Optimal. Leaf size=70 \[ \frac{\text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{a d}+\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)} \]
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Rubi [A] time = 0.143734, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4264, 3818, 3787, 3771, 2639, 2641} \[ \frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}+\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3818
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx\\ &=-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{a}{2}-\frac{1}{2} a \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{a^2}\\ &=-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 a}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx}{2 a}\\ &=-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))}+\frac{\int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 a}+\frac{\int \sqrt{\cos (c+d x)} \, dx}{2 a}\\ &=\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}+\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [C] time = 1.45026, size = 263, normalized size = 3.76 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{2 \left (\sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )+\csc (c)\right )}{d \sqrt{\cos (c+d x)}}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sec (c+d x) \left (\left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )-\left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+e^{2 i (c+d x)}+1\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{a (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.293, size = 200, normalized size = 2.9 \begin{align*}{\frac{1}{da}\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 ({\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) -{\it EllipticE} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) \right ) +2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{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 )} \cos \left (d x + c\right )^{\frac{3}{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{\cos \left (d x + c\right )}}{a \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + a \cos \left (d x + c\right )^{2}}, x\right ) \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 (a \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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