Optimal. Leaf size=125 \[ -\frac {1}{32 a^3 d (-\cot (c+d x)+i)}+\frac {13}{16 a^3 d (\cot (c+d x)+i)}-\frac {23 i}{32 a^3 d (\cot (c+d x)+i)^2}-\frac {1}{3 a^3 d (\cot (c+d x)+i)^3}+\frac {i}{16 a^3 d (\cot (c+d x)+i)^4}+\frac {5 x}{32 a^3} \]
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Rubi [A] time = 0.11, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3088, 848, 88, 203} \[ -\frac {1}{32 a^3 d (-\cot (c+d x)+i)}+\frac {13}{16 a^3 d (\cot (c+d x)+i)}-\frac {23 i}{32 a^3 d (\cot (c+d x)+i)^2}-\frac {1}{3 a^3 d (\cot (c+d x)+i)^3}+\frac {i}{16 a^3 d (\cot (c+d x)+i)^4}+\frac {5 x}{32 a^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 203
Rule 848
Rule 3088
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^5}{(i a+a x)^3 \left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^5}{\left (-\frac {i}{a}+\frac {x}{a}\right )^2 (i a+a x)^5} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{32 a^3 (-i+x)^2}+\frac {i}{4 a^3 (i+x)^5}-\frac {1}{a^3 (i+x)^4}-\frac {23 i}{16 a^3 (i+x)^3}+\frac {13}{16 a^3 (i+x)^2}+\frac {5}{32 a^3 \left (1+x^2\right )}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {1}{32 a^3 d (i-\cot (c+d x))}+\frac {i}{16 a^3 d (i+\cot (c+d x))^4}-\frac {1}{3 a^3 d (i+\cot (c+d x))^3}-\frac {23 i}{32 a^3 d (i+\cot (c+d x))^2}+\frac {13}{16 a^3 d (i+\cot (c+d x))}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{32 a^3 d}\\ &=\frac {5 x}{32 a^3}-\frac {1}{32 a^3 d (i-\cot (c+d x))}+\frac {i}{16 a^3 d (i+\cot (c+d x))^4}-\frac {1}{3 a^3 d (i+\cot (c+d x))^3}-\frac {23 i}{32 a^3 d (i+\cot (c+d x))^2}+\frac {13}{16 a^3 d (i+\cot (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 106, normalized size = 0.85 \[ \frac {132 \sin (2 (c+d x))+60 \sin (4 (c+d x))+20 \sin (6 (c+d x))+3 \sin (8 (c+d x))+108 i \cos (2 (c+d x))+60 i \cos (4 (c+d x))+20 i \cos (6 (c+d x))+3 i \cos (8 (c+d x))+120 c+120 d x}{768 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 76, normalized size = 0.61 \[ \frac {{\left (120 \, d x e^{\left (8 i \, d x + 8 i \, c\right )} - 12 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 120 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 60 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 20 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{768 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 119, normalized size = 0.95 \[ -\frac {-\frac {60 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {60 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac {12 \, {\left (5 \, \tan \left (d x + c\right ) + 7 i\right )}}{a^{3} {\left (i \, \tan \left (d x + c\right ) - 1\right )}} + \frac {-125 i \, \tan \left (d x + c\right )^{4} - 596 \, \tan \left (d x + c\right )^{3} + 1110 i \, \tan \left (d x + c\right )^{2} + 996 \, \tan \left (d x + c\right ) - 405 i}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 137, normalized size = 1.10 \[ \frac {5 i \ln \left (\tan \left (d x +c \right )+i\right )}{64 a^{3} d}+\frac {1}{32 a^{3} d \left (\tan \left (d x +c \right )+i\right )}-\frac {5 i \ln \left (\tan \left (d x +c \right )-i\right )}{64 a^{3} d}+\frac {i}{16 a^{3} d \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {3 i}{32 a^{3} d \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{12 a^{3} d \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{8 a^{3} d \left (\tan \left (d x +c \right )-i\right )} \]
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 = 4.79, size = 164, normalized size = 1.31 \[ \frac {5\,x}{32\,a^3}+\frac {-\frac {27\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{16}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,33{}\mathrm {i}}{8}+\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,9{}\mathrm {i}}{8}+\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{24}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,9{}\mathrm {i}}{8}+\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,33{}\mathrm {i}}{8}-\frac {27\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )}^2\,{\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 228, normalized size = 1.82 \[ \begin {cases} - \frac {\left (100663296 i a^{12} d^{4} e^{22 i c} e^{2 i d x} - 1006632960 i a^{12} d^{4} e^{18 i c} e^{- 2 i d x} - 503316480 i a^{12} d^{4} e^{16 i c} e^{- 4 i d x} - 167772160 i a^{12} d^{4} e^{14 i c} e^{- 6 i d x} - 25165824 i a^{12} d^{4} e^{12 i c} e^{- 8 i d x}\right ) e^{- 20 i c}}{6442450944 a^{15} d^{5}} & \text {for}\: 6442450944 a^{15} d^{5} e^{20 i c} \neq 0 \\x \left (\frac {\left (e^{10 i c} + 5 e^{8 i c} + 10 e^{6 i c} + 10 e^{4 i c} + 5 e^{2 i c} + 1\right ) e^{- 8 i c}}{32 a^{3}} - \frac {5}{32 a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {5 x}{32 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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