Optimal. Leaf size=117 \[ -\frac {16 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}+\frac {24 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{9/2}}{9 a^7 d} \]
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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))^{9/2}}{9 a^7 d}-\frac {12 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}+\frac {24 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}-\frac {16 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^3 \sqrt {a+x} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \text {Subst}\left (\int \left (8 a^3 \sqrt {a+x}-12 a^2 (a+x)^{3/2}+6 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {16 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}+\frac {24 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{9/2}}{9 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 0.73, size = 108, normalized size = 0.92 \begin {gather*} \frac {2 \sec ^6(c+d x) (\cos (4 (c+d x))+i \sin (4 (c+d x))) (77 i+242 i \cos (2 (c+d x))+89 \sec (c+d x) \sin (3 (c+d x))+54 \tan (c+d x))}{315 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 = 0.79, size = 100, normalized size = 0.85
method | result | size |
default | \(-\frac {2 \left (128 i \left (\cos ^{4}\left (d x +c \right )\right )-128 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+226 i \left (\cos ^{2}\left (d x +c \right )\right )+130 \sin \left (d x +c \right ) \cos \left (d x +c \right )-35 i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{315 d \cos \left (d x +c \right )^{4} a^{3}}\) | \(100\) |
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 (35 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 270 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 756 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 840 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3}\right )}}{315 \, a^{7} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 134, normalized size = 1.15 \begin {gather*} -\frac {32 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (16 i \, e^{\left (9 i \, d x + 9 i \, c\right )} + 72 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 126 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 105 i \, e^{\left (3 i \, d x + 3 i \, c\right )}\right )}}{315 \, {\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{8}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \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 = 6.68, size = 306, normalized size = 2.62 \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}}\,512{}\mathrm {i}}{315\,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}}\,256{}\mathrm {i}}{315\,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}}\,64{}\mathrm {i}}{105\,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}}\,32{}\mathrm {i}}{63\,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}}\,32{}\mathrm {i}}{9\,a^3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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