Optimal. Leaf size=56 \[ \frac {2 a \tan (c+d x)}{3 d \sqrt {a+a \sec (c+d x)}}+\frac {2 \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3883, 3877}
\begin {gather*} \frac {2 \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {2 a \tan (c+d x)}{3 d \sqrt {a \sec (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3877
Rule 3883
Rubi steps
\begin {align*} \int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} \, dx &=\frac {2 \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{3} \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a \tan (c+d x)}{3 d \sqrt {a+a \sec (c+d x)}}+\frac {2 \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 36, normalized size = 0.64 \begin {gather*} \frac {2 a (2+\sec (c+d x)) \tan (c+d x)}{3 d \sqrt {a (1+\sec (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 62, normalized size = 1.11
method | result | size |
default | \(-\frac {2 \left (2 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )-1\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}\) | \(62\) |
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 [A]
time = 4.02, size = 60, normalized size = 1.07 \begin {gather*} \frac {2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \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 \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \sec ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 82, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {2} {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{2}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 108, normalized size = 1.93 \begin {gather*} \frac {4\,\sqrt {\frac {a\,\left (\cos \left (c+d\,x\right )+1\right )}{\cos \left (c+d\,x\right )}}\,\left (3\,\sin \left (c+d\,x\right )+4\,\sin \left (2\,c+2\,d\,x\right )+3\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (4\,c+4\,d\,x\right )\right )}{3\,d\,\left (12\,\cos \left (c+d\,x\right )+8\,\cos \left (2\,c+2\,d\,x\right )+4\,\cos \left (3\,c+3\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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