Optimal. Leaf size=49 \[ -\frac{(d \cos (a+b x))^{n+1} \, _2F_1\left (3,\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.0463348, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2565, 364} \[ -\frac{(d \cos (a+b x))^{n+1} \, _2F_1\left (3,\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2565
Rule 364
Rubi steps
\begin{align*} \int (d \cos (a+b x))^n \csc ^5(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^n}{\left (1-\frac{x^2}{d^2}\right )^3} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{(d \cos (a+b x))^{1+n} \, _2F_1\left (3,\frac{1+n}{2};\frac{3+n}{2};\cos ^2(a+b x)\right )}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.193424, size = 86, normalized size = 1.76 \[ \frac{d \csc ^4(a+b x) \left (-\cot ^2(a+b x)\right )^{\frac{1-n}{2}} (d \cos (a+b x))^{n-1} \, _2F_1\left (\frac{1-n}{2},\frac{5-n}{2};\frac{7-n}{2};\csc ^2(a+b x)\right )}{b (n-5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.219, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n} \left ( \csc \left ( bx+a \right ) \right ) ^{5}\, 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 (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{5}\,{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 (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{5}, 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 \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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