Optimal. Leaf size=60 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}-\frac{1}{d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.0621285, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2667, 51, 63, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}-\frac{1}{d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{1}{d \sqrt{a+a \sin (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=-\frac{1}{d \sqrt{a+a \sin (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}-\frac{1}{d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0489939, size = 39, normalized size = 0.65 \[ -\frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 54, normalized size = 0.9 \begin{align*} -2\,{\frac{a}{d} \left ( 1/2\,{\frac{1}{a\sqrt{a+a\sin \left ( dx+c \right ) }}}-1/4\,{\frac{\sqrt{2}}{{a}^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16618, size = 252, normalized size = 4.2 \begin{align*} \frac{\frac{\sqrt{2}{\left (a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac{\frac{2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}}{\sqrt{a}} + \sin \left (d x + c\right ) + 3}{\sin \left (d x + c\right ) - 1}\right )}{\sqrt{a}} - 4 \, \sqrt{a \sin \left (d x + c\right ) + a}}{4 \,{\left (a d \sin \left (d x + c\right ) + a d\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*} \int \frac{\sec{\left (c + d x \right )}}{\sqrt{a \left (\sin{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13295, size = 80, normalized size = 1.33 \begin{align*} -\frac{a{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{2}{\sqrt{a \sin \left (d x + c\right ) + a} a}\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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