Optimal. Leaf size=536 \[ -\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-105 a^6 C+45 a^5 b B-a^4 b^2 (9 A-223 C)-99 a^3 b^3 B+a^2 b^4 (15 A-128 C)+72 a b^5 B-8 b^6 (3 A+C)\right )}{12 b^5 d \left (a^2-b^2\right )^2}-\frac {a \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^5 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 1.90, antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3047, 3049, 3059, 2639, 3002, 2641, 2805} \[ -\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-99 a^3 b^3 B+45 a^5 b B-105 a^6 C+72 a b^5 B-8 b^6 (3 A+C)\right )}{12 b^5 d \left (a^2-b^2\right )^2}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac {a \left (a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^5 d (a-b)^2 (a+b)^3}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3002
Rule 3047
Rule 3049
Rule 3059
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {5}{2} \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \cos (c+d x)-\frac {1}{2} \left (3 A b^2-3 a b B+7 a^2 C-4 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {\sqrt {\cos (c+d x)} \left (\frac {3}{4} \left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \cos (c+d x)-\frac {1}{4} \left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {-\frac {1}{8} a \left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right )+\frac {1}{2} b \left (3 a^3 b B-12 a b^3 B-7 a^4 C+2 b^4 (3 A+C)+a^2 b^2 (3 A+14 C)\right ) \cos (c+d x)+\frac {3}{8} \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\int \frac {\frac {1}{8} a b \left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right )+\frac {1}{8} \left (45 a^5 b B-99 a^3 b^3 B+72 a b^5 B-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-105 a^6 C-8 b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 b^4 \left (a^2-b^2\right )^2}+\frac {\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (a \left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 b^5 \left (a^2-b^2\right )^2}-\frac {\left (45 a^5 b B-99 a^3 b^3 B+72 a b^5 B-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-105 a^6 C-8 b^6 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 b^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (45 a^5 b B-99 a^3 b^3 B+72 a b^5 B-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-105 a^6 C-8 b^6 (3 A+C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 b^5 \left (a^2-b^2\right )^2 d}-\frac {a \left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 (a-b)^2 b^5 (a+b)^3 d}-\frac {\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.24, size = 520, normalized size = 0.97 \[ \frac {\frac {4 \sin (c+d x) \sqrt {\cos (c+d x)} \left (35 a^6 C-15 a^5 b B+3 a^4 A b^2-57 a^4 b^2 C+33 a^3 b^3 B-21 a^2 A b^4+4 C \left (b^3-a^2 b\right )^2 \cos (2 (c+d x))+a b \cos (c+d x) \left (49 a^4 C-21 a^3 b B+a^2 b^2 (9 A-83 C)+39 a b^3 B+b^4 (16 C-27 A)\right )+4 b^6 C\right )}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {-\frac {16 \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (3 A+14 C)-12 a b^3 B+2 b^4 (3 A+C)\right ) \left ((a+b) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{a+b}+\frac {2 \left (35 a^5 C-15 a^4 b B+a^3 b^2 (3 A-73 C)+21 a^2 b^3 B+a b^4 (15 A+56 C)-24 b^5 B\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {6 \sin (c+d x) \left (35 a^5 C-15 a^4 b B+a^3 b^2 (3 A-65 C)+29 a^2 b^3 B+3 a b^4 (8 C-3 A)-8 b^5 B\right ) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac {b}{a};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a b^2 \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 b^3 d} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 15.39, size = 2267, normalized size = 4.23 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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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