Integrand size = 34, antiderivative size = 120 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} c f}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {a} c f}+\frac {\sec (e+f x) \sqrt {a+a \sin (e+f x)}}{a c f} \]
-2*arctanh(cos(f*x+e)*a^(1/2)/(a+a*sin(f*x+e))^(1/2))/c/f/a^(1/2)+1/2*arct anh(1/2*cos(f*x+e)*a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))/c/f*2^(1/2)/a^( 1/2)+sec(f*x+e)*(a+a*sin(f*x+e))^(1/2)/a/c/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.62 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx=-\frac {\left (\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1-\sin (e+f x))\right )-2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-\sin (e+f x)\right )\right ) \sec (e+f x) \sqrt {a (1+\sin (e+f x))}}{a c f} \]
-(((Hypergeometric2F1[-1/2, 1, 1/2, (1 - Sin[e + f*x])/2] - 2*Hypergeometr ic2F1[-1/2, 1, 1/2, 1 - Sin[e + f*x]])*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f* x])])/(a*c*f))
Time = 1.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 3419, 3042, 3215, 3042, 3152, 3464, 3042, 3128, 219, 3252, 219}
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.
| \(\displaystyle \int \frac {\csc (e+f x)}{\sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3419 |
| \(\displaystyle \frac {\int \frac {\csc (e+f x) (2 a c+a \sin (e+f x) c)}{\sqrt {\sin (e+f x) a+a}}dx}{2 a c^2}+\frac {\int \frac {\sqrt {\sin (e+f x) a+a}}{c-c \sin (e+f x)}dx}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a c+a \sin (e+f x) c}{\sin (e+f x) \sqrt {\sin (e+f x) a+a}}dx}{2 a c^2}+\frac {\int \frac {\sqrt {\sin (e+f x) a+a}}{c-c \sin (e+f x)}dx}{2 a}\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \sec ^2(e+f x) (\sin (e+f x) a+a)^{3/2}dx}{2 a^2 c}+\frac {\int \frac {2 a c+a \sin (e+f x) c}{\sin (e+f x) \sqrt {\sin (e+f x) a+a}}dx}{2 a c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{3/2}}{\cos (e+f x)^2}dx}{2 a^2 c}+\frac {\int \frac {2 a c+a \sin (e+f x) c}{\sin (e+f x) \sqrt {\sin (e+f x) a+a}}dx}{2 a c^2}\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle \frac {\int \frac {2 a c+a \sin (e+f x) c}{\sin (e+f x) \sqrt {\sin (e+f x) a+a}}dx}{2 a c^2}+\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a}}{a c f}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {2 c \int \csc (e+f x) \sqrt {\sin (e+f x) a+a}dx-a c \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{2 a c^2}+\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a}}{a c f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx-a c \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{2 a c^2}+\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a}}{a c f}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {2 a c \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}+2 c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx}{2 a c^2}+\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a}}{a c f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx+\frac {\sqrt {2} \sqrt {a} c \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a c^2}+\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a}}{a c f}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {\sqrt {2} \sqrt {a} c \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {4 a c \int \frac {1}{a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}}{2 a c^2}+\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a}}{a c f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\sqrt {2} \sqrt {a} c \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {4 \sqrt {a} c \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}}{2 a c^2}+\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a}}{a c f}\) |
((-4*Sqrt[a]*c*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f + (Sqrt[2]*Sqrt[a]*c*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*S in[e + f*x]])])/f)/(2*a*c^2) + (Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(a* c*f)
3.1.14.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[1/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*( (c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[d^2/(c*(b*c - a*d )) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] + Simp[1/(c* (b*c - a*d)) Int[(b*c - a*d - b*d*Sin[e + f*x])/(Sin[e + f*x]*Sqrt[a + b* Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A *b - a*B)/(b*c - a*d) Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c - A*d)/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Time = 0.39 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.03
| method | result | size |
| default | \(\frac {\left (1+\sin \left (f x +e \right )\right ) \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sqrt {a -a \sin \left (f x +e \right )}+2 a^{\frac {5}{2}}-4 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a -a \sin \left (f x +e \right )}\right )}{2 c \,a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(124\) |
1/2*(1+sin(f*x+e))*(2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^( 1/2))*a^2*(a-a*sin(f*x+e))^(1/2)+2*a^(5/2)-4*arctanh((a-a*sin(f*x+e))^(1/2 )/a^(1/2))*a^2*(a-a*sin(f*x+e))^(1/2))/c/a^(5/2)/cos(f*x+e)/(a+a*sin(f*x+e ))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (104) = 208\).
Time = 0.29 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.73 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx=\frac {\sqrt {2} \sqrt {a} \cos \left (f x + e\right ) \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 2 \, \sqrt {a} \cos \left (f x + e\right ) \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, \sqrt {a \sin \left (f x + e\right ) + a}}{4 \, a c f \cos \left (f x + e\right )} \]
1/4*(sqrt(2)*sqrt(a)*cos(f*x + e)*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2 )*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f* x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 2*sqrt(a)*cos(f*x + e)*log((a*cos (f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)* sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a* cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a) /(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - co s(f*x + e) - 1)) + 4*sqrt(a*sin(f*x + e) + a))/(a*c*f*cos(f*x + e))
\[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx=- \frac {\int \frac {1}{\sqrt {a \sin {\left (e + f x \right )} + a} \sin ^{2}{\left (e + f x \right )} - \sqrt {a \sin {\left (e + f x \right )} + a} \sin {\left (e + f x \right )}}\, dx}{c} \]
-Integral(1/(sqrt(a*sin(e + f*x) + a)*sin(e + f*x)**2 - sqrt(a*sin(e + f*x ) + a)*sin(e + f*x)), x)/c
\[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx=\int { -\frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) - c\right )} \sin \left (f x + e\right )} \,d x } \]
Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx=-\frac {\sqrt {2} {\left (\frac {2 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right )}{c} + \frac {\log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c} - \frac {\log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c} + \frac {2}{c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}\right )}}{4 \, \sqrt {a} f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
-1/4*sqrt(2)*(2*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2 *e))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2*e)))/c + log(sin(-1/4*p i + 1/2*f*x + 1/2*e) + 1)/c - log(-sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/c + 2/(c*sin(-1/4*pi + 1/2*f*x + 1/2*e)))/(sqrt(a)*f*sgn(cos(-1/4*pi + 1/2*f* x + 1/2*e)))
Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c-c\,\sin \left (e+f\,x\right )\right )} \,d x \]