Optimal. Leaf size=146 \[ -\frac {32 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac {64 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}-\frac {48 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}+\frac {16 i (a+i a \tan (c+d x))^{9/2}}{9 a^8 d}-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^9 d} \]
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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))^{11/2}}{11 a^9 d}+\frac {16 i (a+i a \tan (c+d x))^{9/2}}{9 a^8 d}-\frac {48 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}+\frac {64 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}-\frac {32 i (a+i a \tan (c+d x))^{3/2}}{3 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))^{7/2}} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^4 \sqrt {a+x} \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac {i \text {Subst}\left (\int \left (16 a^4 \sqrt {a+x}-32 a^3 (a+x)^{3/2}+24 a^2 (a+x)^{5/2}-8 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac {32 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac {64 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}-\frac {48 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}+\frac {16 i (a+i a \tan (c+d x))^{9/2}}{9 a^8 d}-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^9 d}\\ \end {align*}
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Mathematica [A]
time = 0.99, size = 114, normalized size = 0.78 \begin {gather*} \frac {2 \sec ^9(c+d x) (1584+2552 \cos (2 (c+d x))+1283 \cos (4 (c+d x))-1144 i \sin (2 (c+d x))-1027 i \sin (4 (c+d x))) (\cos (5 (c+d x))+i \sin (5 (c+d x)))}{3465 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 117, normalized size = 0.80
method | result | size |
default | \(\frac {2 \left (-2048 i \left (\cos ^{5}\left (d x +c \right )\right )+2048 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-4876 i \left (\cos ^{3}\left (d x +c \right )\right )-3340 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+1505 i \cos \left (d x +c \right )+315 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3465 d \cos \left (d x +c \right )^{5} a^{4}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 94, normalized size = 0.64 \begin {gather*} -\frac {2 i \, {\left (315 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 3080 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 11880 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} - 22176 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{3} + 18480 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{4}\right )}}{3465 \, a^{9} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 160, normalized size = 1.10 \begin {gather*} -\frac {64 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (128 i \, e^{\left (11 i \, d x + 11 i \, c\right )} + 704 i \, e^{\left (9 i \, d x + 9 i \, c\right )} + 1584 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 1848 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 1155 i \, e^{\left (3 i \, d x + 3 i \, c\right )}\right )}}{3465 \, {\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} 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 = 7.63, size = 370, normalized size = 2.53 \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}}\,8192{}\mathrm {i}}{3465\,a^4\,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}}\,4096{}\mathrm {i}}{3465\,a^4\,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}}\,1024{}\mathrm {i}}{1155\,a^4\,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}}\,512{}\mathrm {i}}{693\,a^4\,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}}\,64{}\mathrm {i}}{99\,a^4\,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}}\,64{}\mathrm {i}}{11\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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