Optimal. Leaf size=117 \[ -\frac {16 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d}+\frac {8 i (a+i a \tan (c+d x))^{15/2}}{5 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{17/2}}{17 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{19/2}}{19 a^7 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 45}
\begin {gather*} \frac {2 i (a+i a \tan (c+d x))^{19/2}}{19 a^7 d}-\frac {12 i (a+i a \tan (c+d x))^{17/2}}{17 a^6 d}+\frac {8 i (a+i a \tan (c+d x))^{15/2}}{5 a^5 d}-\frac {16 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^3 (a+x)^{11/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \text {Subst}\left (\int \left (8 a^3 (a+x)^{11/2}-12 a^2 (a+x)^{13/2}+6 a (a+x)^{15/2}-(a+x)^{17/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {16 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d}+\frac {8 i (a+i a \tan (c+d x))^{15/2}}{5 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{17/2}}{17 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{19/2}}{19 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 1.61, size = 113, normalized size = 0.97 \begin {gather*} -\frac {2 a^2 \sec ^8(c+d x) (\cos (6 c+8 d x)+i \sin (6 c+8 d x)) (-833 i+3262 i \cos (2 (c+d x))+1599 \sec (c+d x) \sin (3 (c+d x))+494 \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}{20995 d (\cos (d x)+i \sin (d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 11.27, size = 171, normalized size = 1.46
method | result | size |
default | \(-\frac {2 \left (4096 i \left (\cos ^{9}\left (d x +c \right )\right )-4096 \sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )+512 i \left (\cos ^{7}\left (d x +c \right )\right )-2560 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+224 i \left (\cos ^{5}\left (d x +c \right )\right )-2016 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+132 i \left (\cos ^{3}\left (d x +c \right )\right )-1716 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-2535 i \cos \left (d x +c \right )+1105 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{20995 d \cos \left (d x +c \right )^{9}}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 0.65 \begin {gather*} \frac {2 i \, {\left (1105 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {19}{2}} - 7410 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {17}{2}} a + 16796 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a^{2} - 12920 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a^{3}\right )}}{20995 \, a^{7} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 190 vs. \(2 (85) = 170\).
time = 0.47, size = 190, normalized size = 1.62 \begin {gather*} -\frac {1024 \, \sqrt {2} {\left (16 i \, a^{2} e^{\left (19 i \, d x + 19 i \, c\right )} + 152 i \, a^{2} e^{\left (17 i \, d x + 17 i \, c\right )} + 646 i \, a^{2} e^{\left (15 i \, d x + 15 i \, c\right )} + 1615 i \, a^{2} e^{\left (13 i \, d x + 13 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{20995 \, {\left (d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = 15.90, size = 626, normalized size = 5.35 \begin {gather*} -\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}}\,16384{}\mathrm {i}}{20995\,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}}\,8192{}\mathrm {i}}{20995\,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}}\,6144{}\mathrm {i}}{20995\,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}}\,1024{}\mathrm {i}}{4199\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\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}}\,536576{}\mathrm {i}}{4199\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}-\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}}\,10484736{}\mathrm {i}}{20995\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}+\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}}\,17262592{}\mathrm {i}}{20995\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6}-\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}}\,1129472{}\mathrm {i}}{1615\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^7}+\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}}\,98304{}\mathrm {i}}{323\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^8}-\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}}\,1024{}\mathrm {i}}{19\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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