Optimal. Leaf size=76 \[ \frac{d \cos ^2(a+b x)^{3/4} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac{3}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0973584, 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} \[ \frac{d \cos ^2(a+b x)^{3/4} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac{3}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2587
Rule 2577
Rubi steps
\begin{align*} \int \frac{\csc ^p(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx &=\left (\csc ^p(a+b x) \sin ^p(a+b x)\right ) \int \frac{\sin ^{-p}(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx\\ &=\frac{d \cos ^2(a+b x)^{3/4} \csc ^{-1+p}(a+b x) \, _2F_1\left (\frac{3}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.216115, size = 68, normalized size = 0.89 \[ -\frac{2 \sqrt{d \cos (a+b x)} \sin ^2(a+b x)^{\frac{p+1}{2}} \csc ^{p+1}(a+b x) \, _2F_1\left (\frac{1}{4},\frac{p+1}{2};\frac{5}{4};\cos ^2(a+b x)\right )}{b d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \csc \left ( bx+a \right ) \right ) ^{p}{\frac{1}{\sqrt{d\cos \left ( bx+a \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 \frac{\csc \left (b x + a\right )^{p}}{\sqrt{d \cos \left (b x + a\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 (\frac{\sqrt{d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{p}}{d \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{p}{\left (a + b x \right )}}{\sqrt{d \cos{\left (a + b 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 \frac{\csc \left (b x + a\right )^{p}}{\sqrt{d \cos \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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