Integrand size = 26, antiderivative size = 55 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {4 i}{a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d} \]
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Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 45} \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d}+\frac {4 i}{a^2 d \sqrt {a+i a \tan (c+d x)}} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {a-x}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {i \text {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*}
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {6 i-2 \tan (c+d x)}{a^2 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 1.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {2 i \left (\sqrt {a +i a \tan \left (d x +c \right )}+\frac {2 a}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d \,a^{3}}\) | \(42\) |
default | \(\frac {2 i \left (\sqrt {a +i a \tan \left (d x +c \right )}+\frac {2 a}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d \,a^{3}}\) | \(42\) |
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Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3} d} \]
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\[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 i \, {\left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{a^{2}} + \frac {2}{\sqrt {i \, a \tan \left (d x + c\right ) + a} a}\right )}}{a d} \]
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\[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )+2{}\mathrm {i}\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}}{a^3\,d} \]
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