Optimal. Leaf size=110 \[ \frac {64 i a^3 \sec ^5(c+d x)}{315 d (a+i a \tan (c+d x))^{5/2}}+\frac {16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^5(c+d x)}{9 d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.18, 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} \[ \frac {16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac {64 i a^3 \sec ^5(c+d x)}{315 d (a+i a \tan (c+d x))^{5/2}}+\frac {2 i a \sec ^5(c+d x)}{9 d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3493
Rule 3494
Rubi steps
\begin {align*} \int \sec ^5(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx &=\frac {2 i a \sec ^5(c+d x)}{9 d \sqrt {a+i a \tan (c+d x)}}+\frac {1}{9} (8 a) \int \frac {\sec ^5(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^5(c+d x)}{9 d \sqrt {a+i a \tan (c+d x)}}+\frac {1}{63} \left (32 a^2\right ) \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac {64 i a^3 \sec ^5(c+d x)}{315 d (a+i a \tan (c+d x))^{5/2}}+\frac {16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^5(c+d x)}{9 d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 77, normalized size = 0.70 \[ \frac {2 \sec ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (55 i \sin (2 (c+d x))+71 \cos (2 (c+d x))+36) (\sin (3 (c+d x))+i \cos (3 (c+d x)))}{315 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 97, normalized size = 0.88 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (2016 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1152 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 256 i\right )}}{315 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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 \[ \int \sqrt {i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.28, size = 114, normalized size = 1.04 \[ \frac {2 \left (128 i \left (\cos ^{5}\left (d x +c \right )\right )+128 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-16 i \left (\cos ^{3}\left (d x +c \right )\right )+48 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-5 i \cos \left (d x +c \right )+35 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{315 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [B] time = 6.08, size = 102, normalized size = 0.93 \[ \frac {32\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,36{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,63{}\mathrm {i}+8{}\mathrm {i}\right )}{315\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4} \]
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 \[ \int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \sec ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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