Integrand size = 23, antiderivative size = 182 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {163 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {163 a^3 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a+a \sin (c+d x)}}-\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d} \] Output:
-163/64*a^(5/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/d-163/6 4*a^3*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-163/96*a^3*cot(d*x+c)*csc(d*x+c) /d/(a+a*sin(d*x+c))^(1/2)-17/24*a^3*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x +c))^(1/2)-1/4*a^2*cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(370\) vs. \(2(182)=364\).
Time = 6.55 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.03 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {a^2 \csc ^{13}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-1030 \cos \left (\frac {1}{2} (c+d x)\right )+3102 \cos \left (\frac {3}{2} (c+d x)\right )-326 \cos \left (\frac {5}{2} (c+d x)\right )-978 \cos \left (\frac {7}{2} (c+d x)\right )+1467 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-1956 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+489 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-1467 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+1956 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-489 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+1030 \sin \left (\frac {1}{2} (c+d x)\right )+3102 \sin \left (\frac {3}{2} (c+d x)\right )+326 \sin \left (\frac {5}{2} (c+d x)\right )-978 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{192 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^4} \] Input:
Integrate[Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]
Output:
-1/192*(a^2*Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(-1030*Cos[(c + d*x)/2] + 3102*Cos[(3*(c + d*x))/2] - 326*Cos[(5*(c + d*x))/2] - 978*Cos[ (7*(c + d*x))/2] + 1467*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 195 6*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 489*Cos[ 4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 1467*Log[1 - C os[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1956*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 489*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x )/2] + Sin[(c + d*x)/2]] + 1030*Sin[(c + d*x)/2] + 3102*Sin[(3*(c + d*x))/ 2] + 326*Sin[(5*(c + d*x))/2] - 978*Sin[(7*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4)
Time = 0.94 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3241, 27, 3042, 3459, 3042, 3251, 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.
| \(\displaystyle \int \csc ^5(c+d x) (a \sin (c+d x)+a)^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^{5/2}}{\sin (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3241 |
| \(\displaystyle -\frac {1}{4} a \int -\frac {1}{2} \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a} (13 \sin (c+d x) a+17 a)dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} a \int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a} (13 \sin (c+d x) a+17 a)dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} a \int \frac {\sqrt {\sin (c+d x) a+a} (13 \sin (c+d x) a+17 a)}{\sin (c+d x)^4}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{8} a \left (\frac {163}{6} a \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {17 a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}\) |
Input:
Int[Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]
Output:
-1/4*(a^2*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/d + (a*((- 17*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (163* a*(-1/2*(a*Cot[c + d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (3*(- ((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))/8
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b *Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a* d))), x] + Simp[b^2/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b* c*(m - 1) - a*d*(m + 2*n + 1))*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[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[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 = 72.50 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (1047 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {11}{2}}-2303 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {9}{2}}+1793 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {7}{2}}-489 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {5}{2}}+489 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{6} \sin \left (d x +c \right )^{4}\right )}{192 a^{\frac {7}{2}} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(162\) |
Input:
int(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/192*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(1047*(-a*(sin(d*x+c)-1))^ (1/2)*a^(11/2)-2303*(-a*(sin(d*x+c)-1))^(3/2)*a^(9/2)+1793*(-a*(sin(d*x+c) -1))^(5/2)*a^(7/2)-489*(-a*(sin(d*x+c)-1))^(7/2)*a^(5/2)+489*arctanh((-a*( sin(d*x+c)-1))^(1/2)/a^(1/2))*a^6*sin(d*x+c)^4)/a^(7/2)/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. 473 vs. \(2 (158) = 316\).
Time = 0.10 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.60 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {489 \, {\left (a^{2} \cos \left (d x + c\right )^{5} + a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) + a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (489 \, a^{2} \cos \left (d x + c\right )^{4} + 326 \, a^{2} \cos \left (d x + c\right )^{3} - 836 \, a^{2} \cos \left (d x + c\right )^{2} - 374 \, a^{2} \cos \left (d x + c\right ) + 299 \, a^{2} + {\left (489 \, a^{2} \cos \left (d x + c\right )^{3} + 163 \, a^{2} \cos \left (d x + c\right )^{2} - 673 \, a^{2} \cos \left (d x + c\right ) - 299 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{768 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right ) + d\right )}} \] Input:
integrate(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")
Output:
1/768*(489*(a^2*cos(d*x + c)^5 + a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c) + a^2 + (a^2*cos(d*x + c)^4 - 2 *a^2*cos(d*x + c)^2 + a^2)*sin(d*x + c))*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*(489*a^2*cos(d*x + c)^4 + 326*a^2*cos(d*x + c)^3 - 836*a^2*cos(d*x + c)^2 - 374*a^2*cos(d*x + c) + 299*a^2 + (489*a^2*cos(d*x + c)^3 + 163*a^2 *cos(d*x + c)^2 - 673*a^2*cos(d*x + c) - 299*a^2)*sin(d*x + c))*sqrt(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*cos(d*x + c)^2 + d)*sin(d*x + c) + d)
Timed out. \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**5*(a+a*sin(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \csc \left (d x + c\right )^{5} \,d x } \] Input:
integrate(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((a*sin(d*x + c) + a)^(5/2)*csc(d*x + c)^5, x)
Time = 0.17 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.25 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (489 \, \sqrt {2} a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (3912 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7172 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4606 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1047 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}\right )} \sqrt {a}}{768 \, d} \] Input:
integrate(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")
Output:
-1/768*sqrt(2)*(489*sqrt(2)*a^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*(3912*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c) )*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 7172*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 4606*a^2*sgn(cos(-1/4*pi + 1/2* d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 1047*a^2*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. \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\sin \left (c+d\,x\right )}^5} \,d x \] Input:
int((a + a*sin(c + d*x))^(5/2)/sin(c + d*x)^5,x)
Output:
int((a + a*sin(c + d*x))^(5/2)/sin(c + d*x)^5, x)
\[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )^{2}d x +2 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )d x \right )+\int \sqrt {\sin \left (d x +c \right )+1}\, \csc \left (d x +c \right )^{5}d x \right ) \] Input:
int(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x)
Output:
sqrt(a)*a**2*(int(sqrt(sin(c + d*x) + 1)*csc(c + d*x)**5*sin(c + d*x)**2,x ) + 2*int(sqrt(sin(c + d*x) + 1)*csc(c + d*x)**5*sin(c + d*x),x) + int(sqr t(sin(c + d*x) + 1)*csc(c + d*x)**5,x))