Optimal. Leaf size=77 \[ \frac {\, _2F_1\left (-\frac {1}{4},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c d (1+m) \sqrt [4]{\cos ^2(a+b x)} \sqrt {d \sec (a+b x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2666, 2657}
\begin {gather*} \frac {(c \sin (a+b x))^{m+1} \, _2F_1\left (-\frac {1}{4},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c d (m+1) \sqrt [4]{\cos ^2(a+b x)} \sqrt {d \sec (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2666
Rubi steps
\begin {align*} \int \frac {(c \sin (a+b x))^m}{(d \sec (a+b x))^{3/2}} \, dx &=\frac {\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx}{d^2 \sqrt {d \cos (a+b x)} \sqrt {d \sec (a+b x)}}\\ &=\frac {\, _2F_1\left (-\frac {1}{4},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c d (1+m) \sqrt [4]{\cos ^2(a+b x)} \sqrt {d \sec (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 17.30, size = 116, normalized size = 1.51 \begin {gather*} \frac {2 c \cos (2 (a+b x)) \, _2F_1\left (\frac {1}{4} (-3-2 m),\frac {1-m}{2};\frac {1}{4} (1-2 m);\sec ^2(a+b x)\right ) (c \sin (a+b x))^{-1+m} \left (-\tan ^2(a+b x)\right )^{\frac {1-m}{2}}}{b d (3+2 m) \sqrt {d \sec (a+b x)} \left (-2+\sec ^2(a+b x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (c \sin \left (b x +a \right )\right )^{m}}{\left (d \sec \left (b x +a \right )\right )^{\frac {3}{2}}}\, dx\]
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 {\left (c \sin {\left (a + b x \right )}\right )^{m}}{\left (d \sec {\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 (c\,\sin \left (a+b\,x\right )\right )}^m}{{\left (\frac {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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