Optimal. Leaf size=131 \[ \frac {2^{m+\frac {1}{2}} \left (\frac {1}{\sec (c+d x)+1}\right )^{m+\frac {1}{2}} (e \tan (c+d x))^{m+1} F_1\left (\frac {m+1}{2};m-\frac {1}{2},1;\frac {m+3}{2};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1) \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.09, 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 = {3889} \[ \frac {2^{m+\frac {1}{2}} \left (\frac {1}{\sec (c+d x)+1}\right )^{m+\frac {1}{2}} (e \tan (c+d x))^{m+1} F_1\left (\frac {m+1}{2};m-\frac {1}{2},1;\frac {m+3}{2};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1) \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3889
Rubi steps
\begin {align*} \int \frac {(e \tan (c+d x))^m}{\sqrt {a+a \sec (c+d x)}} \, dx &=\frac {2^{\frac {1}{2}+m} F_1\left (\frac {1+m}{2};-\frac {1}{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) \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [F] time = 2.35, size = 0, normalized size = 0.00 \[ \int \frac {(e \tan (c+d x))^m}{\sqrt {a+a \sec (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \tan \left (d x + c\right )\right )^{m}}{\sqrt {a \sec \left (d x + c\right ) + a}}, x\right ) \]
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 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.67, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \tan \left (d x +c \right )\right )^{m}}{\sqrt {a +a \sec \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {\left (e \tan {\left (c + d x \right )}\right )^{m}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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