Optimal. Leaf size=301 \[ \frac{32 \sin ^5(c+d x)}{4199 a^8 d}-\frac{320 \sin ^3(c+d x)}{12597 a^8 d}+\frac{160 \sin (c+d x)}{4199 a^8 d}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{112 i \cos ^3(c+d x)}{12597 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8} \]
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Rubi [A] time = 0.384034, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3502, 3500, 2633} \[ \frac{32 \sin ^5(c+d x)}{4199 a^8 d}-\frac{320 \sin ^3(c+d x)}{12597 a^8 d}+\frac{160 \sin (c+d x)}{4199 a^8 d}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{112 i \cos ^3(c+d x)}{12597 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3500
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{19 a}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{110 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{323 a^2}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{323 a^3}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{528 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{4199 a^4}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{336 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{4199 a^5}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{224 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{4199 a^6}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{160 \int \cos ^5(c+d x) \, dx}{4199 a^8}\\ &=\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{160 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{4199 a^8 d}\\ &=\frac{160 \sin (c+d x)}{4199 a^8 d}-\frac{320 \sin ^3(c+d x)}{12597 a^8 d}+\frac{32 \sin ^5(c+d x)}{4199 a^8 d}+\frac{i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac{11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac{22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac{66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos ^3(c+d x)}{12597 a^5 d (a+i a \tan (c+d x))^3}+\frac{48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.31914, size = 161, normalized size = 0.53 \[ -\frac{i \sec ^8(c+d x) (-92378 i \sin (c+d x)-226746 i \sin (3 (c+d x))-266475 i \sin (5 (c+d x))-323323 i \sin (7 (c+d x))+73359 i \sin (9 (c+d x))+2431 i \sin (11 (c+d x))-739024 \cos (c+d x)-604656 \cos (3 (c+d x))-426360 \cos (5 (c+d x))-369512 \cos (7 (c+d x))+65208 \cos (9 (c+d x))+1768 \cos (11 (c+d x)))}{12899328 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 372, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ({\frac{-992\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{16}}}-{\frac{{\frac{32525\,i}{8}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{{\frac{7181\,i}{1024}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+{\frac{{\frac{32417\,i}{4}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}-{\frac{{\frac{i}{1024}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}-{\frac{{\frac{25468\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}+{\frac{4428\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{14}}}-{\frac{{\frac{2177\,i}{16}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{18}}}-{\frac{128}{19\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{19}}}+{\frac{5248}{17\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{17}}}-{\frac{7096}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{15}}}+{\frac{87508}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}-{\frac{18011}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+6215\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-9}-{\frac{72425}{32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{26871}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{54229}{1536\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{509}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512\,i}}+{\frac{{\frac{204605\,i}{192}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{1}{1536\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{3}}}+{\frac{3}{512\,\tan \left ( 1/2\,dx+c/2 \right ) +512\,i}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54902, size = 543, normalized size = 1.8 \begin{align*} \frac{{\left (-4199 i \, e^{\left (22 i \, d x + 22 i \, c\right )} - 138567 i \, e^{\left (20 i \, d x + 20 i \, c\right )} + 692835 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 692835 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 831402 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 831402 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 646646 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 377910 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 159885 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 46189 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 8151 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 663 i\right )} e^{\left (-19 i \, d x - 19 i \, c\right )}}{25798656 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.39548, size = 437, normalized size = 1.45 \begin{align*} \begin{cases} \frac{\left (- 6279106898588469469113471576881812733952 i a^{88} d^{11} e^{103 i c} e^{3 i d x} - 207210527653419492480744562037099820220416 i a^{88} d^{11} e^{101 i c} e^{i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{99 i c} e^{- i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{97 i c} e^{- 3 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{95 i c} e^{- 5 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{93 i c} e^{- 7 i d x} + 966982462382624298243474622839799161028608 i a^{88} d^{11} e^{91 i c} e^{- 9 i d x} + 565119620872962252220212441919363146055680 i a^{88} d^{11} e^{89 i c} e^{- 11 i d x} + 239089070369330183631628340812038254100480 i a^{88} d^{11} e^{87 i c} e^{- 13 i d x} + 69070175884473164160248187345699940073472 i a^{88} d^{11} e^{85 i c} e^{- 15 i d x} + 12188854567848205440043797766888224718848 i a^{88} d^{11} e^{83 i c} e^{- 17 i d x} + 991437931356074126702127091086602010624 i a^{88} d^{11} e^{81 i c} e^{- 19 i d x}\right ) e^{- 100 i c}}{38578832784927556418233169368361857437401088 a^{96} d^{12}} & \text{for}\: 38578832784927556418233169368361857437401088 a^{96} d^{12} e^{100 i c} \neq 0 \\\frac{x \left (e^{22 i c} + 11 e^{20 i c} + 55 e^{18 i c} + 165 e^{16 i c} + 330 e^{14 i c} + 462 e^{12 i c} + 462 e^{10 i c} + 330 e^{8 i c} + 165 e^{6 i c} + 55 e^{4 i c} + 11 e^{2 i c} + 1\right ) e^{- 19 i c}}{2048 a^{8}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19953, size = 406, normalized size = 1.35 \begin{align*} \frac{\frac{4199 \,{\left (18 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 33 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 17\right )}}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{3}} + \frac{12823746 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{18} - 140368371 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} - 879644311 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} + 3693272440 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} - 27403194676 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 51919375300 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 79183835016 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 99750226290 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 82860874122 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 56110430792 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 13462452660 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4616712644 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1197851960 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 27911475 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2143959}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{19}}}{6449664 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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