Optimal. Leaf size=185 \[ \frac {2 \cos ^9(c+d x)}{9 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{10 a^2 d}-\frac {3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {3 \sin (c+d x) \cos ^5(c+d x)}{32 a^2 d}+\frac {3 \sin (c+d x) \cos ^3(c+d x)}{128 a^2 d}+\frac {9 \sin (c+d x) \cos (c+d x)}{256 a^2 d}+\frac {9 x}{256 a^2} \]
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Rubi [A] time = 0.46, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2875, 2873, 2568, 2635, 8, 2565, 270} \[ \frac {2 \cos ^9(c+d x)}{9 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{10 a^2 d}-\frac {3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {3 \sin (c+d x) \cos ^5(c+d x)}{32 a^2 d}+\frac {3 \sin (c+d x) \cos ^3(c+d x)}{128 a^2 d}+\frac {9 \sin (c+d x) \cos (c+d x)}{256 a^2 d}+\frac {9 x}{256 a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2873
Rule 2875
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cos ^4(c+d x) \sin ^4(c+d x)-2 a^2 \cos ^4(c+d x) \sin ^5(c+d x)+a^2 \cos ^4(c+d x) \sin ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a^2}+\frac {\int \cos ^4(c+d x) \sin ^6(c+d x) \, dx}{a^2}-\frac {2 \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx}{a^2}\\ &=-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{8 a^2}+\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{2 a^2}+\frac {2 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {\int \cos ^4(c+d x) \, dx}{16 a^2}+\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{16 a^2}+\frac {2 \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {\int \cos ^4(c+d x) \, dx}{32 a^2}+\frac {3 \int \cos ^2(c+d x) \, dx}{64 a^2}\\ &=\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{128 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int 1 \, dx}{128 a^2}+\frac {3 \int \cos ^2(c+d x) \, dx}{128 a^2}\\ &=\frac {3 x}{128 a^2}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{256 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int 1 \, dx}{256 a^2}\\ &=\frac {9 x}{256 a^2}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{256 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}\\ \end {align*}
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Mathematica [B] time = 8.38, size = 585, normalized size = 3.16 \[ \frac {45360 d x \sin \left (\frac {c}{2}\right )-30240 \sin \left (\frac {c}{2}+d x\right )+30240 \sin \left (\frac {3 c}{2}+d x\right )-1260 \sin \left (\frac {3 c}{2}+2 d x\right )-1260 \sin \left (\frac {5 c}{2}+2 d x\right )-6720 \sin \left (\frac {5 c}{2}+3 d x\right )+6720 \sin \left (\frac {7 c}{2}+3 d x\right )-7560 \sin \left (\frac {7 c}{2}+4 d x\right )-7560 \sin \left (\frac {9 c}{2}+4 d x\right )+4032 \sin \left (\frac {9 c}{2}+5 d x\right )-4032 \sin \left (\frac {11 c}{2}+5 d x\right )+630 \sin \left (\frac {11 c}{2}+6 d x\right )+630 \sin \left (\frac {13 c}{2}+6 d x\right )+720 \sin \left (\frac {13 c}{2}+7 d x\right )-720 \sin \left (\frac {15 c}{2}+7 d x\right )+945 \sin \left (\frac {15 c}{2}+8 d x\right )+945 \sin \left (\frac {17 c}{2}+8 d x\right )-560 \sin \left (\frac {17 c}{2}+9 d x\right )+560 \sin \left (\frac {19 c}{2}+9 d x\right )-126 \sin \left (\frac {19 c}{2}+10 d x\right )-126 \sin \left (\frac {21 c}{2}+10 d x\right )-2520 \cos \left (\frac {c}{2}\right ) (187 c-18 d x)+30240 \cos \left (\frac {c}{2}+d x\right )+30240 \cos \left (\frac {3 c}{2}+d x\right )-1260 \cos \left (\frac {3 c}{2}+2 d x\right )+1260 \cos \left (\frac {5 c}{2}+2 d x\right )+6720 \cos \left (\frac {5 c}{2}+3 d x\right )+6720 \cos \left (\frac {7 c}{2}+3 d x\right )-7560 \cos \left (\frac {7 c}{2}+4 d x\right )+7560 \cos \left (\frac {9 c}{2}+4 d x\right )-4032 \cos \left (\frac {9 c}{2}+5 d x\right )-4032 \cos \left (\frac {11 c}{2}+5 d x\right )+630 \cos \left (\frac {11 c}{2}+6 d x\right )-630 \cos \left (\frac {13 c}{2}+6 d x\right )-720 \cos \left (\frac {13 c}{2}+7 d x\right )-720 \cos \left (\frac {15 c}{2}+7 d x\right )+945 \cos \left (\frac {15 c}{2}+8 d x\right )-945 \cos \left (\frac {17 c}{2}+8 d x\right )+560 \cos \left (\frac {17 c}{2}+9 d x\right )+560 \cos \left (\frac {19 c}{2}+9 d x\right )-126 \cos \left (\frac {19 c}{2}+10 d x\right )+126 \cos \left (\frac {21 c}{2}+10 d x\right )-471240 c \sin \left (\frac {c}{2}\right )+327180 \sin \left (\frac {c}{2}\right )}{1290240 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 100, normalized size = 0.54 \[ \frac {17920 \, \cos \left (d x + c\right )^{9} - 46080 \, \cos \left (d x + c\right )^{7} + 32256 \, \cos \left (d x + c\right )^{5} + 2835 \, d x - 63 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 496 \, \cos \left (d x + c\right )^{7} + 488 \, \cos \left (d x + c\right )^{5} - 30 \, \cos \left (d x + c\right )^{3} - 45 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 257, normalized size = 1.39 \[ \frac {\frac {2835 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (2835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 27405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 139356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 860160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 618660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 430080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 1609650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 516096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 1609650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1290240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 618660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 368640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 139356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 184320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 27405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4096\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{10} a^{2}}}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 619, normalized size = 3.35 \[ \frac {32}{315 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {87 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {32 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {553 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {64 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {491 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {32 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {2555 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {64 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {2555 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {32 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {491 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {64 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {553 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {87 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {9 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 605, normalized size = 3.27 \[ -\frac {\frac {\frac {2835 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40960 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {27405 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {184320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {139356 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {368640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {618660 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1290240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {1609650 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {516096 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {1609650 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {430080 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {618660 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {860160 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {139356 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {27405 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {2835 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - 4096}{a^{2} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {120 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {252 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {120 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {a^{2} \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}} - \frac {2835 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{40320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.63, size = 250, normalized size = 1.35 \[ \frac {9\,x}{256\,a^2}+\frac {\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {553\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}-\frac {491\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {2555\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}-\frac {2555\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {491\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {553\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {32}{315}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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