Optimal. Leaf size=144 \[ -\frac{i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac{i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac{i a^6}{32 d (a-i a \tan (c+d x))}+\frac{a^5 x}{32} \]
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Rubi [A] time = 0.0859356, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3487, 44, 206} \[ -\frac{i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac{i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac{i a^6}{32 d (a-i a \tan (c+d x))}+\frac{a^5 x}{32} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{\left (i a^{11}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^6 (a+x)} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^{11}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 a (a-x)^6}+\frac{1}{4 a^2 (a-x)^5}+\frac{1}{8 a^3 (a-x)^4}+\frac{1}{16 a^4 (a-x)^3}+\frac{1}{32 a^5 (a-x)^2}+\frac{1}{32 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac{i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac{i a^6}{32 d (a-i a \tan (c+d x))}-\frac{\left (i a^6\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{32 d}\\ &=\frac{a^5 x}{32}-\frac{i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac{i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac{i a^8}{24 d (a-i a \tan (c+d x))^3}-\frac{i a^7}{32 d (a-i a \tan (c+d x))^2}-\frac{i a^6}{32 d (a-i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.41093, size = 137, normalized size = 0.95 \[ \frac{a^5 (-100 \sin (c+d x)-225 \sin (3 (c+d x))-120 i d x \sin (5 (c+d x))+12 \sin (5 (c+d x))-500 i \cos (c+d x)-375 i \cos (3 (c+d x))+120 d x \cos (5 (c+d x))-12 i \cos (5 (c+d x))) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{3840 d (\cos (d x)+i \sin (d x))^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.135, size = 331, normalized size = 2.3 \begin{align*}{\frac{1}{d} \left ( i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{10}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{20}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{60}} \right ) +5\,{a}^{5} \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) -10\,i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{10}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{40}} \right ) -10\,{a}^{5} \left ( -1/10\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{\sin \left ( dx+c \right ) }{80} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+7/6\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{7\,dx}{256}}+{\frac{7\,c}{256}} \right ) -{\frac{i}{2}}{a}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{10}+{a}^{5} \left ({\frac{\sin \left ( dx+c \right ) }{10} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{21\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{64}}+{\frac{315\,\cos \left ( dx+c \right ) }{128}} \right ) }+{\frac{63\,dx}{256}}+{\frac{63\,c}{256}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73067, size = 221, normalized size = 1.53 \begin{align*} \frac{120 \,{\left (d x + c\right )} a^{5} + \frac{120 \, a^{5} \tan \left (d x + c\right )^{9} + 560 \, a^{5} \tan \left (d x + c\right )^{7} + 1024 \, a^{5} \tan \left (d x + c\right )^{5} - 640 i \, a^{5} \tan \left (d x + c\right )^{4} - 1840 \, a^{5} \tan \left (d x + c\right )^{3} + 4480 i \, a^{5} \tan \left (d x + c\right )^{2} + 3720 \, a^{5} \tan \left (d x + c\right ) - 1024 i \, a^{5}}{\tan \left (d x + c\right )^{10} + 5 \, \tan \left (d x + c\right )^{8} + 10 \, \tan \left (d x + c\right )^{6} + 10 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}}{3840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08802, size = 248, normalized size = 1.72 \begin{align*} \frac{120 \, a^{5} d x - 12 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 75 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 200 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 300 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 300 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )}}{3840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0506, size = 211, normalized size = 1.47 \begin{align*} \frac{a^{5} x}{32} + \begin{cases} \frac{- 100663296 i a^{5} d^{4} e^{10 i c} e^{10 i d x} - 629145600 i a^{5} d^{4} e^{8 i c} e^{8 i d x} - 1677721600 i a^{5} d^{4} e^{6 i c} e^{6 i d x} - 2516582400 i a^{5} d^{4} e^{4 i c} e^{4 i d x} - 2516582400 i a^{5} d^{4} e^{2 i c} e^{2 i d x}}{32212254720 d^{5}} & \text{for}\: 32212254720 d^{5} \neq 0 \\x \left (\frac{a^{5} e^{10 i c}}{32} + \frac{5 a^{5} e^{8 i c}}{32} + \frac{5 a^{5} e^{6 i c}}{16} + \frac{5 a^{5} e^{4 i c}}{16} + \frac{5 a^{5} e^{2 i c}}{32}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.80842, size = 1157, normalized size = 8.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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