Integrand size = 26, antiderivative size = 113 \[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}+\frac {8 i (a+i a \tan (c+d x))^{3/2}}{a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d} \]
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Time = 0.11 (sec) , antiderivative size = 113, 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} \[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}-\frac {12 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}+\frac {8 i (a+i a \tan (c+d x))^{3/2}}{a^5 d}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{a^4 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {(a-x)^3}{\sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{a^7 d} \\ & = -\frac {i \text {Subst}\left (\int \left (\frac {8 a^3}{\sqrt {a+x}}-12 a^2 \sqrt {a+x}+6 a (a+x)^{3/2}-(a+x)^{5/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d} \\ & = -\frac {16 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}+\frac {8 i (a+i a \tan (c+d x))^{3/2}}{a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.54 \[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {2 \sqrt {a+i a \tan (c+d x)} \left (-177 i-71 \tan (c+d x)+27 i \tan ^2(c+d x)+5 \tan ^3(c+d x)\right )}{35 a^4 d} \]
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Time = 1.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {6 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}-8 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}\right )}{d \,a^{7}}\) | \(82\) |
default | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {6 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}-8 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}\right )}{d \,a^{7}}\) | \(82\) |
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none
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {16 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (16 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 56 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 70 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 35 i \, e^{\left (i \, d x + i \, c\right )}\right )}}{35 \, {\left (a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \]
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Timed out. \[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {2 i \, {\left (5 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 42 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 140 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} - 280 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{3}\right )}}{35 \, a^{7} d} \]
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\[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{8}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
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Time = 7.49 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.14 \[ \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,256{}\mathrm {i}}{35\,a^4\,d}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{35\,a^4\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,96{}\mathrm {i}}{35\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,16{}\mathrm {i}}{7\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3} \]
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