Integrand size = 29, antiderivative size = 186 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {9 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}+\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d} \]
9/4*a^(3/2)*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d-2/5*cos(d *x+c)*(a+a*sin(d*x+c))^(3/2)/d-1/2*cot(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))^ (3/2)/d+73/20*a^2*cos(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-2/5*a*cos(d*x+c)*(a+ a*sin(d*x+c))^(1/2)/d-3/4*a*cot(d*x+c)*(a+a*sin(d*x+c))^(1/2)/d
Time = 6.14 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.73 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \csc ^7\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-118 \cos \left (\frac {1}{2} (c+d x)\right )+130 \cos \left (\frac {3}{2} (c+d x)\right )+36 \cos \left (\frac {5}{2} (c+d x)\right )-10 \cos \left (\frac {7}{2} (c+d x)\right )+2 \cos \left (\frac {9}{2} (c+d x)\right )-45 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-45 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+118 \sin \left (\frac {1}{2} (c+d x)\right )+130 \sin \left (\frac {3}{2} (c+d x)\right )-36 \sin \left (\frac {5}{2} (c+d x)\right )-10 \sin \left (\frac {7}{2} (c+d x)\right )-2 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{20 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^2} \]
-1/20*(a*Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(-118*Cos[(c + d*x) /2] + 130*Cos[(3*(c + d*x))/2] + 36*Cos[(5*(c + d*x))/2] - 10*Cos[(7*(c + d*x))/2] + 2*Cos[(9*(c + d*x))/2] - 45*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 45*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2 ]] + 45*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 45*Cos[2*(c + d*x)] *Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 118*Sin[(c + d*x)/2] + 130 *Sin[(3*(c + d*x))/2] - 36*Sin[(5*(c + d*x))/2] - 10*Sin[(7*(c + d*x))/2] - 2*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)
Time = 1.56 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.24, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3360, 3042, 3230, 3042, 3126, 3042, 3125, 3523, 27, 3042, 3454, 27, 3042, 3460, 3042, 3252, 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.
| \(\displaystyle \int \cos (c+d x) \cot ^3(c+d x) (a \sin (c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^{3/2}}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \sin (c+d x) (\sin (c+d x) a+a)^{3/2}dx+\int \csc ^3(c+d x) (\sin (c+d x) a+a)^{3/2} \left (1-2 \sin ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x) (\sin (c+d x) a+a)^{3/2}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx+\frac {3}{5} \int (\sin (c+d x) a+a)^{3/2}dx-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{5} \int (\sin (c+d x) a+a)^{3/2}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx+\frac {3}{5} \left (\frac {4}{3} a \int \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx+\frac {3}{5} \left (\frac {4}{3} a \int \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {3}{2} \csc ^2(c+d x) (a-3 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{2 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \csc ^2(c+d x) (a-3 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {(a-3 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)^2}dx}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {3 \left (\int -\frac {1}{2} \csc (c+d x) \sqrt {\sin (c+d x) a+a} \left (7 \sin (c+d x) a^2+3 a^2\right )dx-\frac {a^2 \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\right )}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a} \left (7 \sin (c+d x) a^2+3 a^2\right )dx-\frac {a^2 \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\right )}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (-\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a} \left (7 \sin (c+d x) a^2+3 a^2\right )}{\sin (c+d x)}dx-\frac {a^2 \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\right )}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \left (\frac {14 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-3 a^2 \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx\right )-\frac {a^2 \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\right )}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \left (\frac {14 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-3 a^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx\right )-\frac {a^2 \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\right )}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \left (\frac {6 a^3 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\frac {14 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\right )}{4 a}+\frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{5} \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {3 \left (\frac {1}{2} \left (\frac {6 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}+\frac {14 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\right )}{4 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}\) |
(-2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(2*d) + (3*((-8*a^2*Cos[c + d*x])/(3*d*S qrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3* d)))/5 + (3*(-((a^2*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d) + ((6*a^(5/2 )*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (14*a^3*Co s[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/2))/(4*a)
3.5.57.3.1 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[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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*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]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/d^4 Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IGtQ[m, 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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{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] && 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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {a}-40 a^{\frac {3}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \left (\sin ^{2}\left (d x +c \right )\right )-45 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{3} \left (\sin ^{2}\left (d x +c \right )\right )-35 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}+45 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {5}{2}}\right )}{20 a^{\frac {3}{2}} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(178\) |
-1/20*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(3/2)*(8*(-a*(sin(d*x+c)- 1))^(5/2)*sin(d*x+c)^2*a^(1/2)-40*a^(3/2)*(-a*(sin(d*x+c)-1))^(3/2)*sin(d* x+c)^2-45*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^3*sin(d*x+c)^2-35*( -a*(sin(d*x+c)-1))^(3/2)*a^(3/2)+45*(-a*(sin(d*x+c)-1))^(1/2)*a^(5/2))/sin (d*x+c)^2/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. 404 vs. \(2 (158) = 316\).
Time = 0.30 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.17 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {45 \, {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (8 \, a \cos \left (d x + c\right )^{5} - 16 \, a \cos \left (d x + c\right )^{4} + 16 \, a \cos \left (d x + c\right )^{3} + 99 \, a \cos \left (d x + c\right )^{2} - 14 \, a \cos \left (d x + c\right ) - {\left (8 \, a \cos \left (d x + c\right )^{4} + 24 \, a \cos \left (d x + c\right )^{3} + 40 \, a \cos \left (d x + c\right )^{2} - 59 \, a \cos \left (d x + c\right ) - 73 \, a\right )} \sin \left (d x + c\right ) - 73 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{80 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \]
1/80*(45*(a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) + (a*cos(d* x + c)^2 - a)*sin(d*x + c) - a)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d* x + c)^2 + 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d* x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d *x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(8* a*cos(d*x + c)^5 - 16*a*cos(d*x + c)^4 + 16*a*cos(d*x + c)^3 + 99*a*cos(d* x + c)^2 - 14*a*cos(d*x + c) - (8*a*cos(d*x + c)^4 + 24*a*cos(d*x + c)^3 + 40*a*cos(d*x + c)^2 - 59*a*cos(d*x + c) - 73*a)*sin(d*x + c) - 73*a)*sqrt (a*sin(d*x + c) + a))/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c ) + (d*cos(d*x + c)^2 - d)*sin(d*x + c) - d)
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{3} \,d x } \]
Time = 0.77 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.17 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (128 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 320 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {20 \, {\left (14 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}\right )} \sqrt {a}}{80 \, d} \]
1/80*sqrt(2)*(128*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 - 320*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 + 45*sqrt(2)*a*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2* d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))*sgn(cos(- 1/4*pi + 1/2*d*x + 1/2*c)) - 20*(14*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 - 9*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) *sin(-1/4*pi + 1/2*d*x + 1/2*c))/(2*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^ 2)*sqrt(a)/d
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^3} \,d x \]