Optimal. Leaf size=55 \[ \frac {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d}+\frac {4 i}{a^2 d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3487, 43} \[ \frac {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d}+\frac {4 i}{a^2 d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {a-x}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (\frac {2 a}{(a+x)^{3/2}}-\frac {1}{\sqrt {a+x}}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac {4 i}{a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 36, normalized size = 0.65 \[ \frac {-2 \tan (c+d x)+6 i}{a^2 d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 49, normalized size = 0.89 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3} d} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 65, normalized size = 1.18 \[ \frac {2 \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (2 \cos \left (d x +c \right ) \sin \left (d x +c \right )+i+2 i \left (\cos ^{2}\left (d x +c \right )\right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 44, normalized size = 0.80 \[ \frac {2 i \, {\left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{a^{2}} + \frac {2}{\sqrt {i \, a \tan \left (d x + c\right ) + a} a}\right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 72, normalized size = 1.31 \[ \frac {2\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )+2{}\mathrm {i}\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}}{a^3\,d} \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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