Optimal. Leaf size=218 \[ -\frac {2 \left (2 a^2 C-3 b^2 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a C \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac {4 a C \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{15 b d} \]
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Rubi [A] time = 0.33, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3024, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (2 a^2 C-3 b^2 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a C \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac {4 a C \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{15 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 3024
Rubi steps
\begin {align*} \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac {2 \int \sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} b (5 A+3 C)-a C \cos (c+d x)\right ) \, dx}{5 b}\\ &=-\frac {4 a C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac {2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac {4 \int \frac {\frac {1}{4} a b (15 A+7 C)-\frac {1}{4} \left (2 a^2 C-3 b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 b}\\ &=-\frac {4 a C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac {2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac {\left (2 a \left (a^2-b^2\right ) C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 b^2}+\frac {1}{15} \left (15 A+\left (9-\frac {2 a^2}{b^2}\right ) C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx\\ &=-\frac {4 a C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac {2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac {\left (\left (15 A+\left (9-\frac {2 a^2}{b^2}\right ) 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}{15 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (2 a \left (a^2-b^2\right ) C \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^2 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (15 A+\left (9-\frac {2 a^2}{b^2}\right ) C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a \left (a^2-b^2\right ) C \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac {2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 181, normalized size = 0.83 \[ \frac {-2 (a+b) \left (2 a^2 C-15 A b^2-9 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+b C \sin (c+d x) \left (2 a^2+8 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right )+4 a C \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.45, size = 821, normalized size = 3.77 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sqrt {a + b \cos {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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