Optimal. Leaf size=76 \[ \frac{d \sqrt{d \cos (a+b x)} \csc ^{p-1}(a+b x) \, _2F_1\left (-\frac{1}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt [4]{\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.105468, antiderivative size = 76, 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 = {2587, 2577} \[ \frac{d \sqrt{d \cos (a+b x)} \csc ^{p-1}(a+b x) \, _2F_1\left (-\frac{1}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt [4]{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2587
Rule 2577
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{3/2} \csc ^p(a+b x) \, dx &=\left (\csc ^p(a+b x) \sin ^p(a+b x)\right ) \int (d \cos (a+b x))^{3/2} \sin ^{-p}(a+b x) \, dx\\ &=\frac{d \sqrt{d \cos (a+b x)} \csc ^{-1+p}(a+b x) \, _2F_1\left (-\frac{1}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt [4]{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.64128, size = 105, normalized size = 1.38 \[ -\frac{2 (d \cos (a+b x))^{5/2} \sin ^2(a+b x)^{\frac{p-1}{2}} \csc ^{p-1}(a+b x) \left (5 \cos ^2(a+b x) \, _2F_1\left (\frac{9}{4},\frac{p+1}{2};\frac{13}{4};\cos ^2(a+b x)\right )+9 \, _2F_1\left (\frac{5}{4},\frac{p-1}{2};\frac{9}{4};\cos ^2(a+b x)\right )\right )}{45 b d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.279, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}} \left ( \csc \left ( bx+a \right ) \right ) ^{p}\, 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 )^{\frac{3}{2}} \csc \left (b x + a\right )^{p}\,{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 (\sqrt{d \cos \left (b x + a\right )} d \csc \left (b x + a\right )^{p} \cos \left (b x + a\right ), 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 )^{\frac{3}{2}} \csc \left (b x + a\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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