Optimal. Leaf size=80 \[ -\frac{2 \sin (c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} (1-2 n),\frac{1}{4} (5-2 n),\cos ^2(c+d x)\right )}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.0381516, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {20, 3772, 2643} \[ -\frac{2 \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (1-2 n);\frac{1}{4} (5-2 n);\cos ^2(c+d x)\right )}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \, dx &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac{1}{2}+n}(c+d x) \, dx\\ &=\left (\cos ^{\frac{1}{2}+n}(c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{-\frac{1}{2}-n}(c+d x) \, dx\\ &=-\frac{2 \, _2F_1\left (\frac{1}{2},\frac{1}{4} (1-2 n);\frac{1}{4} (5-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt{\sec (c+d x)} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.101078, size = 81, normalized size = 1.01 \[ \frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (n+\frac{1}{2}\right ),\frac{1}{2} \left (n+\frac{5}{2}\right ),\sec ^2(c+d x)\right )}{d \left (n+\frac{1}{2}\right ) \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sec \left ( dx+c \right ) \right ) ^{n}\sqrt{\sec \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 \left (b \sec \left (d x + c\right )\right )^{n} \sqrt{\sec \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 (\left (b \sec \left (d x + c\right )\right )^{n} \sqrt{\sec \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 \left (b \sec{\left (c + d x \right )}\right )^{n} \sqrt{\sec{\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 \left (b \sec \left (d x + c\right )\right )^{n} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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