Integrand size = 29, antiderivative size = 205 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {21 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {149 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d} \]
21/64*a^(3/2)*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d-1/4*cot (d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^(3/2)/d-2*a^2*cos(d*x+c)/d/(a+a*sin( d*x+c))^(1/2)+149/64*a^2*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+19/32*a^2*cot (d*x+c)*csc(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-1/8*a*cot(d*x+c)*csc(d*x+c)^2* (a+a*sin(d*x+c))^(1/2)/d
Time = 2.64 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.91 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \csc ^{13}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (1486 \cos \left (\frac {1}{2} (c+d x)\right )-1030 \cos \left (\frac {3}{2} (c+d x)\right )-754 \cos \left (\frac {5}{2} (c+d x)\right )+426 \cos \left (\frac {7}{2} (c+d x)\right )+128 \cos \left (\frac {9}{2} (c+d x)\right )-63 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+84 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-21 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+63 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-84 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+21 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-1486 \sin \left (\frac {1}{2} (c+d x)\right )-1030 \sin \left (\frac {3}{2} (c+d x)\right )+754 \sin \left (\frac {5}{2} (c+d x)\right )+426 \sin \left (\frac {7}{2} (c+d x)\right )-128 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{64 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^4} \]
-1/64*(a*Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(1486*Cos[(c + d*x )/2] - 1030*Cos[(3*(c + d*x))/2] - 754*Cos[(5*(c + d*x))/2] + 426*Cos[(7*( c + d*x))/2] + 128*Cos[(9*(c + d*x))/2] - 63*Log[1 + Cos[(c + d*x)/2] - Si n[(c + d*x)/2]] + 84*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 21*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2] ] + 63*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 84*Cos[2*(c + d*x)]* Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 21*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 1486*Sin[(c + d*x)/2] - 1030*Sin[( 3*(c + d*x))/2] + 754*Sin[(5*(c + d*x))/2] + 426*Sin[(7*(c + d*x))/2] - 12 8*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - S ec[(c + d*x)/4]^2)^4)
Time = 1.78 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.25, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 3360, 3042, 3242, 27, 2011, 3042, 3252, 219, 3523, 27, 3042, 3454, 27, 3042, 3459, 3042, 3251, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^{3/2}}{\sin (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \csc ^5(c+d x) (\sin (c+d x) a+a)^{3/2} \left (1-2 \sin ^2(c+d x)\right )dx+\int \csc (c+d x) (\sin (c+d x) a+a)^{3/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle 2 \int \frac {\csc (c+d x) \left (\sin (c+d x) a^2+a^2\right )}{2 \sqrt {\sin (c+d x) a+a}}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\csc (c+d x) \left (\sin (c+d x) a^2+a^2\right )}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 2011 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx+a \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle -\frac {2 a^2 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {1}{2} \csc ^4(c+d x) (3 a-13 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{4 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc ^4(c+d x) (3 a-13 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(3 a-13 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)^4}dx}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{3} \int -\frac {3}{2} \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a} \left (23 \sin (c+d x) a^2+19 a^2\right )dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{2} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a} \left (23 \sin (c+d x) a^2+19 a^2\right )dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a} \left (23 \sin (c+d x) a^2+19 a^2\right )}{\sin (c+d x)^3}dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {149}{4} a^2 \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {149}{4} a^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {149}{4} a^2 \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {149}{4} a^2 \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {149}{4} a^2 \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}+\frac {\frac {1}{2} \left (\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {149}{4} a^2 \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{d}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}\) |
(-2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (2*a^2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(4*d) + (-((a^2*Cot[c + d*x]*Csc[c + d* x]^2*Sqrt[a + a*Sin[c + d*x]])/d) + ((19*a^3*Cot[c + d*x]*Csc[c + d*x])/(2 *d*Sqrt[a + a*Sin[c + d*x]]) - (149*a^2*(-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sqrt[a + a*Si n[c + d*x]])))/4)/2)/(8*a)
3.5.59.3.1 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[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) 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] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
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 [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) 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, A, B}, 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 = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-128 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sin ^{4}\left (d x +c \right )\right ) a^{\frac {7}{2}}+21 \left (\sin ^{4}\left (d x +c \right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4}-149 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} \sqrt {a}+461 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}-435 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}+107 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}\right )}{64 a^{\frac {5}{2}} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(188\) |
1/64*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(-128*(-a*(sin(d*x+c)-1))^(1 /2)*sin(d*x+c)^4*a^(7/2)+21*sin(d*x+c)^4*arctanh((-a*(sin(d*x+c)-1))^(1/2) /a^(1/2))*a^4-149*(-a*(sin(d*x+c)-1))^(7/2)*a^(1/2)+461*(-a*(sin(d*x+c)-1) )^(5/2)*a^(3/2)-435*(-a*(sin(d*x+c)-1))^(3/2)*a^(5/2)+107*(-a*(sin(d*x+c)- 1))^(1/2)*a^(7/2))/a^(5/2)/sin(d*x+c)^4/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. 460 vs. \(2 (179) = 358\).
Time = 0.30 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.24 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {21 \, {\left (a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (128 \, a \cos \left (d x + c\right )^{5} + 277 \, a \cos \left (d x + c\right )^{4} - 242 \, a \cos \left (d x + c\right )^{3} - 500 \, a \cos \left (d x + c\right )^{2} + 130 \, a \cos \left (d x + c\right ) - {\left (128 \, a \cos \left (d x + c\right )^{4} - 149 \, a \cos \left (d x + c\right )^{3} - 391 \, a \cos \left (d x + c\right )^{2} + 109 \, a \cos \left (d x + c\right ) + 239 \, a\right )} \sin \left (d x + c\right ) + 239 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{256 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right ) + d\right )}} \]
1/256*(21*(a*cos(d*x + c)^5 + a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 - 2*a* cos(d*x + c)^2 + a*cos(d*x + c) + (a*cos(d*x + c)^4 - 2*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*(128*a*cos(d* x + c)^5 + 277*a*cos(d*x + c)^4 - 242*a*cos(d*x + c)^3 - 500*a*cos(d*x + c )^2 + 130*a*cos(d*x + c) - (128*a*cos(d*x + c)^4 - 149*a*cos(d*x + c)^3 - 391*a*cos(d*x + c)^2 + 109*a*cos(d*x + c) + 239*a)*sin(d*x + c) + 239*a)*s qrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4 - 2*d*cos(d* x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c) + (d*cos(d*x + c)^4 - 2*d*c os(d*x + c)^2 + d)*sin(d*x + c) + d)
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{5} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.20 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (21 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 512 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {4 \, {\left (1192 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1844 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 870 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 107 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}\right )} \sqrt {a}}{256 \, d} \]
1/256*sqrt(2)*(21*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*p i + 1/2*d*x + 1/2*c)) + 512*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4 *pi + 1/2*d*x + 1/2*c) + 4*(1192*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 - 1844*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 + 870*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 - 107*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)^4)*sqrt(a)/d
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^5} \,d x \]