Integrand size = 31, antiderivative size = 191 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {23 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 a^{3/2} d}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{3/2} d}-\frac {9 \cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}} \] Output:
23/8*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(3/2)/d-2*arctan h(1/2*a^(1/2)*cos(d*x+c)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))*2^(1/2)/a^(3/2)/d -9/8*cot(d*x+c)/a/d/(a+a*sin(d*x+c))^(1/2)+7/12*cot(d*x+c)*csc(d*x+c)/a/d/ (a+a*sin(d*x+c))^(1/2)-1/3*cot(d*x+c)*csc(d*x+c)^2/a/d/(a+a*sin(d*x+c))^(1 /2)
Result contains complex when optimal does not.
Time = 2.75 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left ((768+768 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right )-\frac {8 \csc ^9\left (\frac {1}{2} (c+d x)\right ) \left (228 \cos \left (\frac {1}{2} (c+d x)\right )-110 \cos \left (\frac {3}{2} (c+d x)\right )-54 \cos \left (\frac {5}{2} (c+d x)\right )-228 \sin \left (\frac {1}{2} (c+d x)\right )-207 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+207 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-110 \sin \left (\frac {3}{2} (c+d x)\right )+54 \sin \left (\frac {5}{2} (c+d x)\right )+69 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-69 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))\right )}{\left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3}\right )}{192 d (a (1+\sin (c+d x)))^{3/2}} \] Input:
Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*((768 + 768*I)*(-1)^(3/4)*ArcTanh [(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - (8*Csc[(c + d*x)/2]^9*( 228*Cos[(c + d*x)/2] - 110*Cos[(3*(c + d*x))/2] - 54*Cos[(5*(c + d*x))/2] - 228*Sin[(c + d*x)/2] - 207*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]* Sin[c + d*x] + 207*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d* x] - 110*Sin[(3*(c + d*x))/2] + 54*Sin[(5*(c + d*x))/2] + 69*Log[1 + Cos[( c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 69*Log[1 - Cos[(c + d*x )/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3))/(192*d*(a*(1 + Sin[c + d*x]))^(3/2))
Time = 1.49 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.12, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 3353, 3042, 3463, 27, 3042, 3463, 27, 3042, 3463, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a \sin (c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x)^4 (a \sin (c+d x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3353 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x) (a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-a \sin (c+d x)}{\sin (c+d x)^4 \sqrt {\sin (c+d x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^3(c+d x) \left (7 a^2-5 a^2 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^3(c+d x) \left (7 a^2-5 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {7 a^2-5 a^2 \sin (c+d x)}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {3 \csc ^2(c+d x) \left (9 a^3-7 a^3 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3 \int \frac {\csc ^2(c+d x) \left (9 a^3-7 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \int \frac {9 a^3-7 a^3 \sin (c+d x)}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (\frac {\int -\frac {\csc (c+d x) \left (23 a^4-9 a^4 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\int \frac {\csc (c+d x) \left (23 a^4-9 a^4 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\int \frac {23 a^4-9 a^4 \sin (c+d x)}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {23 a^3 \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-32 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {23 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-32 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {23 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {64 a^4 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {23 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {32 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {-\frac {-\frac {3 \left (-\frac {\frac {32 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {46 a^4 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {3 \left (-\frac {\frac {32 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {46 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {9 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{a^2}\) |
Input:
Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]
Output:
(-1/3*(a*Cot[c + d*x]*Csc[c + d*x]^2)/(d*Sqrt[a + a*Sin[c + d*x]]) - ((-7* a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]]) - (3*(-1/2*( (-46*a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (32*Sqrt[2]*a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Si n[c + d*x]])])/d)/a - (9*a^3*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])))/ (4*a))/(6*a))/a^2
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[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_)]^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] && (ILtQ[m, 0] || !IGtQ[ n, 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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 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]
Time = 0.39 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-69 a^{6} \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \sin \left (d x +c \right )^{3}+27 a^{\frac {7}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}}+48 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{6} \sin \left (d x +c \right )^{3}-40 a^{\frac {9}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}}+21 a^{\frac {11}{2}} \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\right )}{24 a^{\frac {15}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(182\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBO SE)
Output:
-1/24/a^(15/2)*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(-69*a^6*arctanh(( -a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+c)^3+27*a^(7/2)*(-a*(sin(d*x+c)- 1))^(5/2)+48*2^(1/2)*arctanh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2) )*a^6*sin(d*x+c)^3-40*a^(9/2)*(-a*(sin(d*x+c)-1))^(3/2)+21*a^(11/2)*(-a*(s in(d*x+c)-1))^(1/2))/sin(d*x+c)^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. 564 vs. \(2 (162) = 324\).
Time = 0.11 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.95 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm="f ricas")
Output:
1/96*(69*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 - (cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sin(d*x + c) + 1)*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)) + 96*sqrt(2)*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 - (a*cos(d*x + c)^ 3 + a*cos(d*x + c)^2 - a*cos(d*x + c) - a)*sin(d*x + c) + a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos( d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) + 4*(27*cos(d*x + c)^3 + 41*cos(d*x + c)^2 - (27*cos(d*x + c)^2 - 14*cos(d *x + c) - 49)*sin(d*x + c) - 35*cos(d*x + c) - 49)*sqrt(a*sin(d*x + c) + a ))/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d - (a^2*d*cos(d*x + c)^3 + a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c))
\[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**2/(a+a*sin(d*x+c))**(3/2),x)
Output:
Integral(cot(c + d*x)**2*csc(c + d*x)**2/(a*(sin(c + d*x) + 1))**(3/2), x)
Timed out. \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm="m axima")
Output:
Timed out
Time = 0.16 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.26 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {2} \sqrt {a} {\left (\frac {69 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {96 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {96 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4 \, {\left (108 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 80 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{96 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm="g iac")
Output:
1/96*sqrt(2)*sqrt(a)*(69*sqrt(2)*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)))/(a^2*sgn( cos(-1/4*pi + 1/2*d*x + 1/2*c))) + 96*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 96*log(-sin(-1/4*pi + 1/2* d*x + 1/2*c) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 4*(108*sin(- 1/4*pi + 1/2*d*x + 1/2*c)^5 - 80*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 + 21*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)^3*a ^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int -\frac {{\sin \left (c+d\,x\right )}^2-1}{{\sin \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int(cot(c + d*x)^2/(sin(c + d*x)^2*(a + a*sin(c + d*x))^(3/2)),x)
Output:
int(-(sin(c + d*x)^2 - 1)/(sin(c + d*x)^4*(a + a*sin(c + d*x))^(3/2)), x)
\[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}}{\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x)
Output:
(sqrt(a)*int((sqrt(sin(c + d*x) + 1)*cot(c + d*x)**2*csc(c + d*x)**2)/(sin (c + d*x)**2 + 2*sin(c + d*x) + 1),x))/a**2