Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right ) \]
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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 {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx &=\int \frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 3.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (e \sin \left (d x +c \right )\right )^{m}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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 (e \sin {\left (c + d x \right )}\right )^{m}}{\left (a + b \sec {\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 [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 = -0.04 \begin {gather*} \int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^m}{{\left (a+\frac {b}{\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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