Optimal. Leaf size=115 \[ \frac {3 A \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {2 \sqrt {2} A \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}} \]
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Rubi [A]
time = 0.15, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4107, 4005,
3859, 209, 3880} \begin {gather*} \frac {3 A \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {2 \sqrt {2} A \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3859
Rule 3880
Rule 4005
Rule 4107
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx &=\frac {A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {-\frac {3 a A}{2}-\frac {1}{2} a A \sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{a}\\ &=\frac {A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}+(2 A) \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx+\frac {(3 A) \int \sqrt {a-a \sec (c+d x)} \, dx}{2 a}\\ &=\frac {A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}+\frac {(3 A) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}-\frac {(4 A) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}\\ &=\frac {3 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {2 \sqrt {2} A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.52, size = 382, normalized size = 3.32 \begin {gather*} A \left (\frac {e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (1+\cos (c+d x)) \left (-3 i d x+3 \sinh ^{-1}\left (e^{i (c+d x)}\right )+4 \sqrt {2} \log \left (1-e^{i (c+d x)}\right )+3 \log \left (1+\sqrt {1+e^{2 i (c+d x)}}\right )-4 \sqrt {2} \log \left (1+e^{i (c+d x)}+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sqrt {2} d \sqrt {a-a \sec (c+d x)}}+\frac {(1+\cos (c+d x)) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (c+d x) \left (\frac {\cos \left (\frac {c}{2}\right ) \cos \left (\frac {d x}{2}\right )}{2 d}+\frac {\cos \left (\frac {3 c}{2}\right ) \cos \left (\frac {3 d x}{2}\right )}{2 d}-\frac {\sin \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )}{2 d}-\frac {\sin \left (\frac {3 c}{2}\right ) \sin \left (\frac {3 d x}{2}\right )}{2 d}\right ) \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{\sqrt {a-a \sec (c+d x)}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 8.53, size = 154, normalized size = 1.34
method | result | size |
default | \(-\frac {A \left (1+\cos \left (d x +c \right )\right ) \left (-2 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+\sqrt {2}\, \cos \left (d x +c \right )-3 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )\right ) \sqrt {\frac {a \left (-1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \sqrt {2}}{2 d \sin \left (d x +c \right ) a}\) | \(154\) |
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 = 1.97, size = 435, normalized size = 3.78 \begin {gather*} \left [\frac {2 \, \sqrt {2} A a \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \sqrt {-\frac {1}{a}} - {\left (3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 3 \, A \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 2 \, {\left (A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{2 \, a d \sin \left (d x + c\right )}, \frac {2 \, \sqrt {2} A \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 3 \, A \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - {\left (A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{a d \sin \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*} A \left (\int \frac {\cos {\left (c + d x \right )}}{\sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx + \int \frac {\cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 114, normalized size = 0.99 \begin {gather*} \frac {\frac {2 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{\sqrt {a}} - \frac {3 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} - \frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A}{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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{\sqrt {a-\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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