Optimal. Leaf size=27 \[ \frac {2 i}{a d \sqrt {a+i a \tan (c+d x)}} \]
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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} \[ \frac {2 i}{a d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {1}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac {2 i}{a d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 27, normalized size = 1.00 \[ \frac {2 i}{a d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 49, normalized size = 1.81 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2} d} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 24, normalized size = 0.89 \[ \frac {2 i}{a d \sqrt {a +i a \tan \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, 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 = 0.24, size = 67, normalized size = 2.48 \[ \frac {\left ({\cos \left (c+d\,x\right )}^2\,2{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )\right )\,\sqrt {\frac {a\,\left (2\,{\cos \left (c+d\,x\right )}^2+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (c+d\,x\right )}^2}}}{a^2\,d} \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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