Integrand size = 29, antiderivative size = 130 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {32 \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 a d} \]
-2*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d/a^(1/2)+32/15*cos( d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-2/5*cos(d*x+c)*sin(d*x+c)^2/d/(a+a*sin(d*x +c))^(1/2)+2/15*cos(d*x+c)*(a+a*sin(d*x+c))^(1/2)/a/d
Time = 0.63 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (60 \cos \left (\frac {1}{2} (c+d x)\right )+5 \cos \left (\frac {3}{2} (c+d x)\right )+3 \cos \left (\frac {5}{2} (c+d x)\right )-30 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-60 \sin \left (\frac {1}{2} (c+d x)\right )+5 \sin \left (\frac {3}{2} (c+d x)\right )-3 \sin \left (\frac {5}{2} (c+d x)\right )\right )}{30 d \sqrt {a (1+\sin (c+d x))}} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(60*Cos[(c + d*x)/2] + 5*Cos[(3*(c + d*x))/2] + 3*Cos[(5*(c + d*x))/2] - 30*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 30*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 60*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] - 3*Sin[(5*(c + d*x))/2]))/(30*d*Sqrt[a *(1 + Sin[c + d*x])])
Leaf count is larger than twice the leaf count of optimal. \(268\) vs. \(2(130)=260\).
Time = 2.06 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.06, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {3042, 3360, 3042, 3257, 25, 3042, 3447, 3042, 3502, 27, 3042, 3230, 3042, 3128, 219, 3525, 27, 3042, 3464, 3042, 3128, 219, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a \sin (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x) \sqrt {a \sin (c+d x)+a}}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \frac {\sin ^3(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {\csc (c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3257 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {\int -\frac {\sin (c+d x) (4 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx+\frac {\int \frac {\sin (c+d x) (4 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x) (4 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int \frac {4 a \sin (c+d x)-a \sin ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx+\frac {\int \frac {4 a \sin (c+d x)-a \sin (c+d x)^2}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int -\frac {a^2-14 a^2 \sin (c+d x)}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}+\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\int \frac {a^2-14 a^2 \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\int \frac {a^2-14 a^2 \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {15 a^2 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {15 a^2 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {30 a^2 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{3 a}}{5 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {2 \int \frac {\csc (c+d x) (2 \sin (c+d x) a+a)}{2 \sqrt {\sin (c+d x) a+a}}dx}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc (c+d x) (2 \sin (c+d x) a+a)}{\sqrt {\sin (c+d x) a+a}}dx}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 \sin (c+d x) a+a}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {2 a \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {-\frac {2 a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{a}+\frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {28 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {15 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}+\frac {-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{a}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {4 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
((-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/d)/a + (4*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[ c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + ((2*Cos[c + d*x] *Sqrt[a + a*Sin[c + d*x]])/(3*d) - ((-15*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[a]* Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/d + (28*a^2*Cos[c + d*x ])/(d*Sqrt[a + a*Sin[c + d*x]]))/(3*a))/(5*a)
3.5.66.3.1 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[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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*d*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^(n - 1)/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]])), x] - Simp[1/(b*(2*n - 1)) Int[((c + d*Sin[e + f*x])^(n - 2)/Sqrt[a + b*Sin[e + f*x]])*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n - 3))*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] && GtQ[n, 1] && IntegerQ[2*n]
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_)]), 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_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A *b - a*B)/(b*c - a*d) Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c - A*d)/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*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]
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]
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)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 )/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (15 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )+3 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}-5 a \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-15 a^{2} \sqrt {a -a \sin \left (d x +c \right )}\right )}{15 a^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(123\) |
-2/15*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(15*a^(5/2)*arctanh((a-a*si n(d*x+c))^(1/2)/a^(1/2))+3*(a-a*sin(d*x+c))^(5/2)-5*a*(a-a*sin(d*x+c))^(3/ 2)-15*a^2*(a-a*sin(d*x+c))^(1/2))/a^3/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. 279 vs. \(2 (112) = 224\).
Time = 0.27 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.15 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {15 \, \sqrt {a} {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - {\left (3 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 13\right )} \sin \left (d x + c\right ) + 14 \, \cos \left (d x + c\right ) + 13\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{30 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]
1/30*(15*sqrt(a)*(cos(d*x + c) + sin(d*x + c) + 1)*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*(3*cos(d*x + c)^3 + 4*cos(d*x + c)^2 - (3*cos(d*x + c)^2 - cos(d*x + c) + 13)*sin(d*x + c) + 14*cos(d*x + c) + 13)*sqrt(a*sin(d*x + c) + a)) /(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)
\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \csc \left (d x + c\right )}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
Time = 0.36 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\frac {15 \, \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (12 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{5} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{15 \, d} \]
-1/15*(15*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(2*sqr t(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d *x + 1/2*c))) - 2*sqrt(2)*(12*a^(9/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 1 0*a^(9/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 15*a^(9/2)*sin(-1/4*pi + 1/2* d*x + 1/2*c))/(a^5*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]