Optimal. Leaf size=229 \[ \frac {x}{256 a^8}+\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{192 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.13, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3560, 8}
\begin {gather*} \frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {x}{256 a^8}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{16 d (a+i a \tan (c+d x))^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3560
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^7} \, dx}{2 a}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^6} \, dx}{4 a^2}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^5} \, dx}{8 a^3}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^4} \, dx}{16 a^4}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^3} \, dx}{32 a^5}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {\int \frac {1}{(a+i a \tan (c+d x))^2} \, dx}{64 a^6}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {\int \frac {1}{a+i a \tan (c+d x)} \, dx}{128 a^7}\\ &=\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {\int 1 \, dx}{256 a^8}\\ &=\frac {x}{256 a^8}+\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{192 a^5 d (a+i a \tan (c+d x))^3}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 148, normalized size = 0.65 \begin {gather*} \frac {\sec ^8(c+d x) (14700 i+25088 i \cos (2 (c+d x))+15680 i \cos (4 (c+d x))+7680 i \cos (6 (c+d x))+105 i \cos (8 (c+d x))+1680 d x \cos (8 (c+d x))-6272 \sin (2 (c+d x))-7840 \sin (4 (c+d x))-5760 \sin (6 (c+d x))+105 \sin (8 (c+d x))+1680 i d x \sin (8 (c+d x)))}{430080 a^8 d (-i+\tan (c+d x))^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 143, normalized size = 0.62
method | result | size |
derivativedivides | \(\frac {\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{8}}+\frac {i}{128 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{512}-\frac {i}{48 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {i}{256 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{28 \left (\tan \left (d x +c \right )-i\right )^{7}}+\frac {1}{80 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {1}{192 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{256 \tan \left (d x +c \right )-256 i}+\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{512}}{d \,a^{8}}\) | \(143\) |
default | \(\frac {\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{8}}+\frac {i}{128 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{512}-\frac {i}{48 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {i}{256 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{28 \left (\tan \left (d x +c \right )-i\right )^{7}}+\frac {1}{80 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {1}{192 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{256 \tan \left (d x +c \right )-256 i}+\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{512}}{d \,a^{8}}\) | \(143\) |
risch | \(\frac {x}{256 a^{8}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{64 a^{8} d}+\frac {7 i {\mathrm e}^{-4 i \left (d x +c \right )}}{256 a^{8} d}+\frac {7 i {\mathrm e}^{-6 i \left (d x +c \right )}}{192 a^{8} d}+\frac {35 i {\mathrm e}^{-8 i \left (d x +c \right )}}{1024 a^{8} d}+\frac {7 i {\mathrm e}^{-10 i \left (d x +c \right )}}{320 a^{8} d}+\frac {7 i {\mathrm e}^{-12 i \left (d x +c \right )}}{768 a^{8} d}+\frac {i {\mathrm e}^{-14 i \left (d x +c \right )}}{448 a^{8} d}+\frac {i {\mathrm e}^{-16 i \left (d x +c \right )}}{4096 a^{8} d}\) | \(152\) |
norman | \(\frac {\frac {35 x \left (\tan ^{8}\left (d x +c \right )\right )}{128 a}+\frac {961 \left (\tan ^{7}\left (d x +c \right )\right )}{8960 d a}+\frac {x}{256 a}+\frac {x \left (\tan ^{2}\left (d x +c \right )\right )}{32 a}+\frac {3371 \left (\tan ^{5}\left (d x +c \right )\right )}{1280 d a}+\frac {255 \tan \left (d x +c \right )}{256 d a}-\frac {1117 \left (\tan ^{3}\left (d x +c \right )\right )}{256 d a}+\frac {7 x \left (\tan ^{10}\left (d x +c \right )\right )}{32 a}+\frac {7 x \left (\tan ^{12}\left (d x +c \right )\right )}{64 a}+\frac {x \left (\tan ^{14}\left (d x +c \right )\right )}{32 a}+\frac {x \left (\tan ^{16}\left (d x +c \right )\right )}{256 a}+\frac {7 x \left (\tan ^{6}\left (d x +c \right )\right )}{32 a}+\frac {64 i \left (\tan ^{4}\left (d x +c \right )\right )}{15 d a}+\frac {7 x \left (\tan ^{4}\left (d x +c \right )\right )}{64 a}+\frac {5053 \left (\tan ^{9}\left (d x +c \right )\right )}{26880 d a}+\frac {383 \left (\tan ^{11}\left (d x +c \right )\right )}{3840 d a}+\frac {23 \left (\tan ^{13}\left (d x +c \right )\right )}{768 d a}+\frac {\tan ^{15}\left (d x +c \right )}{256 d a}-\frac {292 i \left (\tan ^{2}\left (d x +c \right )\right )}{105 d a}+\frac {16 i}{105 d a}-\frac {4 i \left (\tan ^{6}\left (d x +c \right )\right )}{5 d a}}{a^{7} \left (1+\tan ^{2}\left (d x +c \right )\right )^{8}}\) | \(322\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 109, normalized size = 0.48 \begin {gather*} \frac {{\left (1680 \, d x e^{\left (16 i \, d x + 16 i \, c\right )} + 6720 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 11760 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 15680 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 14700 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 9408 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 3920 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 960 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105 i\right )} e^{\left (-16 i \, d x - 16 i \, c\right )}}{430080 \, a^{8} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.45, size = 325, normalized size = 1.42 \begin {gather*} \begin {cases} \frac {\left (22698142121947299840 i a^{56} d^{7} e^{70 i c} e^{- 2 i d x} + 39721748713407774720 i a^{56} d^{7} e^{68 i c} e^{- 4 i d x} + 52962331617877032960 i a^{56} d^{7} e^{66 i c} e^{- 6 i d x} + 49652185891759718400 i a^{56} d^{7} e^{64 i c} e^{- 8 i d x} + 31777398970726219776 i a^{56} d^{7} e^{62 i c} e^{- 10 i d x} + 13240582904469258240 i a^{56} d^{7} e^{60 i c} e^{- 12 i d x} + 3242591731706757120 i a^{56} d^{7} e^{58 i c} e^{- 14 i d x} + 354658470655426560 i a^{56} d^{7} e^{56 i c} e^{- 16 i d x}\right ) e^{- 72 i c}}{1452681095804627189760 a^{64} d^{8}} & \text {for}\: a^{64} d^{8} e^{72 i c} \neq 0 \\x \left (\frac {\left (e^{16 i c} + 8 e^{14 i c} + 28 e^{12 i c} + 56 e^{10 i c} + 70 e^{8 i c} + 56 e^{6 i c} + 28 e^{4 i c} + 8 e^{2 i c} + 1\right ) e^{- 16 i c}}{256 a^{8}} - \frac {1}{256 a^{8}}\right ) & \text {otherwise} \end {cases} + \frac {x}{256 a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.88, size = 132, normalized size = 0.58 \begin {gather*} -\frac {-\frac {840 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{8}} + \frac {840 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{8}} + \frac {-2283 i \, \tan \left (d x + c\right )^{8} - 19944 \, \tan \left (d x + c\right )^{7} + 77364 i \, \tan \left (d x + c\right )^{6} + 175448 \, \tan \left (d x + c\right )^{5} - 258370 i \, \tan \left (d x + c\right )^{4} - 261464 \, \tan \left (d x + c\right )^{3} + 192052 i \, \tan \left (d x + c\right )^{2} + 114152 \, \tan \left (d x + c\right ) - 67819 i}{a^{8} {\left (\tan \left (d x + c\right ) - i\right )}^{8}}}{430080 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.96, size = 198, normalized size = 0.86 \begin {gather*} \frac {x}{256\,a^8}-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,5993{}\mathrm {i}}{26880\,a^8}+\frac {16}{105\,a^8}-\frac {143\,{\mathrm {tan}\left (c+d\,x\right )}^2}{480\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,1193{}\mathrm {i}}{3840\,a^8}+\frac {11\,{\mathrm {tan}\left (c+d\,x\right )}^4}{48\,a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,85{}\mathrm {i}}{768\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{32\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,1{}\mathrm {i}}{256\,a^8}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^8\,1{}\mathrm {i}+8\,{\mathrm {tan}\left (c+d\,x\right )}^7-{\mathrm {tan}\left (c+d\,x\right )}^6\,28{}\mathrm {i}-56\,{\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,70{}\mathrm {i}+56\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,28{}\mathrm {i}-8\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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