Integrand size = 15, antiderivative size = 37 \[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=-\frac {\cos (2 a-c+b x)}{b}-\frac {\text {arctanh}(\cos (c+b x)) \sin ^2(a-c)}{b} \] Output:
-cos(b*x+2*a-c)/b-arctanh(cos(b*x+c))*sin(a-c)^2/b
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41 \[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=\frac {-\cos (2 a-c+b x)+\left (-\log \left (\cos \left (\frac {1}{2} (c+b x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+b x)\right )\right )\right ) \sin ^2(a-c)}{b} \] Input:
Integrate[Csc[c + b*x]*Sin[a + b*x]^2,x]
Output:
(-Cos[2*a - c + b*x] + (-Log[Cos[(c + b*x)/2]] + Log[Sin[(c + b*x)/2]])*Si n[a - c]^2)/b
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.
\(\displaystyle \int \sin ^2(a+b x) \csc (b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \csc (b x+c)dx\) |
Input:
Int[Csc[c + b*x]*Sin[a + b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.73
method | result | size |
risch | \(-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{2 b}-\frac {\cos \left (b x +2 a -c \right )}{b}\) | \(138\) |
default | \(\frac {\frac {8 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{2} \arctan \left (\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}\right )}{\left (4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}+4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}+4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}+4 \sin \left (c \right )^{2} \cos \left (a \right )^{2}\right ) \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}-\frac {2 \left (\left (-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )}}{b}\) | \(279\) |
Input:
int(csc(b*x+c)*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2/b*ln(exp(I*(b*x+a))+exp(I*(a-c)))+1/2/b*ln(exp(I*(b*x+a))+exp(I*(a-c) ))*cos(2*a-2*c)+1/2/b*ln(exp(I*(b*x+a))-exp(I*(a-c)))-1/2/b*ln(exp(I*(b*x+ a))-exp(I*(a-c)))*cos(2*a-2*c)-cos(b*x+2*a-c)/b
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (37) = 74\).
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.46 \[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=-\frac {4 \, \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + 2 \, {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right ) - {\left (\cos \left (-a + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) + {\left (\cos \left (-a + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right )}{2 \, b} \] Input:
integrate(csc(b*x+c)*sin(b*x+a)^2,x, algorithm="fricas")
Output:
-1/2*(4*cos(-a + c)*sin(b*x + c)*sin(-a + c) + 2*(2*cos(-a + c)^2 - 1)*cos (b*x + c) - (cos(-a + c)^2 - 1)*log(1/2*cos(b*x + c) + 1/2) + (cos(-a + c) ^2 - 1)*log(-1/2*cos(b*x + c) + 1/2))/b
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (29) = 58\).
Time = 10.43 (sec) , antiderivative size = 3215, normalized size of antiderivative = 86.89 \[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)*sin(b*x+a)**2,x)
Output:
2*Piecewise((0, Eq(b, 0) & Eq(c, 0)), (sin(b*x)/b, Eq(c, 0)), (0, Eq(b, 0) ), (2*log(tan(c/2) + tan(b*x/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)**4* tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/ 2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(c/2) + tan(b*x/2))*tan(c/2)**3/(b *tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(c/2) + tan(b*x/2))*t an(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*t an(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2*log( tan(c/2) + tan(b*x/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)* *4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b ) - 2*log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)** 4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan( c/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)** 3/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2 )**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - 1/tan(c /2))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*t an(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2 )**2 + b) - 2*tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*t...
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (37) = 74\).
Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.19 \[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=\frac {{\left (\cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - 4 \, \cos \left (b x + 2 \, a - c\right )}{4 \, b} \] Input:
integrate(csc(b*x+c)*sin(b*x+a)^2,x, algorithm="maxima")
Output:
1/4*((cos(-2*a + 2*c) - 1)*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(c) + sin(c)^2) - (cos(-2*a + 2*c) - 1)*log(co s(b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) - 4*cos(b*x + 2*a - c))/b
Leaf count of result is larger than twice the leaf count of optimal. 688 vs. \(2 (37) = 74\).
Time = 0.15 (sec) , antiderivative size = 688, normalized size of antiderivative = 18.59 \[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+c)*sin(b*x+a)^2,x, algorithm="giac")
Output:
2*(2*(tan(1/2*a)^4*tan(1/2*c)^2 - 2*tan(1/2*a)^3*tan(1/2*c)^3 + tan(1/2*a) ^2*tan(1/2*c)^4 + 2*tan(1/2*a)^3*tan(1/2*c) - 4*tan(1/2*a)^2*tan(1/2*c)^2 + 2*tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)^2 - 2*tan(1/2*a)*tan(1/2*c) + tan (1/2*c)^2)*log(abs(tan(1/2*b*x + 1/2*c)))/(tan(1/2*a)^4*tan(1/2*c)^4 + 2*t an(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 + 4* tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 2*tan(1/2*c)^2 + 1) + (4*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c)^3 - 4*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2*a)^4*tan(1/2*c)^4 - 4*tan(1/2 *b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c) + 24*tan(1/2*b*x + 1/2*c)*tan(1/2*a) ^3*tan(1/2*c)^2 + 6*tan(1/2*a)^4*tan(1/2*c)^2 - 24*tan(1/2*b*x + 1/2*c)*ta n(1/2*a)^2*tan(1/2*c)^3 - 16*tan(1/2*a)^3*tan(1/2*c)^3 + 4*tan(1/2*b*x + 1 /2*c)*tan(1/2*a)*tan(1/2*c)^4 + 6*tan(1/2*a)^2*tan(1/2*c)^4 - 4*tan(1/2*b* x + 1/2*c)*tan(1/2*a)^3 - tan(1/2*a)^4 + 24*tan(1/2*b*x + 1/2*c)*tan(1/2*a )^2*tan(1/2*c) + 16*tan(1/2*a)^3*tan(1/2*c) - 24*tan(1/2*b*x + 1/2*c)*tan( 1/2*a)*tan(1/2*c)^2 - 36*tan(1/2*a)^2*tan(1/2*c)^2 + 4*tan(1/2*b*x + 1/2*c )*tan(1/2*c)^3 + 16*tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*c)^4 + 4*tan(1/2*b*x + 1/2*c)*tan(1/2*a) + 6*tan(1/2*a)^2 - 4*tan(1/2*b*x + 1/2*c)*tan(1/2*c) - 16*tan(1/2*a)*tan(1/2*c) + 6*tan(1/2*c)^2 - 1)/((tan(1/2*a)^4*tan(1/2*c) ^4 + 2*tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a )^4 + 4*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 2*t...
Time = 18.44 (sec) , antiderivative size = 223, normalized size of antiderivative = 6.03 \[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}\,\ln \left (-\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )}^2\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}^2}{4\,b}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}\,\ln \left (\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )}^2\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}^2}{4\,b} \] Input:
int(sin(a + b*x)^2/sin(c + b*x),x)
Output:
(exp(c*2i - a*2i)*log(((exp(a*2i)*exp(-c*2i) - 1)^2*1i)/2 + (exp(c*1i)*exp (b*x*1i)*(exp(a*4i)*exp(-c*4i)*1i - exp(a*2i)*exp(-c*2i)*2i + 1i))/2)*(exp (a*2i - c*2i) - 1)^2)/(4*b) - exp(a*2i - c*1i + b*x*1i)/(2*b) - (exp(c*2i - a*2i)*log((exp(c*1i)*exp(b*x*1i)*(exp(a*4i)*exp(-c*4i)*1i - exp(a*2i)*ex p(-c*2i)*2i + 1i))/2 - ((exp(a*2i)*exp(-c*2i) - 1)^2*1i)/2)*(exp(a*2i - c* 2i) - 1)^2)/(4*b) - exp(c*1i - a*2i - b*x*1i)/(2*b)
\[ \int \csc (c+b x) \sin ^2(a+b x) \, dx=\int \csc \left (b x +c \right ) \sin \left (b x +a \right )^{2}d x \] Input:
int(csc(b*x+c)*sin(b*x+a)^2,x)
Output:
int(csc(b*x + c)*sin(a + b*x)**2,x)