Optimal. Leaf size=579 \[ \frac{\left (a^2 b^2 (15 A-101 C)+63 a^4 C-b^4 (45 A-8 C)\right ) \sin (c+d x)}{20 b^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left (-5 a^2 b^2 (A-7 C)-21 a^4 C+b^4 (11 A-8 C)\right ) \sin (c+d x)}{4 b^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}+\frac{\left (-a^2 b^2 (A-15 C)-9 a^4 C+7 A b^4\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{a \left (-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d \left (a^2-b^2\right )^2}-\frac{\left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+63 a^6 C+35 A b^6\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 2.42712, antiderivative size = 579, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {4221, 3048, 3047, 3049, 3059, 2639, 3002, 2641, 2805} \[ \frac{\left (a^2 b^2 (15 A-101 C)+63 a^4 C-b^4 (45 A-8 C)\right ) \sin (c+d x)}{20 b^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left (-5 a^2 b^2 (A-7 C)-21 a^4 C+b^4 (11 A-8 C)\right ) \sin (c+d x)}{4 b^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}+\frac{\left (-a^2 b^2 (A-15 C)-9 a^4 C+7 A b^4\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{a \left (-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d \left (a^2-b^2\right )^2}-\frac{\left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+63 a^6 C+35 A b^6\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3048
Rule 3047
Rule 3049
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (\frac{7}{2} \left (A b^2+a^2 C\right )-2 a b (A+C) \cos (c+d x)-\frac{1}{2} \left (5 A b^2+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^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{5}{4} \left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right )-a b \left (3 A b^2-\left (a^2-4 b^2\right ) C\right ) \cos (c+d x)+\frac{1}{4} \left (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 (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (\frac{3}{8} a \left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right )-\frac{1}{2} b \left (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{15}{8} a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 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 (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{15}{16} a^2 \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right )+\frac{3}{4} a b \left (a^2 b^2 (5 A-32 C)+21 a^4 C-4 b^4 (5 A+C)\right ) \cos (c+d x)-\frac{3}{16} \left (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 (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{15}{16} a^2 b \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right )-\frac{15}{16} a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 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}{15 b^5 \left (a^2-b^2\right )^2}-\frac{\left (\left (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 ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{40 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (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 ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{20 b^5 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (a^2 \left (35 A b^6-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\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 (a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{8 b^6 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (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 ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac{a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 b^6 \left (a^2-b^2\right )^2 d}+\frac{a^2 \left (35 A b^6-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 (a-b)^2 b^6 (a+b)^3 d}-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 7.42168, size = 931, normalized size = 1.61 \[ \frac{-\frac{2 \left (168 b C a^5+40 A b^3 a^3-256 b^3 C a^3-160 A b^5 a-32 b^5 C a\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (105 C a^6+25 A b^2 a^4-211 b^2 C a^4-35 A b^4 a^2+112 b^4 C a^2+40 A b^6+24 b^6 C\right ) \left (F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{\left (315 C a^6+75 A b^2 a^4-561 b^2 C a^4-145 A b^4 a^2+192 b^4 C a^2+40 A b^6+24 b^6 C\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (4 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}-2 b^2 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{80 (a-b)^2 b^4 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left (-\frac{\left (75 C a^6+35 A b^2 a^4-107 b^2 C a^4-65 A b^4 a^2+4 b^4 C a^2-2 b^6 C\right ) \sin (c+d x)}{20 b^5 \left (a^2-b^2\right )^2}-\frac{-C \sin (c+d x) a^6-A b^2 \sin (c+d x) a^4}{2 b^5 \left (b^2-a^2\right ) (a+b \cos (c+d x))^2}+\frac{17 C \sin (c+d x) a^7+9 A b^2 \sin (c+d x) a^5-23 b^2 C \sin (c+d x) a^5-15 A b^4 \sin (c+d x) a^3}{4 b^5 \left (b^2-a^2\right )^2 (a+b \cos (c+d x))}-\frac{a C \sin (2 (c+d x))}{b^4}+\frac{C \sin (3 (c+d x))}{10 b^3}\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 7.436, size = 2466, normalized size = 4.3 \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{C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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