Optimal. Leaf size=168 \[ -\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a d}-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{21 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
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Rubi [A] time = 0.129467, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3819, 3787, 3769, 3771, 2639, 2641} \[ -\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{21 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
Antiderivative was successfully verified.
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Rule 3819
Rule 3787
Rule 3769
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx &=-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{\int \frac{-\frac{7 a}{2}+\frac{5}{2} a \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{5 \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{2 a}+\frac{7 \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{2 a}\\ &=\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{5 \int \sqrt{\sec (c+d x)} \, dx}{6 a}+\frac{21 \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{10 a}\\ &=\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{\left (5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a}+\frac{\left (21 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a}\\ &=\frac{21 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}-\frac{5 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a d}+\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [C] time = 2.77552, size = 347, normalized size = 2.07 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (-\sqrt{\sec (c+d x)} \left (18 (11 \cos (2 c)+17) \csc (c) \cos (d x)+4 \left (10 \sin (2 c) \cos (2 d x)-3 \sin (3 c) \cos (3 d x)-99 \cos (c) \sin (d x)+10 \cos (2 c) \sin (2 d x)-3 \cos (3 c) \sin (3 d x)-30 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )-30 \tan \left (\frac{c}{2}\right )\right )\right )+\frac{8 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (63 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+25 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+63 \left (1+e^{2 i (c+d x)}\right )\right )}{-1+e^{2 i c}}\right )}{60 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.399, size = 229, normalized size = 1.4 \begin{align*} -{\frac{1}{15\,ad}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 25\,{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +63\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) +48\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-30\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+23\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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 (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \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{\sqrt{\sec \left (d x + c\right )}}{a \sec \left (d x + c\right )^{4} + a \sec \left (d x + c\right )^{3}}, 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*} \frac{\int \frac{1}{\sec ^{\frac{7}{2}}{\left (c + d x \right )} + \sec ^{\frac{5}{2}}{\left (c + d x \right )}}\, dx}{a} \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 (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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