Integrand size = 26, antiderivative size = 29 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d} \]
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Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 32} \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d} \]
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Rule 32
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a+x)^{5/2} \, dx,x,i a \tan (c+d x)\right )}{a d} \\ & = -\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d} \]
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Time = 4.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {2 i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 a d}\) | \(24\) |
default | \(-\frac {2 i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 a d}\) | \(24\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {16 i \, \sqrt {2} a^{2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (7 i \, d x + 7 i \, c\right )}}{7 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}} \sec ^{2}{\left (c + d x \right )}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {2 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}{7 \, a d} \]
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\[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{2} \,d x } \]
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Time = 7.43 (sec) , antiderivative size = 242, normalized size of antiderivative = 8.34 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,16{}\mathrm {i}}{7\,d}+\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,48{}\mathrm {i}}{7\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,48{}\mathrm {i}}{7\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,16{}\mathrm {i}}{7\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3} \]
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