Optimal. Leaf size=78 \[ \frac {\sqrt [4]{\cos ^2(a+b x)} \csc ^{-1+p}(a+b x) \text {Hypergeometric2F1}\left (\frac {5}{4},\frac {1-p}{2},\frac {3-p}{2},\sin ^2(a+b x)\right )}{b d (1-p) \sqrt {d \cos (a+b x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 78, 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 = {2667, 2657}
\begin {gather*} \frac {\sqrt [4]{\cos ^2(a+b x)} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac {5}{4},\frac {1-p}{2};\frac {3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) \sqrt {d \cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2667
Rubi steps
\begin {align*} \int \frac {\csc ^p(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx &=\left (\csc ^p(a+b x) \sin ^p(a+b x)\right ) \int \frac {\sin ^{-p}(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx\\ &=\frac {\sqrt [4]{\cos ^2(a+b x)} \csc ^{-1+p}(a+b x) \, _2F_1\left (\frac {5}{4},\frac {1-p}{2};\frac {3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) \sqrt {d \cos (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 10.17, size = 68, normalized size = 0.87 \begin {gather*} \frac {2 \csc ^{-1+p}(a+b x) \text {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+p}{2},\frac {3}{4},\cos ^2(a+b x)\right ) \sin ^2(a+b x)^{\frac {1}{2} (-1+p)}}{b d \sqrt {d \cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\csc ^{p}\left (b x +a \right )}{\left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\csc ^{p}{\left (a + b x \right )}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^p}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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