Optimal. Leaf size=510 \[ \frac{2 \cot (c+d x) \left (a^2 b (40 B-36 C)-48 a^3 C-6 a b^2 (5 A-5 B+2 C)-b^3 (15 A-5 B+9 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{15 b^4 d \sqrt{a+b}}-\frac{2 \tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \tan (c+d x) \sec (c+d x) \left (6 a^2 C-5 a b B+5 A b^2-b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left (a^2-b^2\right )}+\frac{2 \tan (c+d x) \left (20 a^2 b B-24 a^3 C-3 a b^2 (5 A-3 C)-5 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{15 b^3 d \left (a^2-b^2\right )}+\frac{2 \cot (c+d x) \left (-6 a^2 b^2 (5 A-4 C)+40 a^3 b B-48 a^4 C-25 a b^3 B+3 b^4 (5 A+3 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b^5 d \sqrt{a+b}} \]
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Rubi [A] time = 1.34519, antiderivative size = 510, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {4098, 4092, 4082, 4005, 3832, 4004} \[ -\frac{2 \tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \tan (c+d x) \sec (c+d x) \left (6 a^2 C-5 a b B+5 A b^2-b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left (a^2-b^2\right )}+\frac{2 \tan (c+d x) \left (20 a^2 b B-24 a^3 C-3 a b^2 (5 A-3 C)-5 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{15 b^3 d \left (a^2-b^2\right )}+\frac{2 \cot (c+d x) \left (a^2 b (40 B-36 C)-48 a^3 C-6 a b^2 (5 A-5 B+2 C)-b^3 (15 A-5 B+9 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b^4 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \left (-6 a^2 b^2 (5 A-4 C)+40 a^3 b B-48 a^4 C-25 a b^3 B+3 b^4 (5 A+3 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b^5 d \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 4098
Rule 4092
Rule 4082
Rule 4005
Rule 3832
Rule 4004
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{2 \int \frac{\sec ^2(c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+\frac{1}{2} b (b B-a (A+C)) \sec (c+d x)-\frac{1}{2} \left (5 A b^2-5 a b B+6 a^2 C-b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (5 A b^2-5 a b B+6 a^2 C-b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}-\frac{4 \int \frac{\sec (c+d x) \left (-\frac{1}{2} a \left (5 A b^2-5 a b B+6 a^2 C-b^2 C\right )+\frac{1}{4} b \left (5 A b^2-5 a b B+2 a^2 C+3 b^2 C\right ) \sec (c+d x)-\frac{1}{4} \left (20 a^2 b B-5 b^3 B-3 a b^2 (5 A-3 C)-24 a^3 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{5 b^2 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (20 a^2 b B-5 b^3 B-3 a b^2 (5 A-3 C)-24 a^3 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{15 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (5 A b^2-5 a b B+6 a^2 C-b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}-\frac{8 \int \frac{\sec (c+d x) \left (\frac{1}{8} b \left (10 a^2 b B+5 b^3 B-12 a^3 C-3 a b^2 (5 A+C)\right )+\frac{1}{8} \left (40 a^3 b B-25 a b^3 B-6 a^2 b^2 (5 A-4 C)-48 a^4 C+3 b^4 (5 A+3 C)\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 b^3 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (20 a^2 b B-5 b^3 B-3 a b^2 (5 A-3 C)-24 a^3 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{15 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (5 A b^2-5 a b B+6 a^2 C-b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}-\frac{\left (40 a^3 b B-25 a b^3 B-6 a^2 b^2 (5 A-4 C)-48 a^4 C+3 b^4 (5 A+3 C)\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 b^3 \left (a^2-b^2\right )}+\frac{\left (a^2 b (40 B-36 C)-48 a^3 C-6 a b^2 (5 A-5 B+2 C)-b^3 (15 A-5 B+9 C)\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 b^3 (a+b)}\\ &=\frac{2 \left (40 a^3 b B-25 a b^3 B-6 a^2 b^2 (5 A-4 C)-48 a^4 C+3 b^4 (5 A+3 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{15 b^5 \sqrt{a+b} d}+\frac{2 \left (a^2 b (40 B-36 C)-48 a^3 C-6 a b^2 (5 A-5 B+2 C)-b^3 (15 A-5 B+9 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{15 b^4 \sqrt{a+b} d}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (20 a^2 b B-5 b^3 B-3 a b^2 (5 A-3 C)-24 a^3 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{15 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (5 A b^2-5 a b B+6 a^2 C-b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 20.8391, size = 874, normalized size = 1.71 \[ \frac{\left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{4 \left (-48 C a^4+40 b B a^3-30 A b^2 a^2+24 b^2 C a^2-25 b^3 B a+15 A b^4+9 b^4 C\right ) \sin (c+d x)}{15 b^4 \left (b^2-a^2\right )}+\frac{4 \sec (c+d x) (5 b B \sin (c+d x)-9 a C \sin (c+d x))}{15 b^3}+\frac{4 \left (C \sin (c+d x) a^4-b B \sin (c+d x) a^3+A b^2 \sin (c+d x) a^2\right )}{b^3 \left (b^2-a^2\right ) (b+a \cos (c+d x))}+\frac{4 C \sec (c+d x) \tan (c+d x)}{5 b^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))^{3/2}}+\frac{4 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sqrt{\frac{1}{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )}} \left ((a+b) \left (48 C a^4-40 b B a^3+6 b^2 (5 A-4 C) a^2+25 b^3 B a-3 b^4 (5 A+3 C)\right ) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )+b (a+b) \left (-48 C a^3+4 b (10 B+9 C) a^2-6 b^2 (5 A+5 B+2 C) a+b^3 (15 A+5 B+9 C)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )+\left (48 C a^4-40 b B a^3+6 b^2 (5 A-4 C) a^2+25 b^3 B a-3 b^4 (5 A+3 C)\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (-b \tan ^4\left (\frac{1}{2} (c+d x)\right )+a \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )-1\right )^2+b\right )\right ) (b+a \cos (c+d x))^{3/2}}{15 b^4 \left (b^2-a^2\right ) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^{3/2} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 1.534, size = 5857, normalized size = 11.5 \begin{align*} \text{output 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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \sec \left (d x + c\right )^{5} + B \sec \left (d x + c\right )^{4} + A \sec \left (d x + c\right )^{3}\right )} \sqrt{b \sec \left (d x + c\right ) + a}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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