Integrand size = 21, antiderivative size = 64 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {\sqrt {d \cos (a+b x)} \csc (a+b x)}{b d}+\frac {\sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b \sqrt {d \cos (a+b x)}} \] Output:
-(d*cos(b*x+a))^(1/2)*csc(b*x+a)/b/d+cos(b*x+a)^(1/2)*InverseJacobiAM(1/2* a+1/2*b*x,2^(1/2))/b/(d*cos(b*x+a))^(1/2)
Time = 0.36 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {-\cot (a+b x)+\sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b \sqrt {d \cos (a+b x)}} \] Input:
Integrate[Csc[a + b*x]^2/Sqrt[d*Cos[a + b*x]],x]
Output:
(-Cot[a + b*x] + Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(b*Sqrt[d*C os[a + b*x]])
Time = 0.57 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3050, 3042, 3121, 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.
\(\displaystyle \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (a+b x)^2 \sqrt {d \cos (a+b x)}}dx\) |
\(\Big \downarrow \) 3050 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {d \cos (a+b x)}}dx-\frac {\csc (a+b x) \sqrt {d \cos (a+b x)}}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )}}dx-\frac {\csc (a+b x) \sqrt {d \cos (a+b x)}}{b d}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {\sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}}dx}{2 \sqrt {d \cos (a+b x)}}-\frac {\csc (a+b x) \sqrt {d \cos (a+b x)}}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}}dx}{2 \sqrt {d \cos (a+b x)}}-\frac {\csc (a+b x) \sqrt {d \cos (a+b x)}}{b d}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b \sqrt {d \cos (a+b x)}}-\frac {\csc (a+b x) \sqrt {d \cos (a+b x)}}{b d}\) |
Input:
Int[Csc[a + b*x]^2/Sqrt[d*Cos[a + b*x]],x]
Output:
-((Sqrt[d*Cos[a + b*x]]*Csc[a + b*x])/(b*d)) + (Sqrt[Cos[a + b*x]]*Ellipti cF[(a + b*x)/2, 2])/(b*Sqrt[d*Cos[a + b*x]])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a *b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Cos[e + f*x])^ n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 1] && IntegersQ[2*m, 2*n]
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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Leaf count of result is larger than twice the leaf count of optimal. \(187\) vs. \(2(59)=118\).
Time = 4.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.94
method | result | size |
default | \(\frac {\sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, d \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )^{\frac {3}{2}} \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-4 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+4 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}{2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} d +\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d \right )^{\frac {3}{2}} \sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\, b}\) | \(188\) |
Input:
int(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/cos(1/2*b*x+ 1/2*a)/(-2*sin(1/2*b*x+1/2*a)^4*d+sin(1/2*b*x+1/2*a)^2*d)^(3/2)*d*sin(1/2* b*x+1/2*a)*(2*cos(1/2*b*x+1/2*a)*(2*sin(1/2*b*x+1/2*a)^2-1)^(3/2)*(sin(1/2 *b*x+1/2*a)^2)^(1/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))-4*sin(1/2*b*x+1 /2*a)^4+4*sin(1/2*b*x+1/2*a)^2-1)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.44 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {-i \, \sqrt {\frac {1}{2}} \sqrt {d} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {\frac {1}{2}} \sqrt {d} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - \sqrt {d \cos \left (b x + a\right )}}{b d \sin \left (b x + a\right )} \] Input:
integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x, algorithm="fricas")
Output:
(-I*sqrt(1/2)*sqrt(d)*sin(b*x + a)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a)) + I*sqrt(1/2)*sqrt(d)*sin(b*x + a)*weierstrassPInverse( -4, 0, cos(b*x + a) - I*sin(b*x + a)) - sqrt(d*cos(b*x + a)))/(b*d*sin(b*x + a))
\[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )}}{\sqrt {d \cos {\left (a + b x \right )}}}\, dx \] Input:
integrate(csc(b*x+a)**2/(d*cos(b*x+a))**(1/2),x)
Output:
Integral(csc(a + b*x)**2/sqrt(d*cos(a + b*x)), x)
\[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \] Input:
integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x, algorithm="maxima")
Output:
integrate(csc(b*x + a)^2/sqrt(d*cos(b*x + a)), x)
\[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \] Input:
integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x, algorithm="giac")
Output:
integrate(csc(b*x + a)^2/sqrt(d*cos(b*x + a)), x)
Timed out. \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \] Input:
int(1/(sin(a + b*x)^2*(d*cos(a + b*x))^(1/2)),x)
Output:
int(1/(sin(a + b*x)^2*(d*cos(a + b*x))^(1/2)), x)
\[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\cos \left (b x +a \right )}\, \csc \left (b x +a \right )^{2}}{\cos \left (b x +a \right )}d x \right )}{d} \] Input:
int(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x)
Output:
(sqrt(d)*int((sqrt(cos(a + b*x))*csc(a + b*x)**2)/cos(a + b*x),x))/d