Optimal. Leaf size=502 \[ -\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right )}{4 a^2 b d \left (a^2-b^2\right )^2}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right )}{4 a^3 d \left (a^2-b^2\right )^2}-\frac{\left (5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 C-3 a b^5 B+15 A b^6\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 b d (a-b)^2 (a+b)^3}+\frac{\sin (c+d x) \left (-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right )}{4 a^3 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left (-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right )}{4 a^2 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \]
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Rubi [A] time = 1.85821, antiderivative size = 502, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3055, 3059, 2639, 3002, 2641, 2805} \[ -\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right )}{4 a^2 b d \left (a^2-b^2\right )^2}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right )}{4 a^3 d \left (a^2-b^2\right )^2}-\frac{\left (5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 C-3 a b^5 B+15 A b^6\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 b d (a-b)^2 (a+b)^3}+\frac{\sin (c+d x) \left (-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right )}{4 a^3 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left (-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right )}{4 a^2 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3055
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}+\frac{\int \frac{\frac{1}{2} \left (-5 A b^2+a b B+a^2 (4 A-C)\right )-2 a (A b-a B+b C) \cos (c+d x)+\frac{3}{2} \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\int \frac{\frac{1}{4} \left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right )+a \left (A b^3+2 a^3 B+a b^2 B-a^2 b (4 A+3 C)\right ) \cos (c+d x)-\frac{1}{4} \left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\int \frac{\frac{1}{8} \left (-15 A b^5+8 a^5 B-5 a^3 b^2 B+3 a b^4 B+a^2 b^3 (33 A+C)-a^4 b (24 A+7 C)\right )-\frac{1}{2} a \left (5 A b^4+4 a^3 b B-a b^3 B+2 a^4 (A-C)-a^2 b^2 (10 A+C)\right ) \cos (c+d x)-\frac{1}{8} b \left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}-\frac{\int \frac{\frac{1}{8} b \left (15 A b^5-8 a^5 B+5 a^3 b^2 B-3 a b^4 B-a^2 b^3 (33 A+C)+a^4 b (24 A+7 C)\right )+\frac{1}{8} a b \left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^3 b \left (a^2-b^2\right )^2}-\frac{\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}-\frac{\left (15 A b^6-15 a^5 b B+6 a^3 b^3 B-3 a b^5 B+3 a^6 C-a^2 b^4 (38 A+C)+5 a^4 b^2 (7 A+2 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 a^3 b \left (a^2-b^2\right )^2}-\frac{\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{8 a^2 b \left (a^2-b^2\right )^2}\\ &=-\frac{\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 \left (a^2-b^2\right )^2 d}-\frac{\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 b \left (a^2-b^2\right )^2 d}-\frac{\left (15 A b^6-15 a^5 b B+6 a^3 b^3 B-3 a b^5 B+3 a^6 C-a^2 b^4 (38 A+C)+5 a^4 b^2 (7 A+2 C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^3 (a-b)^2 b (a+b)^3 d}+\frac{\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 7.03742, size = 510, normalized size = 1.02 \[ \frac{\frac{2 \sqrt{\cos (c+d x)} \left (b^2 \sin (2 (c+d x)) \left (-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right )+2 a b \sin (c+d x) \left (a^2 b^2 (C-47 A)+a^4 (16 A-7 C)+11 a^3 b B-5 a b^3 B+25 A b^4\right )+16 A \left (a^3-a b^2\right )^2 \tan (c+d x)\right )}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{-\frac{2 \left (a^2 b^3 (95 A+3 C)-a^4 b (56 A+9 C)-19 a^3 b^2 B+16 a^5 B+9 a b^4 B-45 A b^5\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{16 a \left (-a^2 b^2 (10 A+C)+2 a^4 (A-C)+4 a^3 b B-a b^3 B+5 A b^4\right ) \left ((a+b) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{b (a+b)}+\frac{2 \sin (c+d x) \left (-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right ) \left (\left (2 a^2-b^2\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.839, size = 2027, normalized size = 4. \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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