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 = 0.10, antiderivative size = 76, normalized size of antiderivative = 1.00, 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 2577
Rule 2587
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*}
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Mathematica [A] time = 0.22, 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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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{p}}{d \cos \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{p}}{\sqrt {d \cos \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{p}\left (b x +a \right )}{\sqrt {d \cos \left (b x +a \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{p}}{\sqrt {d \cos \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^p}{\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{p}{\left (a + b x \right )}}{\sqrt {d \cos {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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