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 = 0.19, 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 = {3575, 3574}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 3574
Rule 3575
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 = 1.04, size = 108, normalized size = 0.73 \begin {gather*} \frac {2 \sec ^9(c+d x) (i \cos (4 (c+d x))+\sin (4 (c+d x))) (475 i-2242 i \cos (2 (c+d x))+1089 \sec (c+d x) \sin (3 (c+d x))+374 \tan (c+d x))}{12155 a d (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.52, size = 171, normalized size = 1.16
method | result | size |
default | \(\frac {2 \left (4096 i \left (\cos ^{9}\left (d x +c \right )\right )+4096 \sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )-512 i \left (\cos ^{7}\left (d x +c \right )\right )+1536 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )-160 i \left (\cos ^{5}\left (d x +c \right )\right )+1120 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-84 i \left (\cos ^{3}\left (d x +c \right )\right )+924 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1573 i \cos \left (d x +c \right )-715 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{12155 d \cos \left (d x +c \right )^{8} a^{2}}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 764 vs. \(2 (115) = 230\).
time = 0.53, size = 764, normalized size = 5.20 \begin {gather*} -\frac {2 \, {\left (-1767 i \, \sqrt {a} - \frac {6854 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2088 i \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {16438 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5661 i \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {56984 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {13328 i \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {129336 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {7514 i \, \sqrt {a} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {156468 \, \sqrt {a} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {156468 \, \sqrt {a} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {7514 i \, \sqrt {a} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {129336 \, \sqrt {a} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {13328 i \, \sqrt {a} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {56984 \, \sqrt {a} \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} + \frac {5661 i \, \sqrt {a} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {16438 \, \sqrt {a} \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {2088 i \, \sqrt {a} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} - \frac {6854 \, \sqrt {a} \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} + \frac {1767 i \, \sqrt {a} \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}\right )} {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac {3}{2}}}{12155 \, {\left (a^{2} - \frac {10 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {120 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {252 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {120 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {10 \, a^{2} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {a^{2} \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}\right )} d {\left (-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 184, normalized size = 1.25 \begin {gather*} -\frac {512 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-1105 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 510 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 136 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 16 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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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