Optimal. Leaf size=654 \[ \frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right )}{12 b^6 d \left (a^2-b^2\right )^2}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right )}{20 b^3 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left (-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{12 b^4 d \left (a^2-b^2\right )^2} \]
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Rubi [A] time = 2.63888, antiderivative size = 654, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right )}{12 b^6 d \left (a^2-b^2\right )^2}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right )}{20 b^3 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left (-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{12 b^4 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3047
Rule 3049
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{7}{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{7}{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{5}{2}}(c+d x) \left (\frac{7}{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 (5 A b^2-5 a b B+9 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{7}{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 (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac{5}{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{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{5}{4} \left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 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 (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 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 (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^{\frac{7}{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 (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac{5}{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}{8} a \left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right )+\frac{1}{2} b \left (5 a^3 b B-20 a b^3 B-9 a^4 C+2 b^4 (5 A+3 C)+a^2 b^2 (5 A+18 C)\right ) \cos (c+d x)+\frac{5}{8} \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{5 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^{\frac{7}{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 (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac{5}{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{2 \int \frac{\frac{5}{16} a \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right )-\frac{1}{4} b \left (35 a^4 b B-70 a^2 b^3 B-10 b^5 B-3 a^3 b^2 (5 A-32 C)-63 a^5 C+12 a b^4 (5 A+C)\right ) \cos (c+d x)-\frac{3}{16} \left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^{\frac{7}{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 (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac{5}{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{2 \int \frac{-\frac{5}{16} a b \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right )-\frac{5}{16} \left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^5 \left (a^2-b^2\right )^2}-\frac{\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \int \sqrt{\cos (c+d x)} \, dx}{40 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^{\frac{7}{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 (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac{5}{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^2 \left (35 A b^6-35 a^5 b B+86 a^3 b^3 B-63 a b^5 B-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 b^6 \left (a^2-b^2\right )^2}+\frac{\left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 b^6 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac{\left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 b^6 \left (a^2-b^2\right )^2 d}+\frac{a^2 \left (35 A b^6-35 a^5 b B+86 a^3 b^3 B-63 a b^5 B-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 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^6 (a+b)^3 d}+\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^{\frac{7}{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 (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac{5}{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*}
Mathematica [A] time = 7.4406, size = 555, normalized size = 0.85 \[ \frac{4 \sqrt{\cos (c+d x)} \left (\frac{30 a^3 \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{\left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac{15 a^2 \sin (c+d x) \left (7 a^2 b^2 (A-3 C)-11 a^3 b B+15 a^4 C+17 a b^3 B-13 A b^4\right )}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+40 (b B-3 a C) \sin (c+d x)+12 b C \sin (2 (c+d x))\right )+\frac{\frac{2 \left (3 a^4 b^2 (25 A-211 C)-21 a^2 b^4 (5 A-16 C)+365 a^3 b^3 B-175 a^5 b B+315 a^6 C-280 a b^5 B+24 b^6 (5 A+3 C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{16 \left (3 a^3 b^2 (5 A-32 C)+70 a^2 b^3 B-35 a^4 b B+63 a^5 C-12 a b^4 (5 A+C)+10 b^5 B\right ) \left ((a+b) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{a+b}+\frac{6 \sin (c+d x) \left (3 a^4 b^2 (25 A-187 C)+a^2 b^4 (192 C-145 A)+325 a^3 b^3 B-175 a^5 b B+315 a^6 C-120 a b^5 B+8 b^6 (5 A+3 C)\right ) \left (\left (2 a^2-b^2\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{240 b^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.424, size = 2520, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{7}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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