Optimal. Leaf size=128 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 b d}+\frac{2 a^2 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}+\frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.540456, antiderivative size = 128, 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 = {4264, 3851, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{2 a^2 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}+\frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b d}+\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3851
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{7}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx\\ &=\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)} \left (\frac{a}{2}+\frac{1}{2} b \sec (c+d x)-\frac{3}{2} a \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b}\\ &=\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3 a^2}{4}+a b \sec (c+d x)+\frac{1}{4} \left (3 a^2+b^2\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 b^2}\\ &=\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3 a^3}{4}+\frac{1}{4} a^2 b \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^2 b^2}+\frac{\left (a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{b^2}\\ &=\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{a^2 \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{b^2}+\frac{\left (a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{b^2}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{3 b}\\ &=\frac{2 a^2 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}+\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{a \int \sqrt{\cos (c+d x)} \, dx}{b^2}+\frac{\int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b}\\ &=\frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b d}+\frac{2 a^2 \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}+\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.66451, size = 213, normalized size = 1.66 \[ \frac{\frac{6 \sin (c+d x) \left (2 b (a+b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right ),-1\right )-\left (a^2-2 b^2\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{b \sqrt{\sin ^2(c+d x)}}+8 b \left (2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-\frac{2 b \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}\right )+\frac{2 \left (9 a^2+2 b^2\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{4 \sin (c+d x) (b-3 a \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}}{6 b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.096, size = 450, normalized size = 3.5 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \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{1}{{\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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