Optimal. Leaf size=147 \[ \frac {256 i a^4 \sec ^{11}(c+d x)}{12155 d (a+i a \tan (c+d x))^{11/2}}+\frac {64 i a^3 \sec ^{11}(c+d x)}{1105 d (a+i a \tan (c+d x))^{9/2}}+\frac {8 i a^2 \sec ^{11}(c+d x)}{85 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^{11}(c+d x)}{17 d (a+i a \tan (c+d x))^{5/2}} \]
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Rubi [A] time = 0.26, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3494, 3493} \[ \frac {8 i a^2 \sec ^{11}(c+d x)}{85 d (a+i a \tan (c+d x))^{7/2}}+\frac {64 i a^3 \sec ^{11}(c+d x)}{1105 d (a+i a \tan (c+d x))^{9/2}}+\frac {256 i a^4 \sec ^{11}(c+d x)}{12155 d (a+i a \tan (c+d x))^{11/2}}+\frac {2 i a \sec ^{11}(c+d x)}{17 d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3493
Rule 3494
Rubi steps
\begin {align*} \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {2 i a \sec ^{11}(c+d x)}{17 d (a+i a \tan (c+d x))^{5/2}}+\frac {1}{17} (12 a) \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\\ &=\frac {8 i a^2 \sec ^{11}(c+d x)}{85 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^{11}(c+d x)}{17 d (a+i a \tan (c+d x))^{5/2}}+\frac {1}{85} \left (32 a^2\right ) \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\\ &=\frac {64 i a^3 \sec ^{11}(c+d x)}{1105 d (a+i a \tan (c+d x))^{9/2}}+\frac {8 i a^2 \sec ^{11}(c+d x)}{85 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^{11}(c+d x)}{17 d (a+i a \tan (c+d x))^{5/2}}+\frac {\left (128 a^3\right ) \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{9/2}} \, dx}{1105}\\ &=\frac {256 i a^4 \sec ^{11}(c+d x)}{12155 d (a+i a \tan (c+d x))^{11/2}}+\frac {64 i a^3 \sec ^{11}(c+d x)}{1105 d (a+i a \tan (c+d x))^{9/2}}+\frac {8 i a^2 \sec ^{11}(c+d x)}{85 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^{11}(c+d x)}{17 d (a+i a \tan (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.99, size = 108, normalized size = 0.73 \[ \frac {2 \sec ^9(c+d x) (\sin (4 (c+d x))+i \cos (4 (c+d x))) (-2242 i \cos (2 (c+d x))+374 \tan (c+d x)+1089 \sin (3 (c+d x)) \sec (c+d x)+475 i)}{12155 a d (\tan (c+d x)-i) \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.15, size = 184, normalized size = 1.25 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (565760 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 261120 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 69632 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8192 i\right )}}{12155 \, {\left (a^{2} d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a^{2} d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a^{2} d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{11}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 6.82, size = 171, normalized size = 1.16 \[ \frac {2 \left (4096 i \left (\cos ^{9}\left (d x +c \right )\right )+4096 \sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )-512 i \left (\cos ^{7}\left (d x +c \right )\right )+1536 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )-160 i \left (\cos ^{5}\left (d x +c \right )\right )+1120 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-84 i \left (\cos ^{3}\left (d x +c \right )\right )+924 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1573 i \cos \left (d x +c \right )-715 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{12155 d \cos \left (d x +c \right )^{8} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.12, size = 764, normalized size = 5.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.86, size = 301, normalized size = 2.05 \[ \frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,512{}\mathrm {i}}{11\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,1536{}\mathrm {i}}{13\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6}+\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,512{}\mathrm {i}}{5\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^7}-\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,512{}\mathrm {i}}{17\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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