Optimal. Leaf size=315 \[ \frac{\left (a^2-7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{4 b d \left (a^2-b^2\right )^2}-\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{3 a \left (a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 d \left (a^2-b^2\right )^2}+\frac{3 \left (-2 a^2 b^2+a^4+5 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 0.703575, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3845, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\left (a^2-7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b d \left (a^2-b^2\right )^2}+\frac{3 a \left (a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 d \left (a^2-b^2\right )^2}+\frac{3 \left (-2 a^2 b^2+a^4+5 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 3845
Rule 4098
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\frac{a^2 \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{a^2}{2}-2 a b \sec (c+d x)-\frac{1}{2} \left (3 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac{a^2 \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{3 a^2 \left (a^2-3 b^2\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{3}{4} a^2 \left (a^2-3 b^2\right )+a b \left (a^2-4 b^2\right ) \sec (c+d x)+\frac{1}{4} \left (3 a^4-5 a^2 b^2+8 b^4\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{a^2 \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{3 a^2 \left (a^2-3 b^2\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{3}{4} a^3 \left (a^2-3 b^2\right )-\left (-a^2 b \left (a^2-4 b^2\right )+\frac{3}{4} a^2 b \left (a^2-3 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 (a^4-2 a^2 b^2+5 b^4\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{a^2 \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{3 a^2 \left (a^2-3 b^2\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^2-7 b^2\right ) \int \sqrt{\sec (c+d x)} \, dx}{8 b \left (a^2-b^2\right )^2}+\frac{\left (3 a \left (a^2-3 b^2\right )\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 b^2 \left (a^2-b^2\right )^2}+\frac{\left (3 \left (a^4-2 a^2 b^2+5 b^4\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 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{3 \left (a^4-2 a^2 b^2+5 b^4\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-b)^2 b^2 (a+b)^3 d}-\frac{a^2 \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{3 a^2 \left (a^2-3 b^2\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^2-7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{8 b \left (a^2-b^2\right )^2}+\frac{\left (3 a \left (a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{3 a \left (a^2-3 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 b^2 \left (a^2-b^2\right )^2 d}+\frac{\left (a^2-7 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 b \left (a^2-b^2\right )^2 d}+\frac{3 \left (a^4-2 a^2 b^2+5 b^4\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-b)^2 b^2 (a+b)^3 d}-\frac{a^2 \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{3 a^2 \left (a^2-3 b^2\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 = 6.74898, size = 697, normalized size = 2.21 \[ \frac{\frac{2 \left (-19 a^2 b^2+9 a^4+16 b^4\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{b \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (3 a^4-9 a^2 b^2\right ) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 a b \sec ^2(c+d x)+2 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+2 a b\right )}{a^2 b \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (8 a^3 b-32 a b^3\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{a \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}}{16 b^2 d (a-b)^2 (a+b)^2}+\frac{\sqrt{\sec (c+d x)} \left (\frac{3 a \left (3 b^2-a^2\right ) \sin (c+d x)}{4 b^2 \left (b^2-a^2\right )^2}-\frac{a \sin (c+d x)}{2 \left (b^2-a^2\right ) (a \cos (c+d x)+b)^2}+\frac{a^3 \sin (c+d x)-7 a b^2 \sin (c+d x)}{4 b \left (b^2-a^2\right )^2 (a \cos (c+d x)+b)}\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 3.678, size = 1203, normalized size = 3.8 \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{\sec \left (d x + c\right )^{\frac{7}{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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