) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 A b-6 A b^3-5 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 2.43, size = 274, normalized size = 0.83 \begin {gather*} \frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (-2 a^3 A b+6 a A b^3+8 a^4 B-15 a^2 b^2 B+3 b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}-\frac {a b \left (a \left (-a^2 A b+5 A b^3+4 a^3 B-8 a b^2 B\right )+b \left (-2 a^2 A b+6 A b^3+5 a^3 B-9 a b^2 B\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2}\right )}{3 b^3 d (a+b \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(953\) vs.
\(2(369)=738\).
time = 1.23, size = 954, normalized size = 2.88
method | result | size |
default | \(\text {Expression too large to display}\) | \(954\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.26, size = 1193, normalized size = 3.60 \begin {gather*} -\frac {6 \, {\left (4 \, B a^{5} b^{2} - A a^{4} b^{3} - 8 \, B a^{3} b^{4} + 5 \, A a^{2} b^{5} + {\left (5 \, B a^{4} b^{3} - 2 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} + 6 \, A a b^{6}\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^{5} b^{2} - 4 i \, A a^{4} b^{3} - 36 i \, B a^{3} b^{4} + 9 i \, A a^{2} b^{5} + 24 i \, B a b^{6} - 9 i \, A b^{7}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (-16 i \, B a^{6} b + 4 i \, A a^{5} b^{2} + 36 i \, B a^{4} b^{3} - 9 i \, A a^{3} b^{4} - 24 i \, B a^{2} b^{5} + 9 i \, A a b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (16 i \, B a^{7} - 4 i \, A a^{6} b - 36 i \, B a^{5} b^{2} + 9 i \, A a^{4} b^{3} + 24 i \, B a^{3} b^{4} - 9 i \, A a^{2} b^{5}\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^{5} b^{2} + 4 i \, A a^{4} b^{3} + 36 i \, B a^{3} b^{4} - 9 i \, A a^{2} b^{5} - 24 i \, B a b^{6} + 9 i \, A b^{7}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (16 i \, B a^{6} b - 4 i \, A a^{5} b^{2} - 36 i \, B a^{4} b^{3} + 9 i \, A a^{3} b^{4} + 24 i \, B a^{2} b^{5} - 9 i \, A a b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-16 i \, B a^{7} + 4 i \, A a^{6} b + 36 i \, B a^{5} b^{2} - 9 i \, A a^{4} b^{3} - 24 i \, B a^{3} b^{4} + 9 i \, A a^{2} b^{5}\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^{4} b^{3} + 2 i \, A a^{3} b^{4} + 15 i \, B a^{2} b^{5} - 6 i \, A a b^{6} - 3 i \, B b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-8 i \, B a^{5} b^{2} + 2 i \, A a^{4} b^{3} + 15 i \, B a^{3} b^{4} - 6 i \, A a^{2} b^{5} - 3 i \, B a b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-8 i \, B a^{6} b + 2 i \, A a^{5} b^{2} + 15 i \, B a^{4} b^{3} - 6 i \, A a^{3} b^{4} - 3 i \, B a^{2} b^{5}\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^{4} b^{3} - 2 i \, A a^{3} b^{4} - 15 i \, B a^{2} b^{5} + 6 i \, A a b^{6} + 3 i \, B b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (8 i \, B a^{5} b^{2} - 2 i \, A a^{4} b^{3} - 15 i \, B a^{3} b^{4} + 6 i \, A a^{2} b^{5} + 3 i \, B a b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (8 i \, B a^{6} b - 2 i \, A a^{5} b^{2} - 15 i \, B a^{4} b^{3} + 6 i \, A a^{3} b^{4} + 3 i \, B a^{2} b^{5}\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^{4} b^{6} - 2 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} - 2 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{4} - 2 \, a^{4} b^{6} + a^{2} b^{8}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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