Integrand size = 25, antiderivative size = 105 \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}-\frac {2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {4 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a d} \]
2/3*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/a/d-2/3*csc(d*x+c)*(e*csc(d*x+c))^(1/2 )/a/d-4/3*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*El lipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c) ^(1/2)/a/d
Time = 0.96 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {2 (e \csc (c+d x))^{3/2} \left (-1+\cos (c+d x)-2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{3 a d e} \]
(2*(e*Csc[c + d*x])^(3/2)*(-1 + Cos[c + d*x] - 2*EllipticF[(-2*c + Pi - 2* d*x)/4, 2]*Sin[c + d*x]^(3/2)))/(3*a*d*e)
Time = 0.67 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 25, 3318, 25, 3042, 3044, 15, 3047, 3042, 3120}
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.
| \(\displaystyle \int \frac {\sqrt {e \csc (c+d x)}}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {e \sec \left (c+d x-\frac {\pi }{2}\right )}}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{(\sec (c+d x) a+a) \sqrt {\sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{\sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {\cos (c+d x)}{(-\cos (c+d x) a-a) \sqrt {\sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \int -\frac {\cos (c+d x)}{(\cos (c+d x) a+a) \sqrt {\sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos (c+d x)}{(\cos (c+d x) a+a) \sqrt {\sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\sqrt {\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )} \left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx}{a}+\frac {\int -\frac {\cos (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx}{a}-\frac {\int \frac {\cos (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx}{a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {\int \frac {\cos (c+d x)^2}{\sin (c+d x)^{5/2}}dx}{a}-\frac {\int \frac {\cos (c+d x)}{\sin (c+d x)^{5/2}}dx}{a}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {\int \frac {\cos (c+d x)^2}{\sin (c+d x)^{5/2}}dx}{a}-\frac {\int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)}d\sin (c+d x)}{a d}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {\int \frac {\cos (c+d x)^2}{\sin (c+d x)^{5/2}}dx}{a}+\frac {2}{3 a d \sin ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3047 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {-\frac {2}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}}{a}+\frac {2}{3 a d \sin ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {-\frac {2}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}}{a}+\frac {2}{3 a d \sin ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\sin (c+d x)} \left (-\sqrt {e \csc (c+d x)}\right ) \left (\frac {2}{3 a d \sin ^{\frac {3}{2}}(c+d x)}+\frac {-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}}{a}\right )\) |
-(Sqrt[e*Csc[c + d*x]]*(((-4*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d) - (2* Cos[c + d*x])/(3*d*Sin[c + d*x]^(3/2)))/a + 2/(3*a*d*Sin[c + d*x]^(3/2)))* Sqrt[Sin[c + d*x]])
3.3.95.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(a*Cos[e + f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/ (b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Cos[e + f*x] )^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ [m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos [e + f*x]^FracPart[p] Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / ; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 8.08 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.71
| method | result | size |
| default | \(\frac {\sqrt {2}\, \sqrt {\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}}\, \left (1-\cos \left (d x +c \right )\right ) \left (2 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ) \csc \left (d x +c \right )}{3 a d \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}}\) | \(285\) |
1/3/a/d*2^(1/2)*(e/(1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)+sin(d*x+c)) )^(1/2)*(1-cos(d*x+c))*(2*I*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*2^(1/2)*( -I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(I*(-cot(d*x+c)+csc(d*x+c)))^(1/2)*Ell ipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))-(1-cos(d*x+c))^3* csc(d*x+c)^3-csc(d*x+c)+cot(d*x+c))/((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc( d*x+c)^2+1)*csc(d*x+c))^(1/2)/((1-cos(d*x+c))^3*csc(d*x+c)^3+csc(d*x+c)-co t(d*x+c))^(1/2)*csc(d*x+c)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {2 i \, e} {\left (i \, \cos \left (d x + c\right ) + i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {-2 i \, e} {\left (-i \, \cos \left (d x + c\right ) - i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{3 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
-2/3*(sqrt(2*I*e)*(I*cos(d*x + c) + I)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + sqrt(-2*I*e)*(-I*cos(d*x + c) - I)*weierstrassPInv erse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + sqrt(e/sin(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)
\[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sqrt {e \csc {\left (c + d x \right )}}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
\[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \csc \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \csc \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]