Optimal. Leaf size=82 \[ \frac {2^{\frac {3}{4}+m} \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-m;\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{4}-m} (a+a \sin (c+d x))^m}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2768, 72, 71}
\begin {gather*} \frac {2^{m+\frac {3}{4}} (\sin (c+d x)+1)^{\frac {1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-m;\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt {e \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 2768
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {\left (a^2 \sqrt [4]{a-a \sin (c+d x)} \sqrt [4]{a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {5}{4}+m}}{(a-a x)^{5/4}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}\\ &=\frac {\left (2^{-\frac {5}{4}+m} a \sqrt [4]{a-a \sin (c+d x)} (a+a \sin (c+d x))^m \left (\frac {a+a \sin (c+d x)}{a}\right )^{\frac {1}{4}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {5}{4}+m}}{(a-a x)^{5/4}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}\\ &=\frac {2^{\frac {3}{4}+m} \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-m;\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{4}-m} (a+a \sin (c+d x))^m}{d e \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 82, normalized size = 1.00 \begin {gather*} \frac {2^{\frac {3}{4}+m} \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-m;\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{4}-m} (a (1+\sin (c+d x)))^m}{d e \sqrt {e \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sin \left (d x +c \right )\right )^{m}}{\left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
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 (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m}}{\left (e \cos {\left (c + d 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 (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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