Optimal. Leaf size=106 \[ -\frac {4 \sin ^7(c+d x)}{7 a^3 d}+\frac {9 \sin ^5(c+d x)}{5 a^3 d}-\frac {2 \sin ^3(c+d x)}{a^3 d}+\frac {\sin (c+d x)}{a^3 d}+\frac {4 i \cos ^7(c+d x)}{7 a^3 d}-\frac {i \cos ^5(c+d x)}{5 a^3 d} \]
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Rubi [A] time = 0.23, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3092, 3090, 2633, 2565, 30, 2564, 270, 14} \[ -\frac {4 \sin ^7(c+d x)}{7 a^3 d}+\frac {9 \sin ^5(c+d x)}{5 a^3 d}-\frac {2 \sin ^3(c+d x)}{a^3 d}+\frac {\sin (c+d x)}{a^3 d}+\frac {4 i \cos ^7(c+d x)}{7 a^3 d}-\frac {i \cos ^5(c+d x)}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 270
Rule 2564
Rule 2565
Rule 2633
Rule 3090
Rule 3092
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=\frac {i \int \cos ^4(c+d x) (i a \cos (c+d x)+a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {i \int \left (-i a^3 \cos ^7(c+d x)-3 a^3 \cos ^6(c+d x) \sin (c+d x)+3 i a^3 \cos ^5(c+d x) \sin ^2(c+d x)+a^3 \cos ^4(c+d x) \sin ^3(c+d x)\right ) \, dx}{a^6}\\ &=\frac {i \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx}{a^3}-\frac {(3 i) \int \cos ^6(c+d x) \sin (c+d x) \, dx}{a^3}+\frac {\int \cos ^7(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx}{a^3}\\ &=-\frac {i \operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}+\frac {(3 i) \operatorname {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {\operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{a^3 d}\\ &=\frac {3 i \cos ^7(c+d x)}{7 a^3 d}+\frac {\sin (c+d x)}{a^3 d}-\frac {\sin ^3(c+d x)}{a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {\sin ^7(c+d x)}{7 a^3 d}-\frac {i \operatorname {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{a^3 d}\\ &=-\frac {i \cos ^5(c+d x)}{5 a^3 d}+\frac {4 i \cos ^7(c+d x)}{7 a^3 d}+\frac {\sin (c+d x)}{a^3 d}-\frac {2 \sin ^3(c+d x)}{a^3 d}+\frac {9 \sin ^5(c+d x)}{5 a^3 d}-\frac {4 \sin ^7(c+d x)}{7 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 149, normalized size = 1.41 \[ \frac {5 \sin (c+d x)}{16 a^3 d}+\frac {\sin (3 (c+d x))}{8 a^3 d}+\frac {\sin (5 (c+d x))}{20 a^3 d}+\frac {\sin (7 (c+d x))}{112 a^3 d}+\frac {3 i \cos (c+d x)}{16 a^3 d}+\frac {i \cos (3 (c+d x))}{8 a^3 d}+\frac {i \cos (5 (c+d x))}{20 a^3 d}+\frac {i \cos (7 (c+d x))}{112 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.12, size = 63, normalized size = 0.59 \[ \frac {{\left (-35 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 140 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 70 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 28 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{560 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 119, normalized size = 1.12 \[ \frac {\frac {35}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}} + \frac {525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1960 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4025 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4480 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3143 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1176 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 243}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{7}}}{280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 141, normalized size = 1.33 \[ \frac {\frac {2}{16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 i}+\frac {4 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{6}}-\frac {9 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}+\frac {17 i}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{7}}+\frac {38}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}-\frac {15}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {15}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{d \,a^{3}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.17, size = 134, normalized size = 1.26 \[ -\frac {\left (35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,105{}\mathrm {i}-175\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,105{}\mathrm {i}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,77{}\mathrm {i}+43\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-13{}\mathrm {i}\right )\,2{}\mathrm {i}}{35\,a^3\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )\,{\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 201, normalized size = 1.90 \[ \begin {cases} - \frac {\left (71680 i a^{12} d^{4} e^{17 i c} e^{i d x} - 286720 i a^{12} d^{4} e^{15 i c} e^{- i d x} - 143360 i a^{12} d^{4} e^{13 i c} e^{- 3 i d x} - 57344 i a^{12} d^{4} e^{11 i c} e^{- 5 i d x} - 10240 i a^{12} d^{4} e^{9 i c} e^{- 7 i d x}\right ) e^{- 16 i c}}{1146880 a^{15} d^{5}} & \text {for}\: 1146880 a^{15} d^{5} e^{16 i c} \neq 0 \\\frac {x \left (e^{8 i c} + 4 e^{6 i c} + 6 e^{4 i c} + 4 e^{2 i c} + 1\right ) e^{- 7 i c}}{16 a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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