Integrand size = 25, antiderivative size = 133 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} d}-\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \] Output:
2*b*(2*a^2+b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2) ^(5/2)/d-a*sec(d*x+c)/(a^2-b^2)/d/(a+b*sin(d*x+c))+sec(d*x+c)*(2*a^2+b^2-3 *a*b*sin(d*x+c))/(a^2-b^2)^2/d
Time = 1.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.27 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{(a+b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {1}{(a-b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )+\frac {a b^2 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}}{d} \] Input:
Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
((2*b*(2*a^2 + b^2)*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*b^ 2*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/d
Time = 0.64 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3343, 3042, 3345, 27, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {\tan (c+d x) \sec (c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)}{\cos (c+d x)^2 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle -\frac {\int \frac {\sec ^2(c+d x) (b-2 a \sin (c+d x))}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {b-2 a \sin (c+d x)}{\cos (c+d x)^2 (a+b \sin (c+d x))}dx}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle -\frac {-\frac {\int \frac {b \left (2 a^2+b^2\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\sec (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {b \left (2 a^2+b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\sec (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {b \left (2 a^2+b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\sec (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {-\frac {2 b \left (2 a^2+b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\sec (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {4 b \left (2 a^2+b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2-b^2\right )}-\frac {\sec (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{3/2}}-\frac {\sec (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}\) |
Input:
Int[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
-((a*Sec[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))) - ((-2*b*(2*a^2 + b^2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/((a^2 - b^ 2)^(3/2)*d) - (Sec[c + d*x]*(2*a^2 + b^2 - 3*a*b*Sin[c + d*x]))/((a^2 - b^ 2)*d))/(a^2 - b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p *(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Time = 2.18 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 b \left (\frac {\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a b}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{2}+b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(164\) |
| default | \(\frac {-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 b \left (\frac {\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a b}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{2}+b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(164\) |
| risch | \(\frac {4 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-4 i a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 i b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-8 i a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+2 i b^{3} {\mathrm e}^{i \left (d x +c \right )}+6 a \,b^{2}+2 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) \left (a^{2}-b^{2}\right )^{2} d}-\frac {2 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right ) a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {2 i b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(490\) |
Input:
int(sec(d*x+c)*tan(d*x+c)/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)+4*b/(a-b)^2/(a+b)^2*((1/2*b^2*tan(1 /2*d*x+1/2*c)+1/2*a*b)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+1 /2*(2*a^2+b^2)/(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))
Time = 0.11 (sec) , antiderivative size = 557, normalized size of antiderivative = 4.19 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [\frac {2 \, a^{5} - 4 \, a^{3} b^{2} + 2 \, a b^{4} + 6 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )\right )}}, \frac {a^{5} - 2 \, a^{3} b^{2} + a b^{4} + 3 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}{{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )}\right ] \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="fricas")
Output:
[1/2*(2*a^5 - 4*a^3*b^2 + 2*a*b^4 + 6*(a^3*b^2 - a*b^4)*cos(d*x + c)^2 - ( (2*a^2*b^2 + b^4)*cos(d*x + c)*sin(d*x + c) + (2*a^3*b + a*b^3)*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 + 3*(a^3*b^2 - a*b^4)*cos(d*x + c)^2 - ((2*a^2*b^2 + b^4)*cos(d*x + c)*sin(d*x + c) + (2*a^3*b + a*b^3)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(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))]
\[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\tan {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+b*sin(d*x+c))**2,x)
Output:
Integral(tan(c + d*x)*sec(c + d*x)/(a + b*sin(c + d*x))**2, x)
Exception generated. \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="maxima")
Output:
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 = 0.51 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.83 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {{\left (2 \, a^{2} b + b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3} - 2 \, a b^{2}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )} {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}\right )}}{d} \] Input:
integrate(sec(d*x+c)*tan(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
2*((2*a^2*b + b^3)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*ta n(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^4 - 2*a^2*b^2 + b^4)*sqrt(a^ 2 - b^2)) + (2*a^2*b*tan(1/2*d*x + 1/2*c)^3 + b^3*tan(1/2*d*x + 1/2*c)^3 - a^3*tan(1/2*d*x + 1/2*c)^2 + 4*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 3*b^3*tan(1 /2*d*x + 1/2*c) - a^3 - 2*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 = 21.03 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.33 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {6\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{{\left (a^2-b^2\right )}^2}+\frac {2\,a\,\left (a^2+2\,b^2\right )}{{\left (a^2-b^2\right )}^2}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a\,b^2-a^3\right )}{a^4-2\,a^2\,b^2+b^4}-\frac {2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}+\frac {2\,b\,\mathrm {atan}\left (\frac {\frac {b\,\left (2\,a^2+b^2\right )\,\left (2\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}+\frac {2\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}}{4\,a^2\,b+2\,b^3}\right )\,\left (2\,a^2+b^2\right )}{d\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \] Input:
int(tan(c + d*x)/(cos(c + d*x)*(a + b*sin(c + d*x))^2),x)
Output:
((6*b^3*tan(c/2 + (d*x)/2))/(a^2 - b^2)^2 + (2*a*(a^2 + 2*b^2))/(a^2 - b^2 )^2 - (2*tan(c/2 + (d*x)/2)^2*(4*a*b^2 - a^3))/(a^4 + b^4 - 2*a^2*b^2) - ( 2*b*tan(c/2 + (d*x)/2)^3*(2*a^2 + b^2))/(a^2 - b^2)^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)) + (2*b*a tan(((b*(2*a^2 + b^2)*(2*a^4*b + 2*b^5 - 4*a^2*b^3))/((a + b)^(5/2)*(a - b )^(5/2)) + (2*a*b*tan(c/2 + (d*x)/2)*(2*a^2 + b^2)*(a^4 + b^4 - 2*a^2*b^2) )/((a + b)^(5/2)*(a - b)^(5/2)))/(4*a^2*b + 2*b^3))*(2*a^2 + b^2))/(d*(a + b)^(5/2)*(a - b)^(5/2))
Time = 0.20 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.57 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{2}+2 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{4}+4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (d x +c \right ) a^{3} b +2 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (d x +c \right ) a \,b^{3}-\cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b +2 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{3}-\cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{5}-\cos \left (d x +c \right ) a^{5}+2 \cos \left (d x +c \right ) a^{3} b^{2}-\cos \left (d x +c \right ) a \,b^{4}-3 \sin \left (d x +c \right )^{2} a^{3} b^{2}+3 \sin \left (d x +c \right )^{2} a \,b^{4}-\sin \left (d x +c \right ) a^{4} b +2 \sin \left (d x +c \right ) a^{2} b^{3}-\sin \left (d x +c \right ) b^{5}+a^{5}+a^{3} b^{2}-2 a \,b^{4}}{\cos \left (d x +c \right ) d \left (\sin \left (d x +c \right ) a^{6} b -3 \sin \left (d x +c \right ) a^{4} b^{3}+3 \sin \left (d x +c \right ) a^{2} b^{5}-\sin \left (d x +c \right ) b^{7}+a^{7}-3 a^{5} b^{2}+3 a^{3} b^{4}-a \,b^{6}\right )} \] Input:
int(sec(d*x+c)*tan(d*x+c)/(a+b*sin(d*x+c))^2,x)
Output:
(4*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos( c + d*x)*sin(c + d*x)*a**2*b**2 + 2*sqrt(a**2 - b**2)*atan((tan((c + d*x)/ 2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)*b**4 + 4*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**3 *b + 2*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* cos(c + d*x)*a*b**3 - cos(c + d*x)*sin(c + d*x)*a**4*b + 2*cos(c + d*x)*si n(c + d*x)*a**2*b**3 - cos(c + d*x)*sin(c + d*x)*b**5 - cos(c + d*x)*a**5 + 2*cos(c + d*x)*a**3*b**2 - cos(c + d*x)*a*b**4 - 3*sin(c + d*x)**2*a**3* b**2 + 3*sin(c + d*x)**2*a*b**4 - sin(c + d*x)*a**4*b + 2*sin(c + d*x)*a** 2*b**3 - sin(c + d*x)*b**5 + a**5 + a**3*b**2 - 2*a*b**4)/(cos(c + d*x)*d* (sin(c + d*x)*a**6*b - 3*sin(c + d*x)*a**4*b**3 + 3*sin(c + d*x)*a**2*b**5 - sin(c + d*x)*b**7 + a**7 - 3*a**5*b**2 + 3*a**3*b**4 - a*b**6))