Optimal. Leaf size=321 \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac{3 b^2 \left (3 a^2-b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}-\frac{\left (7 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a d \left (a^2-b^2\right )^2}-\frac{3 b \left (3 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d \left (a^2-b^2\right )^2}+\frac{3 \left (-2 a^2 b^2+5 a^4+b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 0.770272, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3238, 3845, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac{3 b^2 \left (3 a^2-b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}-\frac{\left (7 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a d \left (a^2-b^2\right )^2}-\frac{3 b \left (3 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d \left (a^2-b^2\right )^2}+\frac{3 \left (-2 a^2 b^2+5 a^4+b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^2 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 3845
Rule 4098
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx &=\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(b+a \sec (c+d x))^3} \, dx\\ &=\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (\frac{b^2}{2}-2 a b \sec (c+d x)+\frac{1}{2} \left (4 a^2-3 b^2\right ) \sec ^2(c+d x)\right )}{(b+a \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{3 b^2 \left (3 a^2-b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\int \frac{-\frac{3}{4} b^2 \left (3 a^2-b^2\right )-a b \left (4 a^2-b^2\right ) \sec (c+d x)+\frac{1}{4} \left (8 a^4-5 a^2 b^2+3 b^4\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{3 b^2 \left (3 a^2-b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac{\int \frac{-\frac{3}{4} b^3 \left (3 a^2-b^2\right )-\left (-\frac{3}{4} a b^2 \left (3 a^2-b^2\right )+a b^2 \left (4 a^2-b^2\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 (3 \left (5 a^4-2 a^2 b^2+b^4\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{8 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{3 b^2 \left (3 a^2-b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}-\frac{\left (3 b \left (3 a^2-b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 a^2 \left (a^2-b^2\right )^2}-\frac{\left (7 a^2-b^2\right ) \int \sqrt{\sec (c+d x)} \, dx}{8 a \left (a^2-b^2\right )^2}+\frac{\left (3 \left (5 a^4-2 a^2 b^2+b^4\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 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{3 \left (5 a^4-2 a^2 b^2+b^4\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^2 (a-b)^2 (a+b)^3 d}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{3 b^2 \left (3 a^2-b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}-\frac{\left (3 b \left (3 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a^2 \left (a^2-b^2\right )^2}-\frac{\left (\left (7 a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{8 a \left (a^2-b^2\right )^2}\\ &=-\frac{3 b \left (3 a^2-b^2\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^2 \left (a^2-b^2\right )^2 d}-\frac{\left (7 a^2-b^2\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 \left (a^2-b^2\right )^2 d}+\frac{3 \left (5 a^4-2 a^2 b^2+b^4\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^2 (a-b)^2 (a+b)^3 d}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac{3 b^2 \left (3 a^2-b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}\\ \end{align*}
Mathematica [B] time = 6.66215, size = 700, normalized size = 2.18 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{3 b \left (3 a^2-b^2\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2}-\frac{b \sin (c+d x)}{2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{b^3 \sin (c+d x)-7 a^2 b \sin (c+d x)}{4 a \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}\right )}{d}+\frac{-\frac{2 \left (8 a b^3-32 a^3 b\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{b \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{2 \left (-19 a^2 b^2+16 a^4+9 b^4\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left (\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{a \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{\left (3 b^4-9 a^2 b^2\right ) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left (4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b\right )}{a b^2 \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a+b \cos (c+d x))}}{16 a^2 d (a-b)^2 (a+b)^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 7.448, size = 1176, normalized size = 3.7 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sec{\left (c + d x \right )}}}{\left (a + b \cos{\left (c + d x \right )}\right )^{3}}\, 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{\sqrt{\sec \left (d x + c\right )}}{{\left (b \cos \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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