Optimal. Leaf size=103 \[ \frac {a \text {ArcTan}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}} \]
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Rubi [A]
time = 0.11, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3957, 2917,
2644, 335, 218, 212, 209, 2721, 2720} \begin {gather*} \frac {a \text {ArcTan}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {e \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 2644
Rule 2720
Rule 2721
Rule 2917
Rule 3957
Rubi steps
\begin {align*} \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx &=-\int \frac {(-a-a \cos (c+d x)) \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\\ &=a \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx+a \int \frac {\sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac {\left (a \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{\sqrt {e \sin (c+d x)}}\\ &=\frac {2 a F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e}\\ &=\frac {2 a F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}+\frac {a \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}\\ &=\frac {a \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.97, size = 193, normalized size = 1.87 \begin {gather*} \frac {4 a \cos \left (\frac {1}{2} (c+d x)\right ) \left (4 F\left (\left .\text {ArcSin}\left (\frac {1}{\sqrt {\tan \left (\frac {1}{4} (c+d x)\right )}}\right )\right |-1\right )+\sqrt {2} \left (\Pi \left (-1-\sqrt {2};\left .\text {ArcSin}\left (\frac {1}{\sqrt {\tan \left (\frac {1}{4} (c+d x)\right )}}\right )\right |-1\right )-\Pi \left (1-\sqrt {2};\left .\text {ArcSin}\left (\frac {1}{\sqrt {\tan \left (\frac {1}{4} (c+d x)\right )}}\right )\right |-1\right )-\Pi \left (-1+\sqrt {2};\left .\text {ArcSin}\left (\frac {1}{\sqrt {\tan \left (\frac {1}{4} (c+d x)\right )}}\right )\right |-1\right )+\Pi \left (1+\sqrt {2};\left .\text {ArcSin}\left (\frac {1}{\sqrt {\tan \left (\frac {1}{4} (c+d x)\right )}}\right )\right |-1\right )\right )\right )}{d \sqrt {1-\cot ^2\left (\frac {1}{4} (c+d x)\right )} \sqrt {e \sin (c+d x)} \sqrt {\tan \left (\frac {1}{4} (c+d x)\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 117, normalized size = 1.14
method | result | size |
default | \(\frac {\frac {a \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {a \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{\sqrt {e}}-\frac {a \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(117\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 4.06, size = 144, normalized size = 1.40 \begin {gather*} \frac {{\left (4 \, \sqrt {2} \sqrt {-i} a {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 4 \, \sqrt {2} \sqrt {i} a {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, a \arctan \left (\frac {\sin \left (d x + c\right ) - 1}{2 \, \sqrt {\sin \left (d x + c\right )}}\right ) + a \log \left (\frac {\cos \left (d x + c\right )^{2} - 4 \, {\left (\sin \left (d x + c\right ) + 1\right )} \sqrt {\sin \left (d x + c\right )} - 6 \, \sin \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 2}\right )\right )} e^{\left (-\frac {1}{2}\right )}}{4 \, 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*} a \left (\int \frac {1}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {a+\frac {a}{\cos \left (c+d\,x\right )}}{\sqrt {e\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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