Integrand size = 26, antiderivative size = 146 \[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {32 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac {64 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}-\frac {48 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}+\frac {16 i (a+i a \tan (c+d x))^{9/2}}{9 a^8 d}-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^9 d} \]
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Time = 0.11 (sec) , antiderivative size = 146, 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} \[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^9 d}+\frac {16 i (a+i a \tan (c+d x))^{9/2}}{9 a^8 d}-\frac {48 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}+\frac {64 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}-\frac {32 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x)^4 \sqrt {a+x} \, dx,x,i a \tan (c+d x)\right )}{a^9 d} \\ & = -\frac {i \text {Subst}\left (\int \left (16 a^4 \sqrt {a+x}-32 a^3 (a+x)^{3/2}+24 a^2 (a+x)^{5/2}-8 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d} \\ & = -\frac {32 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac {64 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}-\frac {48 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}+\frac {16 i (a+i a \tan (c+d x))^{9/2}}{9 a^8 d}-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^9 d} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.55 \[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {2 (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \left (5419-6396 i \tan (c+d x)-4530 \tan ^2(c+d x)+1820 i \tan ^3(c+d x)+315 \tan ^4(c+d x)\right )}{3465 a^4 d} \]
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Time = 1.61 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}+\frac {8 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {24 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {32 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {16 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,a^{9}}\) | \(101\) |
default | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}+\frac {8 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {24 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {32 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {16 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,a^{9}}\) | \(101\) |
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Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10 \[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {64 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (128 i \, e^{\left (11 i \, d x + 11 i \, c\right )} + 704 i \, e^{\left (9 i \, d x + 9 i \, c\right )} + 1584 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 1848 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 1155 i \, e^{\left (3 i \, d x + 3 i \, c\right )}\right )}}{3465 \, {\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \]
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Timed out. \[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.64 \[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {2 i \, {\left (315 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 3080 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 11880 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} - 22176 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{3} + 18480 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{4}\right )}}{3465 \, a^{9} d} \]
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\[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{10}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
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Time = 9.23 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.53 \[ \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,8192{}\mathrm {i}}{3465\,a^4\,d}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,4096{}\mathrm {i}}{3465\,a^4\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,1024{}\mathrm {i}}{1155\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,512{}\mathrm {i}}{693\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{99\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{11\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \]
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