Optimal. Leaf size=30 \[ \text {Int}\left (\frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}},x\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx &=\int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 4.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sin \left (f x +e \right )\right )^{m}}{\left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sin {\left (e + f x \right )}\right )^{m}}{\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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