Optimal. Leaf size=75 \[ -\frac{b^3 \sin (c+d x) (b \sec (c+d x))^{n-3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3-n}{2},\frac{5-n}{2},\cos ^2(c+d x)\right )}{d (3-n) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0553529, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {16, 3772, 2643} \[ -\frac{b^3 \sin (c+d x) (b \sec (c+d x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\cos ^2(c+d x)\right )}{d (3-n) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (b \sec (c+d x))^n \, dx &=b^2 \int (b \sec (c+d x))^{-2+n} \, dx\\ &=\left (b^2 \left (\frac{\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{2-n} \, dx\\ &=-\frac{\cos ^3(c+d x) \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (3-n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0951624, size = 71, normalized size = 0.95 \[ \frac{\cos ^2(c+d x) \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-2}{2},\frac{n}{2},\sec ^2(c+d x)\right )}{d (n-2)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.968, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( b\sec \left ( dx+c \right ) \right ) ^{n}\, 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} \cos \left (d x + c\right )^{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 (\left (b \sec \left (d x + c\right )\right )^{n} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec{\left (c + d x \right )}\right )^{n} \cos ^{2}{\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} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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