Optimal. Leaf size=111 \[ \frac{2^{n+\frac{3}{2}} \sqrt{\tan (c+d x)} \left (\frac{1}{\sec (c+d x)+1}\right )^{n+\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d} \]
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Rubi [A] time = 0.06267, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {3889} \[ \frac{2^{n+\frac{3}{2}} \sqrt{\tan (c+d x)} \left (\frac{1}{\sec (c+d x)+1}\right )^{n+\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3889
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx &=\frac{2^{\frac{3}{2}+n} F_1\left (\frac{1}{4};-\frac{1}{2}+n,1;\frac{5}{4};-\frac{a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac{a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac{1}{1+\sec (c+d x)}\right )^{\frac{1}{2}+n} (a+a \sec (c+d x))^n \sqrt{\tan (c+d x)}}{d}\\ \end{align*}
Mathematica [B] time = 1.37678, size = 229, normalized size = 2.06 \[ \frac{10 \cos (c+d x) (\cos (c+d x)+1) \sqrt{\tan (c+d x)} (a (\sec (c+d x)+1))^n F_1\left (\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}{d \left (2 (\cos (c+d x)-1) \left (2 F_1\left (\frac{5}{4};n-\frac{1}{2},2;\frac{9}{4};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )+(1-2 n) F_1\left (\frac{5}{4};n+\frac{1}{2},1;\frac{9}{4};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )\right )+5 (\cos (c+d x)+1) F_1\left (\frac{1}{4};n-\frac{1}{2},1;\frac{5}{4};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.288, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n}{\frac{1}{\sqrt{\tan \left ( dx+c \right ) }}}}\, dx \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{{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt{\tan \left (d x + c\right )}}\,{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{{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt{\tan \left (d x + c\right )}}, 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*} \int \frac{\left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{n}}{\sqrt{\tan{\left (c + d x \right )}}}\, dx \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{{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt{\tan \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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