Integrand size = 31, antiderivative size = 137 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {3 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}+\frac {3 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \] Output:
3/8*a^(1/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/d+3/8*a*cot (d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-1/12*a*cot(d*x+c)*csc(d*x+c)/d/(a+a*sin(d *x+c))^(1/2)-1/3*cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(285\) vs. \(2(137)=274\).
Time = 1.53 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.08 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\csc ^{10}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-12 \cos \left (\frac {1}{2} (c+d x)\right )+58 \cos \left (\frac {3}{2} (c+d x)\right )+18 \cos \left (\frac {5}{2} (c+d x)\right )+12 \sin \left (\frac {1}{2} (c+d x)\right )-27 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+27 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+58 \sin \left (\frac {3}{2} (c+d x)\right )-18 \sin \left (\frac {5}{2} (c+d x)\right )+9 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-9 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))\right )}{24 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3} \] Input:
Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]
Output:
-1/24*(Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(-12*Cos[(c + d*x)/2 ] + 58*Cos[(3*(c + d*x))/2] + 18*Cos[(5*(c + d*x))/2] + 12*Sin[(c + d*x)/2 ] - 27*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 27*Log[ 1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 58*Sin[(3*(c + d*x ))/2] - 18*Sin[(5*(c + d*x))/2] + 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d* x)/2]]*Sin[3*(c + d*x)] - 9*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*S in[3*(c + d*x)]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)
Time = 1.02 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {3042, 3353, 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.
\(\displaystyle \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^2 \sqrt {a \sin (c+d x)+a}}{\sin (c+d x)^4}dx\) |
\(\Big \downarrow \) 3353 |
\(\displaystyle \frac {\int \csc ^4(c+d x) (a-a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(a-a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)^4}dx}{a^2}\) |
\(\Big \downarrow \) 3454 |
\(\displaystyle \frac {\frac {1}{3} \int \frac {1}{2} \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a} \left (a^2-3 a^2 \sin (c+d x)\right )dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a} \left (a^2-3 a^2 \sin (c+d x)\right )dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{6} \int \frac {\sqrt {\sin (c+d x) a+a} \left (a^2-3 a^2 \sin (c+d x)\right )}{\sin (c+d x)^3}dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 3459 |
\(\displaystyle \frac {\frac {1}{6} \left (-\frac {9}{4} a^2 \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{6} \left (-\frac {9}{4} a^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 3251 |
\(\displaystyle \frac {\frac {1}{6} \left (-\frac {9}{4} a^2 \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{6} \left (-\frac {9}{4} a^2 \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 3252 |
\(\displaystyle \frac {\frac {1}{6} \left (-\frac {9}{4} a^2 \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{6} \left (-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {9}{4} a^2 \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}\) |
Input:
Int[Cot[c + d*x]^2*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]
Output:
(-1/3*(a^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/d + (-1/2 *(a^3*Cot[c + d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (9*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*Sin[c + d*x]])))/4)/6)/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[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_)]^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^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]
Time = 0.41 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (9 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+9 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \sin \left (d x +c \right )^{3}-8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-9 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}\right )}{24 a^{\frac {7}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(144\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBO SE)
Output:
1/24*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(7/2)*(9*(-a*(sin(d*x+c)-1 ))^(5/2)*a^(3/2)+9*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^4*sin(d*x+ c)^3-8*(-a*(sin(d*x+c)-1))^(3/2)*a^(5/2)-9*(-a*(sin(d*x+c)-1))^(1/2)*a^(7/ 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. 361 vs. \(2 (117) = 234\).
Time = 0.10 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.64 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (9 \, \cos \left (d x + c\right )^{3} + 19 \, \cos \left (d x + c\right )^{2} - {\left (9 \, \cos \left (d x + c\right )^{2} - 10 \, \cos \left (d x + c\right ) - 11\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 11\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="f ricas")
Output:
1/96*(9*(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 )) - 4*(9*cos(d*x + c)^3 + 19*cos(d*x + c)^2 - (9*cos(d*x + c)^2 - 10*cos( d*x + c) - 11)*sin(d*x + c) - cos(d*x + c) - 11)*sqrt(a*sin(d*x + c) + a)) /(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 - (d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)
\[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**2*(a+a*sin(d*x+c))**(1/2),x)
Output:
Integral(sqrt(a*(sin(c + d*x) + 1))*cot(c + d*x)**2*csc(c + d*x)**2, x)
\[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cot \left (d x + c\right )^{2} \csc \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="m axima")
Output:
integrate(sqrt(a*sin(d*x + c) + a)*cot(d*x + c)^2*csc(d*x + c)^2, x)
Time = 0.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.34 \[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (9 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (36 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{96 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="g iac")
Output:
1/96*sqrt(2)*(9*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)))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) + 4*(36*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 16*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*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)^3)*sqrt(a)/d
Timed out. \[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int -\frac {\left ({\sin \left (c+d\,x\right )}^2-1\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^4} \,d x \] Input:
int((cot(c + d*x)^2*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^2,x)
Output:
int(-((sin(c + d*x)^2 - 1)*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^4, x)
\[ \int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}d x \right ) \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**2*csc(c + d*x)**2,x)