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 = 0.11, 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} \[ \frac {d \sqrt {d \cos (a+b x)} \csc ^{p-1}(a+b x) \, _2F_1\left (-\frac {1}{4},\frac {1-p}{2};\frac {3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt [4]{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2587
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*}
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Mathematica [A] time = 0.68, 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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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d \cos \left (b x + a\right )} d \csc \left (b x + a\right )^{p} \cos \left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \left (\csc ^{p}\left (b x +a \right )\right )\, dx \]
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 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \csc \left (b x + a\right )^{p}\,{d x} \]
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 \[ \int {\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}\,{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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