Optimal. Leaf size=518 \[ \frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3465 b^3 d}-\frac {2 \left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d} \]
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Rubi [A] time = 1.29, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac {2 \sin (c+d x) \left (88 a^2 b B-48 a^3 C-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac {2 \sin (c+d x) \left (-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3465 b^3 d}-\frac {2 \left (a^2-b^2\right ) \left (-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 3023
Rule 3049
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {2 \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (2 a C+\frac {1}{2} b (11 A+9 C) \cos (c+d x)+\frac {1}{2} (11 b B-6 a C) \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {4 \int (a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} a (11 b B-6 a C)+\frac {1}{4} b (77 b B-6 a C) \cos (c+d x)+\frac {1}{4} \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{99 b^2}\\ &=\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {8 \int (a+b \cos (c+d x))^{3/2} \left (\frac {3}{8} b \left (165 A b^2-22 a b B+12 a^2 C+135 b^2 C\right )+\frac {1}{8} \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) \cos (c+d x)\right ) \, dx}{693 b^3}\\ &=\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {16 \int \sqrt {a+b \cos (c+d x)} \left (-\frac {3}{16} b \left (22 a^2 b B-539 b^3 B-12 a^3 C-3 a b^2 (209 A+157 C)\right )+\frac {3}{16} \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465 b^3}\\ &=\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {32 \int \frac {\frac {3}{32} b \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+75 b^4 (11 A+9 C)+9 a^2 b^2 (187 A+141 C)\right )+\frac {3}{32} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{10395 b^3}\\ &=\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}-\frac {\left (\left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3465 b^4}+\frac {\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3465 b^4}\\ &=\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {\left (\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 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}{3465 b^4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 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}{3465 b^4 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 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 )}{3465 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}\\ \end {align*}
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Mathematica [A] time = 2.72, size = 407, normalized size = 0.79 \[ \frac {b (a+b \cos (c+d x)) \left (b \left (5 b \left (\sin (3 (c+d x)) \left (12 a^2 C+440 a b B+396 A b^2+513 b^2 C\right )+7 b ((24 a C+22 b B) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )+4 \sin (2 (c+d x)) \left (-36 a^3 C+66 a^2 b B+48 a b^2 (33 A+34 C)+1463 b^3 B\right )\right )+2 \sin (c+d x) \left (192 a^4 C-352 a^3 b B+18 a^2 b^2 (44 A+27 C)+8844 a b^3 B+15 b^4 (506 A+435 C)\right )\right )+16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (-12 a^4 C+22 a^3 b B+9 a^2 b^2 (187 A+141 C)+2046 a b^3 B+75 b^4 (11 A+9 C)\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-\left (48 a^5 C-88 a^4 b B+18 a^3 b^2 (11 A+6 C)-363 a^2 b^3 B-6 a b^4 (451 A+348 C)-1617 b^5 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{27720 b^4 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{5} + {\left (C a + B b\right )} \cos \left (d x + c\right )^{4} + A a \cos \left (d x + c\right )^{2} + {\left (B a + A b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.60, size = 2603, normalized size = 5.03 \[ \text {Expression too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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