\(\int \csc (x) \sin (23 x) \, dx\) [305]

Optimal result

Integrand size = 7, antiderivative size = 86 \[ \int \csc (x) \sin (23 x) \, dx=x+\sin (2 x)+\frac {1}{2} \sin (4 x)+\frac {1}{3} \sin (6 x)+\frac {1}{4} \sin (8 x)+\frac {1}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {1}{7} \sin (14 x)+\frac {1}{8} \sin (16 x)+\frac {1}{9} \sin (18 x)+\frac {1}{10} \sin (20 x)+\frac {1}{11} \sin (22 x) \] Output:

x+sin(2*x)+1/2*sin(4*x)+1/3*sin(6*x)+1/4*sin(8*x)+1/5*sin(10*x)+1/6*sin(12 
*x)+1/7*sin(14*x)+1/8*sin(16*x)+1/9*sin(18*x)+1/10*sin(20*x)+1/11*sin(22*x 
)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \csc (x) \sin (23 x) \, dx=x+\sin (2 x)+\frac {1}{2} \sin (4 x)+\frac {1}{3} \sin (6 x)+\frac {1}{4} \sin (8 x)+\frac {1}{5} \sin (10 x)+\frac {1}{6} \sin (12 x)+\frac {1}{7} \sin (14 x)+\frac {1}{8} \sin (16 x)+\frac {1}{9} \sin (18 x)+\frac {1}{10} \sin (20 x)+\frac {1}{11} \sin (22 x) \] Input:

Integrate[Csc[x]*Sin[23*x],x]
 

Output:

x + Sin[2*x] + Sin[4*x]/2 + Sin[6*x]/3 + Sin[8*x]/4 + Sin[10*x]/5 + Sin[12 
*x]/6 + Sin[14*x]/7 + Sin[16*x]/8 + Sin[18*x]/9 + Sin[20*x]/10 + Sin[22*x] 
/11
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(86)=172\).

Time = 2.01 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.21, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 3.429, Rules used = {3042, 4889, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 2345, 27, 1471, 27, 298, 216}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin (23 x) \csc (x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (23 x)}{\sin (x)}dx\)

\(\Big \downarrow \) 4889

\(\displaystyle \int \frac {-\tan ^{22}(x)+253 \tan ^{20}(x)-8855 \tan ^{18}(x)+100947 \tan ^{16}(x)-490314 \tan ^{14}(x)+1144066 \tan ^{12}(x)-1352078 \tan ^{10}(x)+817190 \tan ^8(x)-245157 \tan ^6(x)+33649 \tan ^4(x)-1771 \tan ^2(x)+23}{\left (\tan ^2(x)+1\right )^{12}}d\tan (x)\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}-\frac {1}{22} \int \frac {2 \left (11 \tan ^{20}(x)-2794 \tan ^{18}(x)+100199 \tan ^{16}(x)-1210616 \tan ^{14}(x)+6604070 \tan ^{12}(x)-19188796 \tan ^{10}(x)[phone] \tan ^8(x)-43050744 \tan ^6(x)[phone] \tan ^4(x)-46117610 \tan ^2(x)+2096899\right )}{\left (\tan ^2(x)+1\right )^{11}}d\tan (x)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}-\frac {1}{11} \int \frac {11 \tan ^{20}(x)-2794 \tan ^{18}(x)+100199 \tan ^{16}(x)-1210616 \tan ^{14}(x)+6604070 \tan ^{12}(x)-19188796 \tan ^{10}(x)[phone] \tan ^8(x)-43050744 \tan ^6(x)[phone] \tan ^4(x)-46117610 \tan ^2(x)+2096899}{\left (\tan ^2(x)+1\right )^{11}}d\tan (x)\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{20} \int \frac {4 \left (-55 \tan ^{18}(x)+14025 \tan ^{16}(x)-515020 \tan ^{14}(x)+6568100 \tan ^{12}(x)-39588450 \tan ^{10}(x)[phone] \tan ^8(x)-305840700 \tan ^6(x)[phone] \tan ^4(x)-749831775 \tan ^2(x)[phone]\right )}{\left (\tan ^2(x)+1\right )^{10}}d\tan (x)-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \int \frac {-55 \tan ^{18}(x)+14025 \tan ^{16}(x)-515020 \tan ^{14}(x)+6568100 \tan ^{12}(x)-39588450 \tan ^{10}(x)[phone] \tan ^8(x)-305840700 \tan ^6(x)[phone] \tan ^4(x)-749831775 \tan ^2(x)[phone]}{\left (\tan ^2(x)+1\right )^{10}}d\tan (x)-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}-\frac {1}{18} \int \frac {2 \left (495 \tan ^{16}(x)-126720 \tan ^{14}(x)+4761900 \tan ^{12}(x)-63874800 \tan ^{10}(x)[phone] \tan ^8(x)-1639962720 \tan ^6(x)[phone] \tan ^4(x)-9082378800 \tan ^2(x)[phone]\right )}{\left (\tan ^2(x)+1\right )^9}d\tan (x)\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}-\frac {1}{9} \int \frac {495 \tan ^{16}(x)-126720 \tan ^{14}(x)+4761900 \tan ^{12}(x)-63874800 \tan ^{10}(x)[phone] \tan ^8(x)-1639962720 \tan ^6(x)[phone] \tan ^4(x)-9082378800 \tan ^2(x)[phone]}{\left (\tan ^2(x)+1\right )^9}d\tan (x)\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (\frac {1}{16} \int \frac {240 \left (-33 \tan ^{14}(x)+8481 \tan ^{12}(x)-325941 \tan ^{10}(x)+4584261 \tan ^8(x)-32595651 \tan ^6(x)[phone] \tan ^4(x)-434761767 \tan ^2(x)[phone]\right )}{\left (\tan ^2(x)+1\right )^8}d\tan (x)-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \int \frac {-33 \tan ^{14}(x)+8481 \tan ^{12}(x)-325941 \tan ^{10}(x)+4584261 \tan ^8(x)-32595651 \tan ^6(x)[phone] \tan ^4(x)-434761767 \tan ^2(x)[phone]}{\left (\tan ^2(x)+1\right )^8}d\tan (x)-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}-\frac {1}{14} \int \frac {2 \left (231 \tan ^{12}(x)-59598 \tan ^{10}(x)+2341185 \tan ^8(x)-34431012 \tan ^6(x)[phone] \tan ^4(x)-1256086062 \tan ^2(x)[phone]\right )}{\left (\tan ^2(x)+1\right )^7}d\tan (x)\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}-\frac {1}{7} \int \frac {231 \tan ^{12}(x)-59598 \tan ^{10}(x)+2341185 \tan ^8(x)-34431012 \tan ^6(x)[phone] \tan ^4(x)-1256086062 \tan ^2(x)[phone]}{\left (\tan ^2(x)+1\right )^7}d\tan (x)\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {1}{12} \int \frac {44 \left (-63 \tan ^{10}(x)+16317 \tan ^8(x)-654822 \tan ^6(x)[phone] \tan ^4(x)-81663435 \tan ^2(x)+8625065\right )}{\left (\tan ^2(x)+1\right )^6}d\tan (x)-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \int \frac {-63 \tan ^{10}(x)+16317 \tan ^8(x)-654822 \tan ^6(x)[phone] \tan ^4(x)-81663435 \tan ^2(x)+8625065}{\left (\tan ^2(x)+1\right )^6}d\tan (x)-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-\frac {1}{10} \int \frac {90 \left (7 \tan ^8(x)-1820 \tan ^6(x)+74578 \tan ^4(x)-1190700 \tan ^2(x)+163935\right )}{\left (\tan ^2(x)+1\right )^5}d\tan (x)\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \int \frac {7 \tan ^8(x)-1820 \tan ^6(x)+74578 \tan ^4(x)-1190700 \tan ^2(x)+163935}{\left (\tan ^2(x)+1\right )^5}d\tan (x)\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-\frac {1}{8} \int \frac {56 \left (-\tan ^6(x)+261 \tan ^4(x)-10915 \tan ^2(x)+2135\right )}{\left (\tan ^2(x)+1\right )^4}d\tan (x)\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \int \frac {-\tan ^6(x)+261 \tan ^4(x)-10915 \tan ^2(x)+2135}{\left (\tan ^2(x)+1\right )^4}d\tan (x)\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {6656 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {1}{6} \int \frac {2 \left (3 \tan ^4(x)-786 \tan ^2(x)+251\right )}{\left (\tan ^2(x)+1\right )^3}d\tan (x)\right )\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {6656 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}-\frac {1}{3} \int \frac {3 \tan ^4(x)-786 \tan ^2(x)+251}{\left (\tan ^2(x)+1\right )^3}d\tan (x)\right )\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {1}{3} \left (\frac {1}{4} \int \frac {12 \left (3-\tan ^2(x)\right )}{\left (\tan ^2(x)+1\right )^2}d\tan (x)-\frac {260 \tan (x)}{\left (\tan ^2(x)+1\right )^2}\right )+\frac {6656 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}\right )\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {1}{3} \left (3 \int \frac {3-\tan ^2(x)}{\left (\tan ^2(x)+1\right )^2}d\tan (x)-\frac {260 \tan (x)}{\left (\tan ^2(x)+1\right )^2}\right )+\frac {6656 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}\right )\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {1}{3} \left (3 \left (\int \frac {1}{\tan ^2(x)+1}d\tan (x)+\frac {2 \tan (x)}{\tan ^2(x)+1}\right )-\frac {260 \tan (x)}{\left (\tan ^2(x)+1\right )^2}\right )+\frac {6656 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}\right )\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{11} \left (\frac {1}{5} \left (\frac {1}{9} \left (15 \left (\frac {1}{7} \left (\frac {11}{3} \left (\frac {10100480 \tan (x)}{\left (\tan ^2(x)+1\right )^5}-9 \left (\frac {178880 \tan (x)}{\left (\tan ^2(x)+1\right )^4}-7 \left (\frac {1}{3} \left (3 \left (\arctan (\tan (x))+\frac {2 \tan (x)}{\tan ^2(x)+1}\right )-\frac {260 \tan (x)}{\left (\tan ^2(x)+1\right )^2}\right )+\frac {6656 \tan (x)}{3 \left (\tan ^2(x)+1\right )^3}\right )\right )\right )-\frac {415607296 \tan (x)}{3 \left (\tan ^2(x)+1\right )^6}\right )+\frac {322500608 \tan (x)}{7 \left (\tan ^2(x)+1\right )^7}\right )-\frac {1009455104 \tan (x)}{\left (\tan ^2(x)+1\right )^8}\right )+\frac {899022848 \tan (x)}{9 \left (\tan ^2(x)+1\right )^9}\right )-\frac {49545216 \tan (x)}{5 \left (\tan ^2(x)+1\right )^{10}}\right )+\frac {2097152 \tan (x)}{11 \left (\tan ^2(x)+1\right )^{11}}\)

Input:

Int[Csc[x]*Sin[23*x],x]
 

Output:

(2097152*Tan[x])/(11*(1 + Tan[x]^2)^11) + ((-49545216*Tan[x])/(5*(1 + Tan[ 
x]^2)^10) + ((899022848*Tan[x])/(9*(1 + Tan[x]^2)^9) + ((-1009455104*Tan[x 
])/(1 + Tan[x]^2)^8 + 15*((322500608*Tan[x])/(7*(1 + Tan[x]^2)^7) + ((-415 
607296*Tan[x])/(3*(1 + Tan[x]^2)^6) + (11*((10100480*Tan[x])/(1 + Tan[x]^2 
)^5 - 9*((178880*Tan[x])/(1 + Tan[x]^2)^4 - 7*((6656*Tan[x])/(3*(1 + Tan[x 
]^2)^3) + ((-260*Tan[x])/(1 + Tan[x]^2)^2 + 3*(ArcTan[Tan[x]] + (2*Tan[x]) 
/(1 + Tan[x]^2)))/3))))/3)/7))/9)/5)/11
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4889

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors 
[Tan[v], x]}, Simp[d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2*x 
^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /;  !FalseQ[v] && FunctionOfQ[N 
onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] &&  !MatchQ[ 
u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I 
ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78

method	result	size
risch	\(x +\sin \left (2 x \right )+\frac {\sin \left (4 x \right )}{2}+\frac {\sin \left (6 x \right )}{3}+\frac {\sin \left (8 x \right )}{4}+\frac {\sin \left (10 x \right )}{5}+\frac {\sin \left (12 x \right )}{6}+\frac {\sin \left (14 x \right )}{7}+\frac {\sin \left (16 x \right )}{8}+\frac {\sin \left (18 x \right )}{9}+\frac {\sin \left (20 x \right )}{10}+\frac {\sin \left (22 x \right )}{11}\)	\(67\)

Input:

int(csc(x)*sin(23*x),x,method=_RETURNVERBOSE)
 

Output:

x+sin(2*x)+1/2*sin(4*x)+1/3*sin(6*x)+1/4*sin(8*x)+1/5*sin(10*x)+1/6*sin(12 
*x)+1/7*sin(14*x)+1/8*sin(16*x)+1/9*sin(18*x)+1/10*sin(20*x)+1/11*sin(22*x 
)
 

Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83 \[ \int \csc (x) \sin (23 x) \, dx=\frac {2}{3465} \, {\left (330301440 \, \cos \left (x\right )^{21} - 1560674304 \, \cos \left (x\right )^{19} + 3146579968 \, \cos \left (x\right )^{17} - 3533092864 \, \cos \left (x\right )^{15} + 2418754560 \, \cos \left (x\right )^{13} - 1039018240 \, \cos \left (x\right )^{11} + 277763200 \, \cos \left (x\right )^{9} - 44272800 \, \cos \left (x\right )^{7} + 3843840 \, \cos \left (x\right )^{5} - 150150 \, \cos \left (x\right )^{3} + 3465 \, \cos \left (x\right )\right )} \sin \left (x\right ) + x \] Input:

integrate(csc(x)*sin(23*x),x, algorithm="fricas")
 

Output:

2/3465*(330301440*cos(x)^21 - 1560674304*cos(x)^19 + 3146579968*cos(x)^17 
- 3533092864*cos(x)^15 + 2418754560*cos(x)^13 - 1039018240*cos(x)^11 + 277 
763200*cos(x)^9 - 44272800*cos(x)^7 + 3843840*cos(x)^5 - 150150*cos(x)^3 + 
 3465*cos(x))*sin(x) + x
 

Sympy [A] (verification not implemented)

Time = 73.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.16 \[ \int \csc (x) \sin (23 x) \, dx=x - \frac {1024 \sin ^{11}{\left (2 x \right )}}{11} + \frac {2560 \sin ^{9}{\left (2 x \right )}}{9} - \frac {2304 \sin ^{7}{\left (2 x \right )}}{7} + \frac {896 \sin ^{5}{\left (2 x \right )}}{5} - \frac {140 \sin ^{3}{\left (2 x \right )}}{3} + 6 \sin {\left (2 x \right )} + \frac {8 \sin ^{5}{\left (4 x \right )}}{5} - \frac {8 \sin ^{3}{\left (4 x \right )}}{3} + \frac {3 \sin {\left (4 x \right )}}{2} + \frac {\sin {\left (8 x \right )}}{4} + \frac {\sin {\left (16 x \right )}}{8} \] Input:

integrate(csc(x)*sin(23*x),x)
 

Output:

x - 1024*sin(2*x)**11/11 + 2560*sin(2*x)**9/9 - 2304*sin(2*x)**7/7 + 896*s 
in(2*x)**5/5 - 140*sin(2*x)**3/3 + 6*sin(2*x) + 8*sin(4*x)**5/5 - 8*sin(4* 
x)**3/3 + 3*sin(4*x)/2 + sin(8*x)/4 + sin(16*x)/8
 

Maxima [F(-1)]

Timed out. \[ \int \csc (x) \sin (23 x) \, dx=\text {Timed out} \] Input:

integrate(csc(x)*sin(23*x),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \csc (x) \sin (23 x) \, dx=x + \frac {1}{11} \, \sin \left (22 \, x\right ) + \frac {1}{10} \, \sin \left (20 \, x\right ) + \frac {1}{9} \, \sin \left (18 \, x\right ) + \frac {1}{8} \, \sin \left (16 \, x\right ) + \frac {1}{7} \, \sin \left (14 \, x\right ) + \frac {1}{6} \, \sin \left (12 \, x\right ) + \frac {1}{5} \, \sin \left (10 \, x\right ) + \frac {1}{4} \, \sin \left (8 \, x\right ) + \frac {1}{3} \, \sin \left (6 \, x\right ) + \frac {1}{2} \, \sin \left (4 \, x\right ) + \sin \left (2 \, x\right ) \] Input:

integrate(csc(x)*sin(23*x),x, algorithm="giac")
 

Output:

x + 1/11*sin(22*x) + 1/10*sin(20*x) + 1/9*sin(18*x) + 1/8*sin(16*x) + 1/7* 
sin(14*x) + 1/6*sin(12*x) + 1/5*sin(10*x) + 1/4*sin(8*x) + 1/3*sin(6*x) + 
1/2*sin(4*x) + sin(2*x)
 

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \csc (x) \sin (23 x) \, dx=x+\sin \left (2\,x\right )+\frac {\sin \left (4\,x\right )}{2}+\frac {\sin \left (6\,x\right )}{3}+\frac {\sin \left (8\,x\right )}{4}+\frac {\sin \left (10\,x\right )}{5}+\frac {\sin \left (12\,x\right )}{6}+\frac {\sin \left (14\,x\right )}{7}+\frac {\sin \left (16\,x\right )}{8}+\frac {\sin \left (18\,x\right )}{9}+\frac {\sin \left (20\,x\right )}{10}+\frac {\sin \left (22\,x\right )}{11} \] Input:

int(sin(23*x)/sin(x),x)
                                                                                    
                                                                                    
 

Output:

x + sin(2*x) + sin(4*x)/2 + sin(6*x)/3 + sin(8*x)/4 + sin(10*x)/5 + sin(12 
*x)/6 + sin(14*x)/7 + sin(16*x)/8 + sin(18*x)/9 + sin(20*x)/10 + sin(22*x) 
/11
 

Reduce [F]

\[ \int \csc (x) \sin (23 x) \, dx=\int \csc \left (x \right ) \sin \left (23 x \right )d x \] Input:

int(csc(x)*sin(23*x),x)
 

Output:

int(csc(x)*sin(23*x),x)