Integrand size = 31, antiderivative size = 233 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {1724 a^2 \cos (c+d x)}{6435 d \sqrt {a+a \sin (c+d x)}}-\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {3448 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{45045 d}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d} \]
-1724/15015*cos(d*x+c)*(a+a*sin(d*x+c))^(3/2)/d+2/13*cos(d*x+c)*sin(d*x+c) ^4*(a+a*sin(d*x+c))^(3/2)/d-1724/6435*a^2*cos(d*x+c)/d/(a+a*sin(d*x+c))^(1 /2)-862/9009*a^2*cos(d*x+c)*sin(d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)-38/1287* a^2*cos(d*x+c)*sin(d*x+c)^4/d/(a+a*sin(d*x+c))^(1/2)+3448/45045*a*cos(d*x+ c)*(a+a*sin(d*x+c))^(1/2)/d+6/143*a*cos(d*x+c)*sin(d*x+c)^4*(a+a*sin(d*x+c ))^(1/2)/d
Time = 2.79 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.52 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \sqrt {a (1+\sin (c+d x))} (281816-194160 \cos (2 (c+d x))+22680 \cos (4 (c+d x))+381174 \sin (c+d x)-77665 \sin (3 (c+d x))+3465 \sin (5 (c+d x)))}{360360 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
-1/360360*(a*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d *x])]*(281816 - 194160*Cos[2*(c + d*x)] + 22680*Cos[4*(c + d*x)] + 381174* Sin[c + d*x] - 77665*Sin[3*(c + d*x)] + 3465*Sin[5*(c + d*x)]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 1.53 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.18, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.613, Rules used = {3042, 3358, 3042, 3455, 27, 3042, 3455, 27, 3042, 3460, 3042, 3249, 3042, 3238, 27, 3042, 3230, 3042, 3125}
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.
| \(\displaystyle \int \sin ^3(c+d x) \cos ^2(c+d x) (a \sin (c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^2 (a \sin (c+d x)+a)^{3/2}dx\) |
| \(\Big \downarrow \) 3358 |
| \(\displaystyle \frac {\int \sin ^3(c+d x) (a-a \sin (c+d x)) (\sin (c+d x) a+a)^{5/2}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin (c+d x)^3 (a-a \sin (c+d x)) (\sin (c+d x) a+a)^{5/2}dx}{a^2}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{13} \int \frac {1}{2} \sin ^3(c+d x) (\sin (c+d x) a+a)^{3/2} \left (5 a^2-3 a^2 \sin (c+d x)\right )dx+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{13} \int \sin ^3(c+d x) (\sin (c+d x) a+a)^{3/2} \left (5 a^2-3 a^2 \sin (c+d x)\right )dx+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{13} \int \sin (c+d x)^3 (\sin (c+d x) a+a)^{3/2} \left (5 a^2-3 a^2 \sin (c+d x)\right )dx+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \sin ^3(c+d x) \sqrt {\sin (c+d x) a+a} \left (19 \sin (c+d x) a^3+31 a^3\right )dx+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \int \sin ^3(c+d x) \sqrt {\sin (c+d x) a+a} \left (19 \sin (c+d x) a^3+31 a^3\right )dx+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \int \sin (c+d x)^3 \sqrt {\sin (c+d x) a+a} \left (19 \sin (c+d x) a^3+31 a^3\right )dx+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \int \sin ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \int \sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}dx-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \int \sin ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \int \sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3238 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \left (\frac {2 \int \frac {1}{2} (3 a-2 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{5 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}\right )-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \left (\frac {\int (3 a-2 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{5 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}\right )-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \left (\frac {\int (3 a-2 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{5 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}\right )-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \left (\frac {\frac {7}{3} a \int \sqrt {\sin (c+d x) a+a}dx+\frac {4 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{5 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}\right )-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{13} \left (\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \left (\frac {\frac {7}{3} a \int \sqrt {\sin (c+d x) a+a}dx+\frac {4 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{5 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}\right )-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )+\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}\right )+\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}}{a^2}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac {1}{13} \left (\frac {6 a^3 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}+\frac {1}{11} \left (\frac {431}{9} a^3 \left (\frac {6}{7} \left (\frac {\frac {4 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {14 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{5 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}\right )-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {38 a^4 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}\right )\right )}{a^2}\) |
((2*a^2*Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(13*d) + ( (6*a^3*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*d) + ((-3 8*a^4*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (431*a ^3*((-2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c + d*x]]) + (6 *((-2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*a*d) + ((-14*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (4*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d))/(5*a)))/7))/9)/11)/13)/a^2
3.4.29.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[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[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-Cos[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*(b*(m + 1) - a*Si n[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && ! LtQ[m, -2^(-1)]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[2*n*((b*c + a*d)/(b*( 2*n + 1))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), 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] && GtQ[n, 0] && IntegerQ[2*n]
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/b^2 Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -1]
Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.42
| method | result | size |
| default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{2} \left (3465 \left (\sin ^{5}\left (d x +c \right )\right )+11340 \left (\sin ^{4}\left (d x +c \right )\right )+15085 \left (\sin ^{3}\left (d x +c \right )\right )+12930 \left (\sin ^{2}\left (d x +c \right )\right )+10344 \sin \left (d x +c \right )+6896\right )}{45045 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(97\) |
-2/45045*(1+sin(d*x+c))*a^2*(sin(d*x+c)-1)^2*(3465*sin(d*x+c)^5+11340*sin( d*x+c)^4+15085*sin(d*x+c)^3+12930*sin(d*x+c)^2+10344*sin(d*x+c)+6896)/cos( d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.81 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (3465 \, a \cos \left (d x + c\right )^{7} - 4410 \, a \cos \left (d x + c\right )^{6} - 14140 \, a \cos \left (d x + c\right )^{5} + 7330 \, a \cos \left (d x + c\right )^{4} + 15299 \, a \cos \left (d x + c\right )^{3} - 568 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) - {\left (3465 \, a \cos \left (d x + c\right )^{6} + 7875 \, a \cos \left (d x + c\right )^{5} - 6265 \, a \cos \left (d x + c\right )^{4} - 13595 \, a \cos \left (d x + c\right )^{3} + 1704 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) + 4544 \, a\right )} \sin \left (d x + c\right ) + 4544 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45045 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
-2/45045*(3465*a*cos(d*x + c)^7 - 4410*a*cos(d*x + c)^6 - 14140*a*cos(d*x + c)^5 + 7330*a*cos(d*x + c)^4 + 15299*a*cos(d*x + c)^3 - 568*a*cos(d*x + c)^2 + 2272*a*cos(d*x + c) - (3465*a*cos(d*x + c)^6 + 7875*a*cos(d*x + c)^ 5 - 6265*a*cos(d*x + c)^4 - 13595*a*cos(d*x + c)^3 + 1704*a*cos(d*x + c)^2 + 2272*a*cos(d*x + c) + 4544*a)*sin(d*x + c) + 4544*a)*sqrt(a*sin(d*x + c ) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)
Timed out. \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{3} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.82 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {16 \, \sqrt {2} {\left (27720 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 114660 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 190190 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 160875 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72072 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15015 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )} \sqrt {a}}{45045 \, d} \]
-16/45045*sqrt(2)*(27720*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^13 - 114660*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin( -1/4*pi + 1/2*d*x + 1/2*c)^11 + 190190*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c ))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 160875*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 + 72072*a*sgn(cos(-1/4*pi + 1/2* d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 15015*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3)*sqrt(a)/d
Timed out. \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]