Optimal. Leaf size=28 \[ \text {Int}\left (\sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m,x\right ) \]
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Rubi [A] time = 0.06, 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 = {} \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx &=\int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx\\ \end {align*}
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Mathematica [A] time = 0.71, size = 0, normalized size = 0.00 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}, x\right ) \]
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 \[ \int \sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.65, size = 0, normalized size = 0.00 \[ \int \sqrt {a +b \sec \left (d x +c \right )}\, \left (e \tan \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int \sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \left (e \tan {\left (c + d x \right )}\right )^{m} \sqrt {a + b \sec {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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