Integrand size = 21, antiderivative size = 96 \[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=-\frac {d (d \cos (a+b x))^{5/2} \csc (a+b x)}{b}-\frac {5 d^4 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b \sqrt {d \cos (a+b x)}}-\frac {5 d^3 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b} \] Output:
-d*(d*cos(b*x+a))^(5/2)*csc(b*x+a)/b-5/3*d^4*cos(b*x+a)^(1/2)*InverseJacob iAM(1/2*a+1/2*b*x,2^(1/2))/b/(d*cos(b*x+a))^(1/2)-5/3*d^3*(d*cos(b*x+a))^( 1/2)*sin(b*x+a)/b
Time = 0.61 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.76 \[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=\frac {d^3 \sqrt {d \cos (a+b x)} \left (\sqrt {\cos (a+b x)} (-4+\cos (2 (a+b x))) \csc (a+b x)-5 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )\right )}{3 b \sqrt {\cos (a+b x)}} \] Input:
Integrate[(d*Cos[a + b*x])^(7/2)*Csc[a + b*x]^2,x]
Output:
(d^3*Sqrt[d*Cos[a + b*x]]*(Sqrt[Cos[a + b*x]]*(-4 + Cos[2*(a + b*x)])*Csc[ a + b*x] - 5*EllipticF[(a + b*x)/2, 2]))/(3*b*Sqrt[Cos[a + b*x]])
Time = 0.76 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3047, 3042, 3115, 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.
\(\displaystyle \int \csc ^2(a+b x) (d \cos (a+b x))^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \cos (a+b x))^{7/2}}{\sin (a+b x)^2}dx\) |
\(\Big \downarrow \) 3047 |
\(\displaystyle -\frac {5}{2} d^2 \int (d \cos (a+b x))^{3/2}dx-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5}{2} d^2 \int \left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {5}{2} d^2 \left (\frac {1}{3} d^2 \int \frac {1}{\sqrt {d \cos (a+b x)}}dx+\frac {2 d \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b}\right )-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5}{2} d^2 \left (\frac {1}{3} d^2 \int \frac {1}{\sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )}}dx+\frac {2 d \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b}\right )-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle -\frac {5}{2} d^2 \left (\frac {d^2 \sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}}dx}{3 \sqrt {d \cos (a+b x)}}+\frac {2 d \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b}\right )-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5}{2} d^2 \left (\frac {d^2 \sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {d \cos (a+b x)}}+\frac {2 d \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b}\right )-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle -\frac {5}{2} d^2 \left (\frac {2 d^2 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b \sqrt {d \cos (a+b x)}}+\frac {2 d \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b}\right )-\frac {d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}\) |
Input:
Int[(d*Cos[a + b*x])^(7/2)*Csc[a + b*x]^2,x]
Output:
-((d*(d*Cos[a + b*x])^(5/2)*Csc[a + b*x])/b) - (5*d^2*((2*d^2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(3*b*Sqrt[d*Cos[a + b*x]]) + (2*d*Sqrt[d *Cos[a + b*x]]*Sin[a + b*x])/(3*b)))/2
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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 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. \(215\) vs. \(2(85)=170\).
Time = 7.86 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.25
method | result | size |
default | \(-\frac {\sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, d^{5} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (-32 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}+10 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )^{\frac {3}{2}} \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+64 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}-28 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-4 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+3\right )}{6 \left (-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} d +\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d \right )^{\frac {3}{2}} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\, b}\) | \(216\) |
Input:
int((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/6*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*d^5/(-2*sin (1/2*b*x+1/2*a)^4*d+sin(1/2*b*x+1/2*a)^2*d)^(3/2)/cos(1/2*b*x+1/2*a)*sin(1 /2*b*x+1/2*a)*(-32*sin(1/2*b*x+1/2*a)^8+10*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))+64*sin(1/2*b*x+1/2*a)^6-28*sin(1/2*b*x+1/2*a)^4-4*sin(1/2*b* x+1/2*a)^2+3)/(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 = 108, normalized size of antiderivative = 1.12 \[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=\frac {5 i \, \sqrt {\frac {1}{2}} d^{\frac {7}{2}} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 5 i \, \sqrt {\frac {1}{2}} d^{\frac {7}{2}} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (2 \, d^{3} \cos \left (b x + a\right )^{2} - 5 \, d^{3}\right )} \sqrt {d \cos \left (b x + a\right )}}{3 \, b \sin \left (b x + a\right )} \] Input:
integrate((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x, algorithm="fricas")
Output:
1/3*(5*I*sqrt(1/2)*d^(7/2)*sin(b*x + a)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a)) - 5*I*sqrt(1/2)*d^(7/2)*sin(b*x + a)*weierstrassPI nverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)) + (2*d^3*cos(b*x + a)^2 - 5*d ^3)*sqrt(d*cos(b*x + a)))/(b*sin(b*x + a))
Timed out. \[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate((d*cos(b*x+a))**(7/2)*csc(b*x+a)**2,x)
Output:
Timed out
\[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \csc \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x, algorithm="maxima")
Output:
integrate((d*cos(b*x + a))^(7/2)*csc(b*x + a)^2, x)
\[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \csc \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x, algorithm="giac")
Output:
integrate((d*cos(b*x + a))^(7/2)*csc(b*x + a)^2, x)
Timed out. \[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \] Input:
int((d*cos(a + b*x))^(7/2)/sin(a + b*x)^2,x)
Output:
int((d*cos(a + b*x))^(7/2)/sin(a + b*x)^2, x)
\[ \int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx=\sqrt {d}\, \left (\int \sqrt {\cos \left (b x +a \right )}\, \cos \left (b x +a \right )^{3} \csc \left (b x +a \right )^{2}d x \right ) d^{3} \] Input:
int((d*cos(b*x+a))^(7/2)*csc(b*x+a)^2,x)
Output:
sqrt(d)*int(sqrt(cos(a + b*x))*cos(a + b*x)**3*csc(a + b*x)**2,x)*d**3