Optimal. Leaf size=356 \[ \frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^2 d}-\frac {2 a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
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Rubi [A] time = 0.65, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3050, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^2 d}-\frac {2 a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (3 a^2 b^2 (21 A+11 C)+8 a^4 C+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 3023
Rule 3050
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {2 \int (a+b \cos (c+d x))^{3/2} \left (a C+\frac {1}{2} b (9 A+7 C) \cos (c+d x)-2 a C \cos ^2(c+d x)\right ) \, dx}{9 b}\\ &=-\frac {8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {4 \int (a+b \cos (c+d x))^{3/2} \left (-\frac {3}{2} a b C+\frac {1}{4} \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x)\right ) \, dx}{63 b^2}\\ &=\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {8 \int \sqrt {a+b \cos (c+d x)} \left (\frac {3}{8} b \left (63 A b^2-2 a^2 C+49 b^2 C\right )+\frac {3}{8} a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \cos (c+d x)\right ) \, dx}{315 b^2}\\ &=\frac {2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {16 \int \frac {\frac {3}{8} a b \left (126 A b^2+\left (a^2+93 b^2\right ) C\right )+\frac {3}{16} \left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{945 b^2}\\ &=\frac {2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}-\frac {\left (a \left (a^2-b^2\right ) \left (63 A b^2+8 a^2 C+39 b^2 C\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{315 b^3}+\frac {\left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{315 b^3}\\ &=\frac {2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {\left (\left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 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^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a \left (a^2-b^2\right ) \left (63 A b^2+8 a^2 C+39 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}{315 b^3 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 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^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \left (63 A b^2+8 a^2 C+39 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 )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}\\ \end {align*}
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Mathematica [A] time = 1.37, size = 269, normalized size = 0.76 \[ \frac {b (a+b \cos (c+d x)) \left (b \left (2 \left (6 a^2 C+126 A b^2+133 b^2 C\right ) \sin (2 (c+d x))+5 b C (20 a \sin (3 (c+d x))+7 b \sin (4 (c+d x)))\right )-4 a \left (8 a^2 C-252 A b^2-201 b^2 C\right ) \sin (c+d x)\right )+8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (2 a b^2 \left (C \left (a^2+93 b^2\right )+126 A b^2\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{1260 b^3 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{4} + C a \cos \left (d x + c\right )^{3} + A b \cos \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.78, size = 1527, normalized size = 4.29 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
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 \[ \int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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