Optimal. Leaf size=82 \[ \frac {2^{m+\frac {3}{4}} (\sin (c+d x)+1)^{\frac {1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-m;\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.10, 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 = {2689, 70, 69} \[ \frac {2^{m+\frac {3}{4}} (\sin (c+d x)+1)^{\frac {1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-m;\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
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 ) \operatorname {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 ) \operatorname {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 = 0.10, size = 82, normalized size = 1.00 \[ \frac {2^{m+\frac {3}{4}} (\sin (c+d x)+1)^{\frac {1}{4}-m} (a (\sin (c+d x)+1))^m \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-m;\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, 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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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