Optimal. Leaf size=27 \[ -\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d} \]
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Rubi [A] time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3487, 32} \[ -\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 32, normalized size = 1.19 \[ \frac {2 (\tan (c+d x)-i)}{d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 37, normalized size = 1.37 \[ -\frac {2 i \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.00, size = 55, normalized size = 2.04 \[ -\frac {2 i \, \sqrt {\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 24, normalized size = 0.89 \[ -\frac {2 i \sqrt {a +i a \tan \left (d x +c \right )}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 21, normalized size = 0.78 \[ -\frac {2 i \, \sqrt {i \, a \tan \left (d x + c\right ) + a}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 47, normalized size = 1.74 \[ -\frac {\sqrt {\frac {a\,\left (2\,{\cos \left (c+d\,x\right )}^2+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (c+d\,x\right )}^2}}\,2{}\mathrm {i}}{a\,d} \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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