Integrand size = 23, antiderivative size = 221 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {283 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {1729 \cos (c+d x)}{120 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {787 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{240 a^3 d} \]
1/4*cos(d*x+c)*sin(d*x+c)^4/d/(a+a*sin(d*x+c))^(5/2)+21/16*cos(d*x+c)*sin( d*x+c)^3/a/d/(a+a*sin(d*x+c))^(3/2)+283/32*arctanh(1/2*cos(d*x+c)*a^(1/2)* 2^(1/2)/(a+a*sin(d*x+c))^(1/2))/a^(5/2)/d*2^(1/2)-1729/120*cos(d*x+c)/a^2/ d/(a+a*sin(d*x+c))^(1/2)-157/80*cos(d*x+c)*sin(d*x+c)^2/a^2/d/(a+a*sin(d*x +c))^(1/2)+787/240*cos(d*x+c)*(a+a*sin(d*x+c))^(1/2)/a^3/d
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (2547 \cos \left (\frac {1}{2} (c+d x)\right )+3603 \cos \left (\frac {3}{2} (c+d x)\right )-872 \cos \left (\frac {5}{2} (c+d x)\right )+52 \cos \left (\frac {7}{2} (c+d x)\right )+12 \cos \left (\frac {9}{2} (c+d x)\right )-2547 \sin \left (\frac {1}{2} (c+d x)\right )+(8490+8490 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+3603 \sin \left (\frac {3}{2} (c+d x)\right )+872 \sin \left (\frac {5}{2} (c+d x)\right )+52 \sin \left (\frac {7}{2} (c+d x)\right )-12 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{480 d (a (1+\sin (c+d x)))^{5/2}} \]
-1/480*((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(2547*Cos[(c + d*x)/2] + 360 3*Cos[(3*(c + d*x))/2] - 872*Cos[(5*(c + d*x))/2] + 52*Cos[(7*(c + d*x))/2 ] + 12*Cos[(9*(c + d*x))/2] - 2547*Sin[(c + d*x)/2] + (8490 + 8490*I)*(-1) ^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d *x)/2] + Sin[(c + d*x)/2])^4 + 3603*Sin[(3*(c + d*x))/2] + 872*Sin[(5*(c + d*x))/2] + 52*Sin[(7*(c + d*x))/2] - 12*Sin[(9*(c + d*x))/2]))/(d*(a*(1 + Sin[c + d*x]))^(5/2))
Time = 1.34 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.12, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 3244, 27, 3042, 3456, 27, 3042, 3462, 27, 3042, 3447, 3042, 3502, 27, 3042, 3230, 3042, 3128, 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.
| \(\displaystyle \int \frac {\sin ^5(c+d x)}{(a \sin (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5}{(a \sin (c+d x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\int \frac {\sin ^3(c+d x) (8 a-13 a \sin (c+d x))}{2 (\sin (c+d x) a+a)^{3/2}}dx}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\int \frac {\sin ^3(c+d x) (8 a-13 a \sin (c+d x))}{(\sin (c+d x) a+a)^{3/2}}dx}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\int \frac {\sin (c+d x)^3 (8 a-13 a \sin (c+d x))}{(\sin (c+d x) a+a)^{3/2}}dx}{8 a^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\int \frac {\sin ^2(c+d x) \left (126 a^2-157 a^2 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{2 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\int \frac {\sin ^2(c+d x) \left (126 a^2-157 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\int \frac {\sin (c+d x)^2 \left (126 a^2-157 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {2 \int -\frac {\sin (c+d x) \left (628 a^3-787 a^3 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{5 a}+\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {\sin (c+d x) \left (628 a^3-787 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {\sin (c+d x) \left (628 a^3-787 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {628 a^3 \sin (c+d x)-787 a^3 \sin ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {628 a^3 \sin (c+d x)-787 a^3 \sin (c+d x)^2}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {2 \int -\frac {787 a^4-3458 a^4 \sin (c+d x)}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}+\frac {1574 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {1574 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\int \frac {787 a^4-3458 a^4 \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {1574 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\int \frac {787 a^4-3458 a^4 \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {1574 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {4245 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {6916 a^4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {1574 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {4245 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {6916 a^4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {1574 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {6916 a^4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {8490 a^4 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{3 a}}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac {\frac {\frac {314 a^2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {1574 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {6916 a^4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {4245 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{4 a^2}-\frac {21 a \sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}\) |
(Cos[c + d*x]*Sin[c + d*x]^4)/(4*d*(a + a*Sin[c + d*x])^(5/2)) - ((-21*a*C os[c + d*x]*Sin[c + d*x]^3)/(2*d*(a + a*Sin[c + d*x])^(3/2)) + ((314*a^2*C os[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) - ((1574*a^2*Co s[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - ((-4245*Sqrt[2]*a^(7/2)*ArcTa nh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/d + (6916*a ^4*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/(3*a))/(5*a))/(4*a^2))/(8*a ^2)
3.1.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], 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] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{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] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Sin[e + f*x])^m*(c + d*S in[e + f*x])^(n - 1)*Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 0.89 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(\frac {\left (\left (3840 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {5}{2}}+320 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}+192 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {a}-4245 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+\sin \left (d x +c \right ) \left (-7680 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {5}{2}}-640 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-384 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {a}+8490 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right )-9780 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {5}{2}}+470 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-384 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {a}+8490 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{480 a^{\frac {11}{2}} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(323\) |
1/480*((3840*(a-a*sin(d*x+c))^(1/2)*a^(5/2)+320*(a-a*sin(d*x+c))^(3/2)*a^( 3/2)+192*(a-a*sin(d*x+c))^(5/2)*a^(1/2)-4245*2^(1/2)*arctanh(1/2*(a-a*sin( d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^3)*cos(d*x+c)^2+sin(d*x+c)*(-7680*(a-a*si n(d*x+c))^(1/2)*a^(5/2)-640*(a-a*sin(d*x+c))^(3/2)*a^(3/2)-384*(a-a*sin(d* x+c))^(5/2)*a^(1/2)+8490*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2 )/a^(1/2))*a^3)-9780*(a-a*sin(d*x+c))^(1/2)*a^(5/2)+470*(a-a*sin(d*x+c))^( 3/2)*a^(3/2)-384*(a-a*sin(d*x+c))^(5/2)*a^(1/2)+8490*2^(1/2)*arctanh(1/2*( a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^3)*(-a*(sin(d*x+c)-1))^(1/2)/a^(1 1/2)/(1+sin(d*x+c))/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (190) = 380\).
Time = 0.32 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.72 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {4245 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (96 \, \cos \left (d x + c\right )^{5} + 256 \, \cos \left (d x + c\right )^{4} - 1760 \, \cos \left (d x + c\right )^{3} + 2475 \, \cos \left (d x + c\right )^{2} - {\left (96 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} - 1920 \, \cos \left (d x + c\right )^{2} - 4395 \, \cos \left (d x + c\right ) - 60\right )} \sin \left (d x + c\right ) + 4335 \, \cos \left (d x + c\right ) - 60\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{960 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
1/960*(4245*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*co s(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(96*cos(d*x + c)^5 + 256*cos(d*x + c)^4 - 1760*cos(d*x + c)^3 + 2 475*cos(d*x + c)^2 - (96*cos(d*x + c)^4 - 160*cos(d*x + c)^3 - 1920*cos(d* x + c)^2 - 4395*cos(d*x + c) - 60)*sin(d*x + c) + 4335*cos(d*x + c) - 60)* sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))
Timed out. \[ \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sin \left (d x + c\right )^{5}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Time = 0.64 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\frac {4245 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4245 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {30 \, \sqrt {2} {\left (37 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {256 \, \sqrt {2} {\left (6 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{15} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{960 \, d} \]
-1/960*(4245*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(5/2)*sgn( cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 4245*sqrt(2)*log(-sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(5/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) + 30*sqrt(2)* (37*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 35*sqrt(a)*sin(-1/4*pi + 1/ 2*d*x + 1/2*c))/((sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^2*a^3*sgn(cos(-1/4 *pi + 1/2*d*x + 1/2*c))) - 256*sqrt(2)*(6*a^(25/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 5*a^(25/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 + 30*a^(25/2)*sin( -1/4*pi + 1/2*d*x + 1/2*c))/(a^15*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^5}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]