Optimal. Leaf size=92 \[ \frac {\sqrt {\frac {1}{\sec (c+d x)+1}} \sqrt {a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )|\frac {a-b}{a+b}\right )}{d \sqrt {\frac {a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}} \]
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Rubi [A] time = 0.16, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2829, 3968} \[ \frac {\sqrt {\frac {1}{\sec (c+d x)+1}} \sqrt {a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )|\frac {a-b}{a+b}\right )}{d \sqrt {\frac {a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}} \]
Antiderivative was successfully verified.
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Rule 2829
Rule 3968
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sec (c+d x)}}{1+\cos (c+d x)} \, dx &=\int \frac {\sec (c+d x) \sqrt {a+b \sec (c+d x)}}{1+\sec (c+d x)} \, dx\\ &=\frac {E\left (\sin ^{-1}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {a+b \sec (c+d x)}}{d \sqrt {\frac {a+b \sec (c+d x)}{(a+b) (1+\sec (c+d x))}}}\\ \end {align*}
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Mathematica [A] time = 7.22, size = 85, normalized size = 0.92 \[ \frac {\sqrt {\frac {1}{\sec (c+d x)+1}} \sqrt {a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )}{d \sqrt {\frac {a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a}}{\cos \left (d x + c\right ) + 1}, x\right ) \]
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 \[ \int \frac {\sqrt {b \sec \left (d x + c\right ) + a}}{\cos \left (d x + c\right ) + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 150, normalized size = 1.63 \[ -\frac {\EllipticE \left (\frac {\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {a \cos \left (d x +c \right )+b}{\left (1+\cos \left (d x +c \right )\right ) \left (a +b \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )-1\right ) \sqrt {\frac {a \cos \left (d x +c \right )+b}{\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right )^{2} \left (-a -b \right )}{d \left (a \cos \left (d x +c \right )+b \right ) \sin \left (d x +c \right )^{2}} \]
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 \[ \int \frac {\sqrt {b \sec \left (d x + c\right ) + a}}{\cos \left (d x + c\right ) + 1}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{\cos \left (c+d\,x\right )+1} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sec {\left (c + d x \right )}}}{\cos {\left (c + d x \right )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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