Optimal. Leaf size=99 \[ \frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}-\frac {2 \csc (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{a d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]
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Rubi [A]
time = 0.16, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3963, 3957,
2918, 2644, 30, 2647, 2719} \begin {gather*} -\frac {2 \csc (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{a d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rule 2647
Rule 2719
Rule 2918
Rule 3957
Rule 3963
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx &=\frac {\int \frac {\sqrt {\sin (c+d x)}}{a+a \sec (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {\int \frac {\cos (c+d x) \sqrt {\sin (c+d x)}}{-a-a \cos (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \frac {\cos (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{a \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{a \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {2 \int \sqrt {\sin (c+d x)} \, dx}{a \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\sin (c+d x)\right )}{a d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}-\frac {2 \csc (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{a d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.42, size = 95, normalized size = 0.96 \begin {gather*} \frac {6 (2 i+\cot (c+d x)-\csc (c+d x))-4 \sqrt {1-e^{2 i (c+d x)}} (i+\cot (c+d x)) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 i (c+d x)}\right )}{3 a d \sqrt {e \csc (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 536, normalized size = 5.41
method | result | size |
default | \(-\frac {\left (4 \cos \left (d x +c \right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (d x +c \right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+4 \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-2 \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}+\sqrt {2}\, \cos \left (d x +c \right )-\sqrt {2}\right ) \sqrt {2}}{a d \sqrt {\frac {e}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right )}\) | \(536\) |
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 = 0.88, size = 78, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (\sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + \frac {\cos \left (d x + c\right ) - 1}{\sqrt {\sin \left (d x + c\right )}}\right )} e^{\left (-\frac {1}{2}\right )}}{a 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*} \frac {\int \frac {1}{\sqrt {e \csc {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \csc {\left (c + d x \right )}}}\, dx}{a} \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 {\cos \left (c+d\,x\right )}{a\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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