Optimal. Leaf size=507 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)+16 a^3 b^2 B+2 a^5 B-15 a b^4 B+21 A b^5\right )}{3 a^5 d \left (a^2-b^2\right )}+\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{\sin (c+d x) \left (a^2 (-(2 A-5 C))-5 a b B+7 A b^2\right )}{5 a^2 d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left (-a^2 (4 A b-3 b C)+2 a^3 B-5 a b^2 B+7 A b^3\right )}{3 a^3 d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+20 a^3 b B-25 a b^3 B+35 A b^4\right )}{5 a^4 d \left (a^2-b^2\right )}-\frac{b^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (-3 a^2 b^2 (3 A-C)+7 a^3 b B-5 a^4 C-5 a b^3 B+7 A b^4\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^5 d (a-b) (a+b)^2} \]
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Rubi [A] time = 1.51598, antiderivative size = 507, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{\sin (c+d x) \left (a^2 (-(2 A-5 C))-5 a b B+7 A b^2\right )}{5 a^2 d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left (-a^2 (4 A b-3 b C)+2 a^3 B-5 a b^2 B+7 A b^3\right )}{3 a^3 d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)+16 a^3 b^2 B+2 a^5 B-15 a b^4 B+21 A b^5\right )}{3 a^5 d \left (a^2-b^2\right )}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+20 a^3 b B-25 a b^3 B+35 A b^4\right )}{5 a^4 d \left (a^2-b^2\right )}-\frac{b^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (-3 a^2 b^2 (3 A-C)+7 a^3 b B-5 a^4 C-5 a b^3 B+7 A b^4\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^5 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4100
Rule 4104
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{\int \frac{\frac{1}{2} \left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right )+a (A b-a B+b C) \sec (c+d x)-\frac{5}{2} \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac{\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}+\frac{2 \int \frac{\frac{5}{4} \left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right )+\frac{1}{2} a \left (2 A b^2-5 a b B+a^2 (3 A+5 C)\right ) \sec (c+d x)-\frac{3}{4} b \left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{5 a^2 \left (a^2-b^2\right )}\\ &=-\frac{\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{4 \int \frac{\frac{3}{8} \left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right )+\frac{1}{4} a \left (14 A b^3-5 a^3 B-10 a b^2 B+a^2 b (A+15 C)\right ) \sec (c+d x)-\frac{5}{8} b \left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{15 a^3 \left (a^2-b^2\right )}\\ &=-\frac{\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{4 \int \frac{\frac{3}{8} a \left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right )-\left (\frac{3}{8} b \left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right )-\frac{1}{4} a^2 \left (14 A b^3-5 a^3 B-10 a b^2 B+a^2 b (A+15 C)\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )}-\frac{\left (b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )}\\ &=-\frac{\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}+\frac{\left (21 A b^5+2 a^5 B+16 a^3 b^2 B-15 a b^4 B-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)\right ) \int \sqrt{\sec (c+d x)} \, dx}{6 a^5 \left (a^2-b^2\right )}-\frac{\left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{10 a^4 \left (a^2-b^2\right )}-\frac{\left (b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{2 a^5 \left (a^2-b^2\right )}\\ &=-\frac{b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^5 (a-b) (a+b)^2 d}-\frac{\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}+\frac{\left (\left (21 A b^5+2 a^5 B+16 a^3 b^2 B-15 a b^4 B-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a^5 \left (a^2-b^2\right )}-\frac{\left (\left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a^4 \left (a^2-b^2\right )}\\ &=-\frac{\left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a^4 \left (a^2-b^2\right ) d}+\frac{\left (21 A b^5+2 a^5 B+16 a^3 b^2 B-15 a b^4 B-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a^5 \left (a^2-b^2\right ) d}-\frac{b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^5 (a-b) (a+b)^2 d}-\frac{\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 7.59615, size = 976, normalized size = 1.93 \[ \frac{\left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{2 \left (-20 B a^4+4 A b a^3+60 b C a^3-40 b^2 B a^2+56 A b^3 a\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (-18 A a^4-30 C a^4+40 b B a^3-32 A b^2 a^2+15 b^2 C a^2-25 b^3 B a+35 A b^4\right ) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}-\frac{2 \left (-18 A a^4-30 C a^4+60 b B a^3-72 A b^2 a^2+45 b^2 C a^2-75 b^3 B a+105 A b^4\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (\Pi \left (-\frac{b}{a};\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 b \sec ^2(c+d x) a+2 b a+2 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+(a-2 b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 b^2 \Pi \left (-\frac{b}{a};\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^2 b (b+a \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 )}\right ) (b+a \cos (c+d x))^2}{30 a^3 (b-a) (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{\left (A a^4-A b^2 a^2+10 b^2 C a^2-10 b^3 B a+10 A b^4\right ) \sin (c+d x)}{5 a^4 \left (a^2-b^2\right )}-\frac{2 \left (A \sin (c+d x) b^5-a B \sin (c+d x) b^4+a^2 C \sin (c+d x) b^3\right )}{a^4 \left (a^2-b^2\right ) (b+a \cos (c+d x))}+\frac{2 (a B-2 A b) \sin (2 (c+d x))}{3 a^3}+\frac{A \sin (3 (c+d x))}{5 a^2}\right ) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 8.982, size = 1377, normalized size = 2.7 \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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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