Optimal. Leaf size=301 \[ \frac {32 \sin ^5(c+d x)}{4199 a^8 d}-\frac {320 \sin ^3(c+d x)}{12597 a^8 d}+\frac {160 \sin (c+d x)}{4199 a^8 d}+\frac {64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac {112 i \cos ^3(c+d x)}{12597 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac {11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8} \]
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Rubi [A] time = 0.38, antiderivative size = 301, normalized size of antiderivative = 1.00, 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 2633
Rule 3500
Rule 3502
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*}
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Mathematica [A] time = 1.52, 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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fricas [A] time = 0.57, size = 140, normalized size = 0.47 \[ \frac {{\left (-4199 i \, e^{\left (22 i \, d x + 22 i \, c\right )} - 138567 i \, e^{\left (20 i \, d x + 20 i \, c\right )} + 692835 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 692835 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 831402 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 831402 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 646646 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 377910 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 159885 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 46189 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 8151 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 663 i\right )} e^{\left (-19 i \, d x - 19 i \, c\right )}}{25798656 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.93, size = 301, normalized size = 1.00 \[ \frac {\frac {4199 \, {\left (18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 33 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 17\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}^{3}} + \frac {12823746 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{18} - 140368371 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 879644311 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 3693272440 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 27403194676 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 51919375300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 79183835016 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 99750226290 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 82860874122 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 56110430792 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 13462452660 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4616712644 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1197851960 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27911475 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2143959}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{19}}}{6449664 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 372, normalized size = 1.24 \[ \frac {\frac {128 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{18}}-\frac {1}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {3}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}-\frac {32525 i}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{8}}+\frac {32417 i}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{10}}+\frac {7181 i}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {i}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}-\frac {1984 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{16}}-\frac {50936 i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{12}}-\frac {2177 i}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}+\frac {8856 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{14}}+\frac {204605 i}{96 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{6}}-\frac {256}{19 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{19}}+\frac {10496}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{17}}-\frac {14192}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{15}}+\frac {175016}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{13}}-\frac {18011}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{11}}+\frac {12430}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{9}}-\frac {72425}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{7}}+\frac {26871}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}-\frac {54229}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {509}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{d \,a^{8}} \]
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 = 9.52, size = 308, normalized size = 1.02 \[ -\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {46189\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {46189\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {20995\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {20995\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {221255\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}+\frac {221255\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}-\frac {66861\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}+\frac {2093\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{32}-\frac {221\,\cos \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{128}+\frac {221\,\cos \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{128}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,309861{}\mathrm {i}}{256}-\frac {\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,665911{}\mathrm {i}}{512}+\frac {\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,665911{}\mathrm {i}}{512}-\frac {\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,194821{}\mathrm {i}}{128}+\frac {\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,194821{}\mathrm {i}}{128}-\frac {\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1825043{}\mathrm {i}}{1024}+\frac {\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1825043{}\mathrm {i}}{1024}-\frac {\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,1074183{}\mathrm {i}}{512}+\frac {\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )\,37895{}\mathrm {i}}{512}-\frac {\sin \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )\,2431{}\mathrm {i}}{1024}+\frac {\sin \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )\,2431{}\mathrm {i}}{1024}\right )}{12597\,a^8\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^{19}\,{\left (\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.30, size = 437, normalized size = 1.45 \[ \begin {cases} \frac {\left (- 6279106898588469469113471576881812733952 i a^{88} d^{11} e^{103 i c} e^{3 i d x} - 207210527653419492480744562037099820220416 i a^{88} d^{11} e^{101 i c} e^{i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{99 i c} e^{- i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{97 i c} e^{- 3 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{95 i c} e^{- 5 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{93 i c} e^{- 7 i d x} + 966982462382624298243474622839799161028608 i a^{88} d^{11} e^{91 i c} e^{- 9 i d x} + 565119620872962252220212441919363146055680 i a^{88} d^{11} e^{89 i c} e^{- 11 i d x} + 239089070369330183631628340812038254100480 i a^{88} d^{11} e^{87 i c} e^{- 13 i d x} + 69070175884473164160248187345699940073472 i a^{88} d^{11} e^{85 i c} e^{- 15 i d x} + 12188854567848205440043797766888224718848 i a^{88} d^{11} e^{83 i c} e^{- 17 i d x} + 991437931356074126702127091086602010624 i a^{88} d^{11} e^{81 i c} e^{- 19 i d x}\right ) e^{- 100 i c}}{38578832784927556418233169368361857437401088 a^{96} d^{12}} & \text {for}\: 38578832784927556418233169368361857437401088 a^{96} d^{12} e^{100 i c} \neq 0 \\\frac {x \left (e^{22 i c} + 11 e^{20 i c} + 55 e^{18 i c} + 165 e^{16 i c} + 330 e^{14 i c} + 462 e^{12 i c} + 462 e^{10 i c} + 330 e^{8 i c} + 165 e^{6 i c} + 55 e^{4 i c} + 11 e^{2 i c} + 1\right ) e^{- 19 i c}}{2048 a^{8}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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