Integrand size = 27, antiderivative size = 212 \[ \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 a^4 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d}-\frac {2 a^2 \left (a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2} d}+\frac {\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}+\frac {a^3 \cos (c+d x)}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))} \]
2*a^4*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b/(a^2-b^2)^(5/2)/d -2*a^2*(a^2-3*b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b/(a^2 -b^2)^(5/2)/d+1/2*cos(d*x+c)/(a+b)^2/d/(1-sin(d*x+c))+1/2*cos(d*x+c)/(a-b) ^2/d/(1+sin(d*x+c))+a^3*cos(d*x+c)/(a^2-b^2)^2/d/(a+b*sin(d*x+c))
Time = 0.89 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.76 \[ \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {6 a^2 b \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{(a+b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {1}{(a-b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )+\frac {a^3 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}}{d} \]
((6*a^2*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5 /2) + Sin[(c + d*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) ) - 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (a^3*Cos[c + d* x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/d
Time = 0.54 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3376, 2009}
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.
| \(\displaystyle \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3}{\cos (c+d x)^2 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3376 |
| \(\displaystyle \int \left (-\frac {a^2 \left (a^2-3 b^2\right )}{b \left (b^2-a^2\right )^2 (a+b \sin (c+d x))}+\frac {a^3}{b \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {1}{2 (a+b)^2 (\sin (c+d x)-1)}-\frac {1}{2 (a-b)^2 (\sin (c+d x)+1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^2 \left (a^2-3 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b d \left (a^2-b^2\right )^{5/2}}+\frac {2 a^4 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b d \left (a^2-b^2\right )^{5/2}}+\frac {a^3 \cos (c+d x)}{d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}\) |
(2*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(5 /2)*d) - (2*a^2*(a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b ^2]])/(b*(a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d *x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (a^3*Cos[c + d*x ])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))
3.15.65.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(d*sin[ e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; Fr eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && ( LtQ[m, -1] || (EqQ[m, -1] && GtQ[p, 0]))
Time = 0.73 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(\frac {-\frac {4 a^{2} \left (\frac {-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}-\frac {3 b \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(155\) |
| default | \(\frac {-\frac {4 a^{2} \left (\frac {-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}-\frac {3 b \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(155\) |
| risch | \(-\frac {2 i \left (3 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+i a^{3} b +2 i a \,b^{3}+a^{4} {\mathrm e}^{i \left (d x +c \right )}+3 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}-b^{4} {\mathrm e}^{i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) \left (a^{2}-b^{2}\right )^{2} d}+\frac {3 i b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {3 i b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(350\) |
1/d*(-4*a^2/(a-b)^2/(a+b)^2*((-1/2*b*tan(1/2*d*x+1/2*c)-1/2*a)/(tan(1/2*d* x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)-3/2*b/(a^2-b^2)^(1/2)*arctan(1/2*(2 *a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))-1/(a+b)^2/(tan(1/2*d*x+1/2*c) -1)+1/(a-b)^2/(tan(1/2*d*x+1/2*c)+1))
Time = 0.29 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.52 \[ \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [\frac {2 \, a^{5} - 4 \, a^{3} b^{2} + 2 \, a b^{4} + 2 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{2} b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a^{3} b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )\right )}}, \frac {a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{2} b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a^{3} b \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}{{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )}\right ] \]
[1/2*(2*a^5 - 4*a^3*b^2 + 2*a*b^4 + 2*(a^5 + a^3*b^2 - 2*a*b^4)*cos(d*x + c)^2 - 3*(a^2*b^2*cos(d*x + c)*sin(d*x + c) + a^3*b*cos(d*x + c))*sqrt(-a^ 2 + b^2)*log(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^ 2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^ 2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 2*(a^4*b - 2*a^2*b^3 + b^5)*sin(d*x + c))/((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*d*cos(d*x + c )*sin(d*x + c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)), (a ^5 - 2*a^3*b^2 + a*b^4 + (a^5 + a^3*b^2 - 2*a*b^4)*cos(d*x + c)^2 - 3*(a^2 *b^2*cos(d*x + c)*sin(d*x + c) + a^3*b*cos(d*x + c))*sqrt(a^2 - b^2)*arcta n(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - (a^4*b - 2*a^2*b ^3 + b^5)*sin(d*x + c))/((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*d*cos(d*x + c)*sin(d*x + c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c))]
Timed out. \[ \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.44 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.05 \[ \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a^{2} b}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{3} - a b^{2}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )} {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}\right )}}{d} \]
2*(3*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/ 2*c) + b)/sqrt(a^2 - b^2)))*a^2*b/((a^4 - 2*a^2*b^2 + b^4)*sqrt(a^2 - b^2) ) + (3*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 3*a*b^2*tan(1/2*d*x + 1/2*c)^2 - a^2 *b*tan(1/2*d*x + 1/2*c) - 2*b^3*tan(1/2*d*x + 1/2*c) - 2*a^3 - a*b^2)/((a* tan(1/2*d*x + 1/2*c)^4 + 2*b*tan(1/2*d*x + 1/2*c)^3 - 2*b*tan(1/2*d*x + 1/ 2*c) - a)*(a^4 - 2*a^2*b^2 + b^4)))/d
Time = 14.57 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.30 \[ \int \frac {\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2\,a\,\left (2\,a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}-\frac {6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{{\left (a^2-b^2\right )}^2}+\frac {2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^2+2\,b^2\right )}{{\left (a^2-b^2\right )}^2}-\frac {6\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a^4-2\,a^2\,b^2+b^4}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}+\frac {6\,a^2\,b\,\mathrm {atan}\left (\frac {\frac {a^2\,b\,\left (2\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )}{2\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}+\frac {a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}}{a^2\,b}\right )}{d\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \]
((2*a*(2*a^2 + b^2))/(a^2 - b^2)^2 - (6*a^2*b*tan(c/2 + (d*x)/2)^3)/(a^2 - b^2)^2 + (2*b*tan(c/2 + (d*x)/2)*(a^2 + 2*b^2))/(a^2 - b^2)^2 - (6*a*b^2* tan(c/2 + (d*x)/2)^2)/(a^4 + b^4 - 2*a^2*b^2))/(d*(a + 2*b*tan(c/2 + (d*x) /2) - a*tan(c/2 + (d*x)/2)^4 - 2*b*tan(c/2 + (d*x)/2)^3)) + (6*a^2*b*atan( ((a^2*b*(2*a^4*b + 2*b^5 - 4*a^2*b^3))/(2*(a + b)^(5/2)*(a - b)^(5/2)) + ( a^3*b*tan(c/2 + (d*x)/2)*(a^4 + b^4 - 2*a^2*b^2))/((a + b)^(5/2)*(a - b)^( 5/2)))/(a^2*b)))/(d*(a + b)^(5/2)*(a - b)^(5/2))