Optimal. Leaf size=80 \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right )}{d (7-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.0289639, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {20, 2643} \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right )}{d (7-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^n}{\cos ^{\frac{9}{2}}(c+d x)} \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac{9}{2}+n}(c+d x) \, dx\\ &=\frac{2 (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-7+2 n);\frac{1}{4} (-3+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7-2 n) \cos ^{\frac{7}{2}}(c+d x) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0903199, size = 80, normalized size = 1. \[ -\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (n-\frac{7}{2}\right );\frac{1}{2} \left (n-\frac{3}{2}\right );\cos ^2(c+d x)\right )}{d \left (n-\frac{7}{2}\right ) \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{9}{2}}}}\, 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 (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac{9}{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{\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac{9}{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{\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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