Optimal. Leaf size=280 \[ \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 C-6 a b B+3 A b^2+b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-8 a^3 C+6 a^2 b B-a b^2 (3 A-5 C)-3 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 C \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b^2 d} \]
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Rubi [A] time = 0.52, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3031, 3023, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 C-6 a b B+3 A b^2+b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (6 a^2 b B-8 a^3 C-a b^2 (3 A-5 C)-3 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 C \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 3023
Rule 3031
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx &=\frac {2 a \left (A b^2-a (b B-a C)\right ) \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} b \left (A b^2-a (b B-a C)\right )+\frac {1}{2} \left (2 a^2 b B-b^3 B-a b^2 (A-C)-2 a^3 C\right ) \cos (c+d x)+\frac {1}{2} b \left (a^2-b^2\right ) C \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=\frac {2 a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \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 b^2-3 a b B+2 a^2 C+b^2 C\right )+\frac {1}{4} b \left (6 a^2 b B-3 b^3 B-a b^2 (3 A-5 C)-8 a^3 C\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 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {\left (6 a^2 b B-3 b^3 B-a b^2 (3 A-5 C)-8 a^3 C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3 b^3 \left (a^2-b^2\right )}+\frac {\left (3 A b^2-6 a b B+8 a^2 C+b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3}\\ &=\frac {2 a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {\left (\left (6 a^2 b B-3 b^3 B-a b^2 (3 A-5 C)-8 a^3 C\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 (3 A b^2-6 a b B+8 a^2 C+b^2 C\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 b B-3 b^3 B-a b^2 (3 A-5 C)-8 a^3 C\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 (3 A b^2-6 a b B+8 a^2 C+b^2 C\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 \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}\\ \end {align*}
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Mathematica [A] time = 1.79, size = 236, normalized size = 0.84 \[ \frac {2 \left (b \sin (c+d x) \left (\frac {a \left (-4 a^2 C+3 a b B-3 A b^2+b^2 C\right )}{b^2-a^2}+b C \cos (c+d x)\right )-\frac {\sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (2 a^2 C-3 a b B+3 A b^2+b^2 C\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (8 a^3 C-6 a^2 b B+a b^2 (3 A-5 C)+3 b^3 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{(a-b) (a+b)}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{3} + B \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 8.51, size = 1036, normalized size = 3.70 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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