Optimal. Leaf size=110 \[ \frac {64 i a^3 \sec ^{13}(c+d x)}{3315 d (a+i a \tan (c+d x))^{13/2}}+\frac {16 i a^2 \sec ^{13}(c+d x)}{255 d (a+i a \tan (c+d x))^{11/2}}+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 110, 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 = {3494, 3493} \[ \frac {16 i a^2 \sec ^{13}(c+d x)}{255 d (a+i a \tan (c+d x))^{11/2}}+\frac {64 i a^3 \sec ^{13}(c+d x)}{3315 d (a+i a \tan (c+d x))^{13/2}}+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 3493
Rule 3494
Rubi steps
\begin {align*} \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}+\frac {1}{17} (8 a) \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{9/2}} \, dx\\ &=\frac {16 i a^2 \sec ^{13}(c+d x)}{255 d (a+i a \tan (c+d x))^{11/2}}+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}+\frac {1}{255} \left (32 a^2\right ) \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{11/2}} \, dx\\ &=\frac {64 i a^3 \sec ^{13}(c+d x)}{3315 d (a+i a \tan (c+d x))^{13/2}}+\frac {16 i a^2 \sec ^{13}(c+d x)}{255 d (a+i a \tan (c+d x))^{11/2}}+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 92, normalized size = 0.84 \[ -\frac {2 \sec ^{12}(c+d x) (247 i \sin (2 (c+d x))+263 \cos (2 (c+d x))+68) (\cos (3 (c+d x))-i \sin (3 (c+d x)))}{3315 a^3 d (\tan (c+d x)-i)^3 \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 173, normalized size = 1.57 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (130560 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 34816 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 4096 i\right )}}{3315 \, {\left (a^{4} d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a^{4} d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a^{4} d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{13}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 6.86, size = 171, normalized size = 1.55 \[ \frac {2 \left (2048 i \left (\cos ^{9}\left (d x +c \right )\right )+2048 \sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )-256 i \left (\cos ^{7}\left (d x +c \right )\right )+768 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )-80 i \left (\cos ^{5}\left (d x +c \right )\right )+560 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-2252 i \left (\cos ^{3}\left (d x +c \right )\right )-1748 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+871 i \cos \left (d x +c \right )+195 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3315 d \cos \left (d x +c \right )^{8} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.08, size = 902, normalized size = 8.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.28, size = 105, normalized size = 0.95 \[ \frac {512\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,68{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,255{}\mathrm {i}+8{}\mathrm {i}\right )}{3315\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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