Optimal. Leaf size=378 \[ -\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^4 d}+\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^3 d}-\frac {2 a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{63 b^2 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d} \]
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Rubi [A] time = 0.93, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3050, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^3 d}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^4 d}-\frac {2 a \left (4 a^2 b^2 (42 A+19 C)+128 a^4 C+3 b^4 (49 A+37 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (12 a^2 b^2 (14 A+9 C)+128 a^4 C+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{63 b^2 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 3023
Rule 3049
Rule 3050
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx &=\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac {2 \int \frac {\cos ^2(c+d x) \left (3 a C+\frac {1}{2} b (9 A+7 C) \cos (c+d x)-4 a C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx}{9 b}\\ &=-\frac {16 a C \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac {4 \int \frac {\cos (c+d x) \left (-8 a^2 C+\frac {1}{2} a b C \cos (c+d x)+\frac {1}{4} \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx}{63 b^2}\\ &=\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac {16 a C \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac {8 \int \frac {\frac {1}{4} a \left (48 a^2 C+7 b^2 (9 A+7 C)\right )+\frac {1}{8} b \left (189 A b^2-16 a^2 C+147 b^2 C\right ) \cos (c+d x)-\frac {3}{4} a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{315 b^3}\\ &=-\frac {4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac {16 a C \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac {16 \int \frac {\frac {3}{8} a b \left (21 A b^2+16 a^2 C+18 b^2 C\right )+\frac {3}{16} \left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{945 b^4}\\ &=-\frac {4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac {16 a C \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac {\left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{315 b^5}-\frac {\left (a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{315 b^5}\\ &=-\frac {4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac {16 a C \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac {\left (\left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 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}{315 b^5 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 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}{315 b^5 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 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 )}{315 b^5 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac {16 a C \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d}\\ \end {align*}
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Mathematica [A] time = 1.45, size = 272, normalized size = 0.72 \[ \frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (32 a^3 b C+6 a b^3 (7 A+6 C)\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )-b (a+b \cos (c+d x)) \left (32 a \left (2 C \left (8 a^2+9 b^2\right )+21 A b^2\right ) \sin (c+d x)-b \left (2 \left (96 a^2 C+126 A b^2+133 b^2 C\right ) \sin (2 (c+d x))+5 b C (7 b \sin (4 (c+d x))-16 a \sin (3 (c+d x)))\right )\right )}{1260 b^5 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.56, size = 1527, normalized size = 4.04 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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