Optimal. Leaf size=29 \[ \frac {2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 29, 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} \[ \frac {2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \]
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))^{5/2}} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {1}{(a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac {2 i}{3 a d (a+i a \tan (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 39, normalized size = 1.34 \[ \frac {2}{3 a^2 d (\tan (c+d x)-i) \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 61, normalized size = 2.10 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{6 \, a^{3} d} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 24, normalized size = 0.83 \[ \frac {2 i}{3 a d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 21, normalized size = 0.72 \[ \frac {2 i}{3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 23, normalized size = 0.79 \[ \frac {2{}\mathrm {i}}{3\,a\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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