Integrand size = 21, antiderivative size = 102 \[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {24 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^4 \sqrt {\cos (a+b x)}}-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}} \] Output:
24/5*(d*cos(b*x+a))^(1/2)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))/b/d^4/cos( b*x+a)^(1/2)-12/5*sin(b*x+a)/b/d^3/(d*cos(b*x+a))^(1/2)+2/5*sin(b*x+a)^3/b /d/(d*cos(b*x+a))^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.64 \[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {\cos ^3(a+b x) \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {9}{4},\frac {5}{2},\frac {7}{2},\sin ^2(a+b x)\right ) \sin ^5(a+b x)}{5 b (d \cos (a+b x))^{7/2}} \] Input:
Integrate[Sin[a + b*x]^4/(d*Cos[a + b*x])^(7/2),x]
Output:
(Cos[a + b*x]^3*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[9/4, 5/2, 7/2, Si n[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*(d*Cos[a + b*x])^(7/2))
Time = 0.84 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3046, 3042, 3046, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (a+b x)^4}{(d \cos (a+b x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3046 |
| \(\displaystyle \frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{3/2}}dx}{5 d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \int \frac {\sin (a+b x)^2}{(d \cos (a+b x))^{3/2}}dx}{5 d^2}\) |
| \(\Big \downarrow \) 3046 |
| \(\displaystyle \frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {2 \int \sqrt {d \cos (a+b x)}dx}{d^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {2 \int \sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )}dx}{d^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {2 \sqrt {d \cos (a+b x)} \int \sqrt {\cos (a+b x)}dx}{d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {2 \sqrt {d \cos (a+b x)} \int \sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}dx}{d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{b d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}\) |
Input:
Int[Sin[a + b*x]^4/(d*Cos[a + b*x])^(7/2),x]
Output:
(2*Sin[a + b*x]^3)/(5*b*d*(d*Cos[a + b*x])^(5/2)) - (6*((-4*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(b*d^2*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + b*x]])))/(5*d^2)
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Sin[e + f *x])^(m - 2)*(b*Cos[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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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. \(365\) vs. \(2(90)=180\).
Time = 5.14 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.59
| method | result | size |
| default | \(\frac {8 \sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (14 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6} \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-12 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-14 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+12 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+3 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} d +\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d}}{5 d^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}-12 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+6 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\, b}\) | \(366\) |
Input:
int(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x,method=_RETURNVERBOSE)
Output:
8/5*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/d^4/sin(1/2* b*x+1/2*a)^3/(8*sin(1/2*b*x+1/2*a)^6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x +1/2*a)^2-1)*(14*sin(1/2*b*x+1/2*a)^6*cos(1/2*b*x+1/2*a)-12*(sin(1/2*b*x+1 /2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticE(cos(1/2*b*x+1/2* a),2^(1/2))*sin(1/2*b*x+1/2*a)^4-14*cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)^ 4+12*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*Ellipti cE(cos(1/2*b*x+1/2*a),2^(1/2))*sin(1/2*b*x+1/2*a)^2+3*cos(1/2*b*x+1/2*a)*s in(1/2*b*x+1/2*a)^2-3*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2 -1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2)))*(-2*sin(1/2*b*x+1/2*a)^4* d+sin(1/2*b*x+1/2*a)^2*d)^(1/2)/(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 = 120, normalized size of antiderivative = 1.18 \[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=-\frac {2 \, {\left (-12 i \, \sqrt {\frac {1}{2}} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 12 i \, \sqrt {\frac {1}{2}} \sqrt {d} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {d \cos \left (b x + a\right )} {\left (7 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )\right )}}{5 \, b d^{4} \cos \left (b x + a\right )^{3}} \] Input:
integrate(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x, algorithm="fricas")
Output:
-2/5*(-12*I*sqrt(1/2)*sqrt(d)*cos(b*x + a)^3*weierstrassZeta(-4, 0, weiers trassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) + 12*I*sqrt(1/2)*sqrt (d)*cos(b*x + a)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b *x + a) - I*sin(b*x + a))) + sqrt(d*cos(b*x + a))*(7*cos(b*x + a)^2 - 1)*s in(b*x + a))/(b*d^4*cos(b*x + a)^3)
Timed out. \[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(sin(b*x+a)**4/(d*cos(b*x+a))**(7/2),x)
Output:
Timed out
\[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{4}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x, algorithm="maxima")
Output:
integrate(sin(b*x + a)^4/(d*cos(b*x + a))^(7/2), x)
\[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{4}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x, algorithm="giac")
Output:
integrate(sin(b*x + a)^4/(d*cos(b*x + a))^(7/2), x)
Timed out. \[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^4}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \] Input:
int(sin(a + b*x)^4/(d*cos(a + b*x))^(7/2),x)
Output:
int(sin(a + b*x)^4/(d*cos(a + b*x))^(7/2), x)
\[ \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\cos \left (b x +a \right )}\, \sin \left (b x +a \right )^{4}}{\cos \left (b x +a \right )^{4}}d x \right )}{d^{4}} \] Input:
int(sin(b*x+a)^4/(d*cos(b*x+a))^(7/2),x)
Output:
(sqrt(d)*int((sqrt(cos(a + b*x))*sin(a + b*x)**4)/cos(a + b*x)**4,x))/d**4