Optimal. Leaf size=378 \[ -\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^2 d}+\frac {2 \left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
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Rubi [A] time = 0.79, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {3029, 2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac {2 \left (18 a^2 b B-8 a^3 C-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^2 d}+\frac {2 \left (a^2-b^2\right ) \left (18 a^2 b B-8 a^3 C-39 a b^2 C-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-33 a^2 b^2 C+18 a^3 b B-8 a^4 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2990
Rule 3023
Rule 3029
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (B+C \cos (c+d x)) \, 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 {7}{2} b C \cos (c+d x)+\frac {1}{2} (9 b B-4 a C) \cos ^2(c+d x)\right ) \, dx}{9 b}\\ &=\frac {2 (9 b B-4 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}{4} b (15 b B-2 a C)-\frac {1}{4} \left (18 a b B-8 a^2 C-49 b^2 C\right ) \cos (c+d x)\right ) \, dx}{63 b^2}\\ &=-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 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 (57 a b B-2 a^2 C+49 b^2 C\right )-\frac {3}{8} \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \cos (c+d x)\right ) \, dx}{315 b^2}\\ &=-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 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}{16} b \left (153 a^2 b B+75 b^3 B+2 a^3 C+186 a b^2 C\right )-\frac {3}{16} \left (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{945 b^2}\\ &=-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 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 (a^2-b^2\right ) \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{315 b^3}-\frac {\left (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{315 b^3}\\ &=-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 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 (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 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 (\left (a^2-b^2\right ) \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a 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 (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 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 \left (a^2-b^2\right ) \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a 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 (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 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 = 1.52, size = 291, normalized size = 0.77 \[ \frac {b (a+b \cos (c+d x)) \left (b \left (2 \left (6 a^2 C+144 a b B+133 b^2 C\right ) \sin (2 (c+d x))+5 b (2 (10 a C+9 b B) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right )+\left (-32 a^3 C+72 a^2 b B+804 a b^2 C+690 b^3 B\right ) \sin (c+d x)\right )+8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (2 a^3 C+153 a^2 b B+186 a b^2 C+75 b^3 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (8 a^4 C-18 a^3 b B+33 a^2 b^2 C+246 a b^3 B+147 b^4 C\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{1260 b^3 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{4} + B a \cos \left (d x + c\right )^{2} + {\left (C a + B b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.83, size = 1635, normalized size = 4.33 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
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 \[ \int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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