Optimal. Leaf size=147 \[ \frac {256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac {64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac {24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A]
time = 0.19, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3575, 3574}
\begin {gather*} \frac {256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac {64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac {24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3574
Rule 3575
Rubi steps
\begin {align*} \int \sec ^7(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx &=\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}+\frac {1}{13} (12 a) \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}+\frac {1}{143} \left (96 a^2\right ) \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac {64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac {24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}+\frac {1}{429} \left (128 a^3\right ) \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\\ &=\frac {256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac {64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac {24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i a \sec ^7(c+d x)}{13 d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 95, normalized size = 0.65 \begin {gather*} \frac {2 \sec ^6(c+d x) (390 \cos (c+d x)+445 \cos (3 (c+d x))+7 i (26 \sin (c+d x)+59 \sin (3 (c+d x)))) (i \cos (4 (c+d x))+\sin (4 (c+d x))) \sqrt {a+i a \tan (c+d x)}}{3003 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.14, size = 141, normalized size = 0.96
method | result | size |
default | \(\frac {2 \left (1024 i \left (\cos ^{7}\left (d x +c \right )\right )+1024 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )-128 i \left (\cos ^{5}\left (d x +c \right )\right )+384 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-40 i \left (\cos ^{3}\left (d x +c \right )\right )+280 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-21 i \cos \left (d x +c \right )+231 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3003 d \cos \left (d x +c \right )^{6}}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A]
time = 0.43, size = 132, normalized size = 0.90 \begin {gather*} -\frac {128 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-429 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 286 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 104 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 16 i\right )}}{3003 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + 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 \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \sec ^{7}{\left (c + d x \right )}\, 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 = 8.35, size = 289, normalized size = 1.97 \begin {gather*} \frac {{\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}}\,128{}\mathrm {i}}{7\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {{\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}}\,128{}\mathrm {i}}{3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {{\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}}\,384{}\mathrm {i}}{11\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {{\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}}\,128{}\mathrm {i}}{13\,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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