Optimal. Leaf size=131 \[ \frac {2^{-\frac {1}{2}+m} F_1\left (\frac {1+m}{2};-\frac {3}{2}+m,1;\frac {3+m}{2};-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{-\frac {1}{2}+m} (e \tan (c+d x))^{1+m}}{d e (1+m) (a+a \sec (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {3974}
\begin {gather*} \frac {2^{m-\frac {1}{2}} \left (\frac {1}{\sec (c+d x)+1}\right )^{m-\frac {1}{2}} (e \tan (c+d x))^{m+1} F_1\left (\frac {m+1}{2};m-\frac {3}{2},1;\frac {m+3}{2};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1) (a \sec (c+d x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3974
Rubi steps
\begin {align*} \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx &=\frac {2^{-\frac {1}{2}+m} F_1\left (\frac {1+m}{2};-\frac {3}{2}+m,1;\frac {3+m}{2};-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{-\frac {1}{2}+m} (e \tan (c+d x))^{1+m}}{d e (1+m) (a+a \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [F]
time = 48.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (e \tan \left (d x +c \right )\right )^{m}}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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 (e \tan {\left (c + d x \right )}\right )^{m}}{\left (a \left (\sec {\left (c + d x \right )} + 1\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 (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m}{{\left (a+\frac {a}{\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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