Optimal. Leaf size=47 \[ \frac {\sqrt {2} \text {ArcSin}\left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{d} \]
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Rubi [A]
time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4307, 2860,
222} \begin {gather*} \frac {\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \text {ArcSin}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 2860
Rule 4307
Rubi steps
\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\cos (c+d x)}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx\\ &=-\frac {\left (\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,-\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )}{d}\\ &=\frac {\sqrt {2} \sin ^{-1}\left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 68, normalized size = 1.45 \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\sec (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 82, normalized size = 1.74
method | result | size |
default | \(\frac {\sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sqrt {2+2 \cos \left (d x +c \right )}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )-1\right )}{d \sin \left (d x +c \right )^{2}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 39, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right )}{d} \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 {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {\cos {\left (c + d x \right )} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.02 \begin {gather*} \int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {\cos \left (c+d\,x\right )+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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