Optimal. Leaf size=469 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 C+3 a b^3 B+A b^4\right )}{4 a^2 b d \left (a^2-b^2\right )^2}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right )}{4 a b^2 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 C+3 a b^5 B+A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 b^2 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 1.17975, antiderivative size = 469, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 C+3 a b^3 B+A b^4\right )}{4 a^2 b d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right )}{4 a b^2 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 C+3 a b^5 B+A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 b^2 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 4098
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{\int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{2} \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)+\frac{1}{2} \left (A b^2-a b B-3 a^2 C+4 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\frac{1}{4} \left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sec (c+d x)+\frac{1}{4} \left (a^3 b B-7 a b^3 B+a^2 b^2 (3 A-5 C)+3 a^4 C+b^4 (3 A+8 C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\frac{1}{4} a \left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right )-\left (-a b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right )+\frac{1}{4} b \left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{2 a^2 b^2 \left (a^2-b^2\right )^2}-\frac{\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 C)\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 a^2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \int \sqrt{\sec (c+d x)} \, dx}{8 a^2 b \left (a^2-b^2\right )^2}-\frac{\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 a b^2 \left (a^2-b^2\right )^2}-\frac{\left (\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 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}{8 a^2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 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)}}{4 a^2 (a-b)^2 b^2 (a+b)^3 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\left (\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{8 a^2 b \left (a^2-b^2\right )^2}-\frac{\left (\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a b^2 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (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 a^2 b \left (a^2-b^2\right )^2 d}-\frac{\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 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)}}{4 a^2 (a-b)^2 b^2 (a+b)^3 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [B] time = 7.23851, size = 1051, normalized size = 2.24 \[ \frac{\sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{2 \left (16 B b^4-24 a A b^3-32 a C b^3+8 a^2 B b^2+8 a^3 C b\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 (9 C a^4+3 b B a^3+A b^2 a^2-19 b^2 C a^2-9 b^3 B a+5 A b^4+16 b^4 C\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 (3 C a^4+b B a^3-5 A b^2 a^2-9 b^2 C a^2+5 b^3 B a-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))^3}{8 (a-b)^2 b^2 (a+b)^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}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{\left (-3 C a^4-b B a^3+5 A b^2 a^2+9 b^2 C a^2-5 b^3 B a+A b^4\right ) \sin (c+d x)}{2 a b^2 \left (b^2-a^2\right )^2}+\frac{C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)}{a \left (a^2-b^2\right ) (b+a \cos (c+d x))^2}+\frac{C \sin (c+d x) a^4+3 b B \sin (c+d x) a^3-7 A b^2 \sin (c+d x) a^2-7 b^2 C \sin (c+d x) a^2+3 b^3 B \sin (c+d x) a+A b^4 \sin (c+d x)}{2 a b \left (b^2-a^2\right )^2 (b+a \cos (c+d x))}\right ) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 10.419, size = 1879, 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(-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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \sec \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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