Integrand size = 26, antiderivative size = 59 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]
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Time = 0.09 (sec) , antiderivative size = 59, 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} \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}-\frac {4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x) (a+x)^{9/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {i \text {Subst}\left (\int \left (2 a (a+x)^{9/2}-(a+x)^{11/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 a^3 (15-11 i \tan (c+d x)) (-i+\tan (c+d x))^5 \sqrt {a+i a \tan (c+d x)}}{143 d} \]
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Time = 187.49 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}-\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}\right )}{d \,a^{3}}\) | \(44\) |
default | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}-\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}\right )}{d \,a^{3}}\) | \(44\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (43) = 86\).
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.14 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {128 \, \sqrt {2} {\left (2 i \, a^{3} e^{\left (13 i \, d x + 13 i \, c\right )} + 13 i \, a^{3} e^{\left (11 i \, d x + 11 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{143 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Timed out. \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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none
Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 i \, {\left (11 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 26 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a\right )}}{143 \, a^{3} d} \]
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\[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{4} \,d x } \]
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Time = 8.68 (sec) , antiderivative size = 434, normalized size of antiderivative = 7.36 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {a^3\,\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}}{143\,d}-\frac {a^3\,\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}}{143\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}+\frac {a^3\,\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}}\,4480{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {a^3\,\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}}\,11520{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {a^3\,\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}}\,12800{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}-\frac {a^3\,\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}}\,6784{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}+\frac {a^3\,\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}}{13\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6} \]
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