Optimal. Leaf size=111 \[ -\frac {i a \cos ^8(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a x}{128} \]
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Rubi [A] time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3486, 2635, 8} \[ -\frac {i a \cos ^8(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a x}{128} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3486
Rubi steps
\begin {align*} \int \cos ^8(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {i a \cos ^8(c+d x)}{8 d}+a \int \cos ^8(c+d x) \, dx\\ &=-\frac {i a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} (7 a) \int \cos ^6(c+d x) \, dx\\ &=-\frac {i a \cos ^8(c+d x)}{8 d}+\frac {7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} (35 a) \int \cos ^4(c+d x) \, dx\\ &=-\frac {i a \cos ^8(c+d x)}{8 d}+\frac {35 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} (35 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {i a \cos ^8(c+d x)}{8 d}+\frac {35 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} (35 a) \int 1 \, dx\\ &=\frac {35 a x}{128}-\frac {i a \cos ^8(c+d x)}{8 d}+\frac {35 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {7 a \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 68, normalized size = 0.61 \[ \frac {a \left (672 \sin (2 (c+d x))+168 \sin (4 (c+d x))+32 \sin (6 (c+d x))+3 \sin (8 (c+d x))-384 i \cos ^8(c+d x)+840 c+840 d x\right )}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 104, normalized size = 0.94 \[ \frac {{\left (840 \, a d x e^{\left (6 i \, d x + 6 i \, c\right )} - 3 i \, a e^{\left (14 i \, d x + 14 i \, c\right )} - 28 i \, a e^{\left (12 i \, d x + 12 i \, c\right )} - 126 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 420 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 252 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 42 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 151, normalized size = 1.36 \[ \frac {{\left (840 \, a d x e^{\left (6 i \, d x + 2 i \, c\right )} + 84 i \, a e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 84 i \, a e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 3 i \, a e^{\left (14 i \, d x + 10 i \, c\right )} - 28 i \, a e^{\left (12 i \, d x + 8 i \, c\right )} - 126 i \, a e^{\left (10 i \, d x + 6 i \, c\right )} - 420 i \, a e^{\left (8 i \, d x + 4 i \, c\right )} + 42 i \, a e^{\left (2 i \, d x - 2 i \, c\right )} + 252 i \, a e^{\left (4 i \, d x\right )} + 4 i \, a e^{\left (-4 i \, c\right )}\right )} e^{\left (-6 i \, d x - 2 i \, c\right )}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 73, normalized size = 0.66 \[ \frac {-\frac {i a \left (\cos ^{8}\left (d x +c \right )\right )}{8}+a \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 103, normalized size = 0.93 \[ \frac {105 \, {\left (d x + c\right )} a + \frac {105 \, a \tan \left (d x + c\right )^{7} + 385 \, a \tan \left (d x + c\right )^{5} + 511 \, a \tan \left (d x + c\right )^{3} + 279 \, a \tan \left (d x + c\right ) - 48 i \, a}{\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{2} + 1}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.85, size = 152, normalized size = 1.37 \[ \frac {35\,a\,x}{128}+\frac {\frac {35\,a\,{\mathrm {tan}\left (c+d\,x\right )}^6}{128}+\frac {35{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^5}{128}+\frac {35\,a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{48}+\frac {35{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{48}+\frac {77\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{128}+\frac {77{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{128}+\frac {a}{8}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^7+{\mathrm {tan}\left (c+d\,x\right )}^6\,1{}\mathrm {i}+3\,{\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,3{}\mathrm {i}+3\,{\mathrm {tan}\left (c+d\,x\right )}^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}+\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 282, normalized size = 2.54 \[ \frac {35 a x}{128} + \begin {cases} - \frac {\left (10133099161583616 i a d^{6} e^{20 i c} e^{8 i d x} + 94575592174780416 i a d^{6} e^{18 i c} e^{6 i d x} + 425590164786511872 i a d^{6} e^{16 i c} e^{4 i d x} + 1418633882621706240 i a d^{6} e^{14 i c} e^{2 i d x} - 851180329573023744 i a d^{6} e^{10 i c} e^{- 2 i d x} - 141863388262170624 i a d^{6} e^{8 i c} e^{- 4 i d x} - 13510798882111488 i a d^{6} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{10376293541461622784 d^{7}} & \text {for}\: 10376293541461622784 d^{7} e^{12 i c} \neq 0 \\x \left (- \frac {35 a}{128} + \frac {\left (a e^{14 i c} + 7 a e^{12 i c} + 21 a e^{10 i c} + 35 a e^{8 i c} + 35 a e^{6 i c} + 21 a e^{4 i c} + 7 a e^{2 i c} + a\right ) e^{- 6 i c}}{128}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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