Optimal. Leaf size=622 \[ -\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}+\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}+\frac {2 \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 1.67, antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-2 a^2 b^2 (A-6 C)+5 a^3 b B-8 a^4 C-9 a b^3 B+6 A b^4\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (15 A-71 C)+30 a^3 b B-48 a^4 C-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}+\frac {2 \sin (c+d x) \left (-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+40 a^4 b B-64 a^5 C+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}+\frac {2 \left (-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+80 a^4 b B-128 a^5 C+a b^4 (45 A+17 C)-5 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-140 a^3 b^3 B+80 a^5 b B-128 a^6 C+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 3023
Rule 3047
Rule 3049
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 \int \frac {\cos ^2(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+\frac {3}{2} b (b B-a (A+C)) \cos (c+d x)-\frac {1}{2} \left (5 A b^2-5 a b B+8 a^2 C-3 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {4 \int \frac {\cos (c+d x) \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C+\frac {1}{4} b \left (a^2 b B+3 b^3 B+2 a^3 C-2 a b^2 (2 A+3 C)\right ) \cos (c+d x)-\frac {1}{4} \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}+\frac {8 \int \frac {-\frac {1}{4} a \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right )+\frac {1}{8} b \left (10 a^3 b B-30 a b^3 B-16 a^4 C+3 b^4 (5 A+3 C)+a^2 b^2 (5 A+27 C)\right ) \cos (c+d x)+\frac {3}{8} \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac {2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}+\frac {16 \int \frac {-\frac {3}{16} b \left (20 a^4 b B-35 a^2 b^3 B-5 b^5 B-2 a^3 b^2 (5 A-22 C)-32 a^5 C+2 a b^4 (15 A+4 C)\right )-\frac {3}{16} \left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{45 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac {2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{15 b^5 \left (a^2-b^2\right )^2}+\frac {\left (80 a^4 b B-80 a^2 b^3 B-5 b^5 B-4 a^3 b^2 (10 A-29 C)-128 a^5 C+a b^4 (45 A+17 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 b^5 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac {2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (\left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 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 b^5 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (80 a^4 b B-80 a^2 b^3 B-5 b^5 B-4 a^3 b^2 (10 A-29 C)-128 a^5 C+a b^4 (45 A+17 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}{15 b^5 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\\ &=-\frac {2 \left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (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^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (80 a^4 b B-80 a^2 b^3 B-5 b^5 B-4 a^3 b^2 (10 A-29 C)-128 a^5 C+a b^4 (45 A+17 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 )}{15 b^5 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac {2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}\\ \end {align*}
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Mathematica [A] time = 5.78, size = 422, normalized size = 0.68 \[ \frac {b \left (\frac {10 a^3 \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{a^2-b^2}-\frac {10 a^2 \sin (c+d x) \left (11 a^4 C-8 a^3 b B+5 a^2 b^2 (A-3 C)+12 a b^3 B-9 A b^4\right ) (a+b \cos (c+d x))}{\left (a^2-b^2\right )^2}+2 (5 b B-14 a C) \sin (c+d x) (a+b \cos (c+d x))^2+3 b C \sin (2 (c+d x)) (a+b \cos (c+d x))^2\right )+\frac {2 \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (32 a^5 C-20 a^4 b B+2 a^3 b^2 (5 A-22 C)+35 a^2 b^3 B-2 a b^4 (15 A+4 C)+5 b^5 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (128 a^6 C-80 a^5 b B+4 a^4 b^2 (10 A-53 C)+140 a^3 b^3 B+5 a^2 b^4 (11 C-15 A)-40 a b^5 B+3 b^6 (5 A+3 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}}{15 b^5 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{5} + B \cos \left (d x + c\right )^{4} + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 17.58, size = 1780, normalized size = 2.86 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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