Optimal. Leaf size=36 \[ \frac {2 i}{d \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.14, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3515, 3488} \[ \frac {2 i}{d \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3488
Rule 3515
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {\int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {2 i}{d \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 36, normalized size = 1.00 \[ \frac {2 i}{d \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 58, normalized size = 1.61 \[ \frac {2 i \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{a d e} \]
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 \[ \int \frac {1}{\sqrt {e \cos \left (d x + c\right )} \sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.43, size = 69, normalized size = 1.92 \[ -\frac {2 i \sqrt {e \cos \left (d x +c \right )}\, \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (i \sin \left (d x +c \right )-\cos \left (d x +c \right )\right )}{d e a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 76, normalized size = 2.11 \[ \frac {2 i \, \sqrt {-\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}}{\sqrt {a} d \sqrt {e} \sqrt {-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {e \cos {\left (c + d x \right )}} \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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