Optimal. Leaf size=146 \[ -\frac {32 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}+\frac {64 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}-\frac {16 i (a+i a \tan (c+d x))^{9/2}}{3 a^7 d}+\frac {16 i (a+i a \tan (c+d x))^{11/2}}{11 a^8 d}-\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^9 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 146, 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))^{13/2}}{13 a^9 d}+\frac {16 i (a+i a \tan (c+d x))^{11/2}}{11 a^8 d}-\frac {16 i (a+i a \tan (c+d x))^{9/2}}{3 a^7 d}+\frac {64 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}-\frac {32 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^4 (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac {i \text {Subst}\left (\int \left (16 a^4 (a+x)^{3/2}-32 a^3 (a+x)^{5/2}+24 a^2 (a+x)^{7/2}-8 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac {32 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}+\frac {64 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}-\frac {16 i (a+i a \tan (c+d x))^{9/2}}{3 a^7 d}+\frac {16 i (a+i a \tan (c+d x))^{11/2}}{11 a^8 d}-\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^9 d}\\ \end {align*}
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Mathematica [A]
time = 0.95, size = 116, normalized size = 0.79 \begin {gather*} \frac {2 \sec ^9(c+d x) (2288 i+4264 i \cos (2 (c+d x))+3131 i \cos (4 (c+d x))+2600 \sin (2 (c+d x))+2875 \sin (4 (c+d x))) (\cos (5 (c+d x))+i \sin (5 (c+d x)))}{15015 a^2 d (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.84, size = 127, normalized size = 0.87
method | result | size |
default | \(-\frac {2 \left (4096 i \left (\cos ^{6}\left (d x +c \right )\right )-4096 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+512 i \left (\cos ^{4}\left (d x +c \right )\right )-2560 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+6230 i \left (\cos ^{2}\left (d x +c \right )\right )+3990 \sin \left (d x +c \right ) \cos \left (d x +c \right )-1155 i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{15015 d \cos \left (d x +c \right )^{6} a^{3}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 94, normalized size = 0.64 \begin {gather*} -\frac {2 i \, {\left (1155 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 10920 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 40040 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2} - 68640 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} + 48048 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4}\right )}}{15015 \, a^{9} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 175, normalized size = 1.20 \begin {gather*} -\frac {128 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (128 i \, e^{\left (13 i \, d x + 13 i \, c\right )} + 832 i \, e^{\left (11 i \, d x + 11 i \, c\right )} + 2288 i \, e^{\left (9 i \, d x + 9 i \, c\right )} + 3432 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 3003 i \, e^{\left (5 i \, d x + 5 i \, c\right )}\right )}}{15015 \, {\left (a^{3} d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} 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 = 8.76, size = 434, normalized size = 2.97 \begin {gather*} -\frac {\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}}{15015\,a^3\,d}-\frac {\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}}{15015\,a^3\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\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}}\,2048{}\mathrm {i}}{5005\,a^3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {\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}}{3003\,a^3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\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}}{429\,a^3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\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}}\,1792{}\mathrm {i}}{143\,a^3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {\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\,a^3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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