\(\int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\) [84]

Optimal result

Integrand size = 23, antiderivative size = 224 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {39 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 a^{5/2} d}+\frac {219 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {63 \cot (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}} \] Output:

-39/4*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(5/2)/d+219/32* 
arctanh(1/2*a^(1/2)*cos(d*x+c)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))*2^(1/2)/a^( 
5/2)/d+1/4*cot(d*x+c)*csc(d*x+c)/d/(a+a*sin(d*x+c))^(5/2)+19/16*cot(d*x+c) 
*csc(d*x+c)/a/d/(a+a*sin(d*x+c))^(3/2)+63/16*cot(d*x+c)/a^2/d/(a+a*sin(d*x 
+c))^(1/2)-31/16*cot(d*x+c)*csc(d*x+c)/a^2/d/(a+a*sin(d*x+c))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.36 (sec) , antiderivative size = 680, normalized size of antiderivative = 3.04 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]
 

Output:

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-16*Sin[(c + d*x)/2] + 8*(Cos[(c + 
 d*x)/2] + Sin[(c + d*x)/2]) - 108*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Si 
n[(c + d*x)/2])^2 + 54*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 40*(Cos[( 
c + d*x)/2] + Sin[(c + d*x)/2])^4 - (438 + 438*I)*(-1)^(3/4)*ArcTanh[(1/2 
+ I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d* 
x)/2])^4 + 20*Cot[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - C 
sc[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 156*Log[1 + Co 
s[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 
 + 156*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Si 
n[(c + d*x)/2])^4 + Sec[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2 
])^4 + (2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] - Sin 
[(c + d*x)/4])^2 - (40*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/ 
2])^4)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - (2*(Cos[(c + d*x)/2] + Sin[ 
(c + d*x)/2])^4)/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (40*Sin[(c + d* 
x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] + Sin[(c 
+ d*x)/4]) + 20*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Tan[(c + d*x)/4])) 
/(32*d*(a*(1 + Sin[c + d*x]))^(5/2))
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.08, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 3245, 27, 3042, 3457, 27, 3042, 3463, 27, 3042, 3463, 25, 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.

\(\displaystyle \int \frac {\csc ^3(c+d x)}{(a \sin (c+d x)+a)^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (c+d x)^3 (a \sin (c+d x)+a)^{5/2}}dx\)

\(\Big \downarrow \) 3245

\(\displaystyle \frac {\int \frac {\csc ^3(c+d x) (12 a-7 a \sin (c+d x))}{2 (\sin (c+d x) a+a)^{3/2}}dx}{4 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\csc ^3(c+d x) (12 a-7 a \sin (c+d x))}{(\sin (c+d x) a+a)^{3/2}}dx}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {12 a-7 a \sin (c+d x)}{\sin (c+d x)^3 (\sin (c+d x) a+a)^{3/2}}dx}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {\int \frac {\csc ^3(c+d x) \left (124 a^2-95 a^2 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{2 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\csc ^3(c+d x) \left (124 a^2-95 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {124 a^2-95 a^2 \sin (c+d x)}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3463

\(\displaystyle \frac {\frac {\frac {\int -\frac {6 \csc ^2(c+d x) \left (42 a^3-31 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {3 \int \frac {\csc ^2(c+d x) \left (42 a^3-31 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {3 \int \frac {42 a^3-31 a^3 \sin (c+d x)}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3463

\(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\int -\frac {\csc (c+d x) \left (52 a^4-21 a^4 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {\int \frac {\csc (c+d x) \left (52 a^4-21 a^4 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {\int \frac {52 a^4-21 a^4 \sin (c+d x)}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3464

\(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {52 a^3 \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-73 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {52 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-73 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3128

\(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {52 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {146 a^4 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {52 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {73 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {\frac {73 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {104 a^4 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {-\frac {62 a^2 \cot (c+d x) \csc (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \left (-\frac {\frac {73 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {104 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{a}-\frac {42 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a}}{4 a^2}+\frac {19 a \cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}}{8 a^2}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}\)

Input:

Int[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]
 

Output:

(Cot[c + d*x]*Csc[c + d*x])/(4*d*(a + a*Sin[c + d*x])^(5/2)) + ((19*a*Cot[ 
c + d*x]*Csc[c + d*x])/(2*d*(a + a*Sin[c + d*x])^(3/2)) + ((-62*a^2*Cot[c 
+ d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (3*(-(((-104*a^(7/2)*A 
rcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (73*Sqrt[2]*a 
^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]) 
/d)/a) - (42*a^3*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])))/a)/(4*a^2))/ 
(8*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3128

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]
 

rule 3245

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*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( 
a*(2*m + 1)*(b*c - a*d))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + 
f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + 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] && LtQ[m, -1] &&  !GtQ[n, 0] && (Intege 
rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
 

rule 3252

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]
 

rule 3457

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*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( 
n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b 
*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 
2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ 
[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] 
 &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3463

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])
 

rule 3464

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(189)=378\).

Time = 0.38 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.80

method	result	size
default	\(\frac {\left (219 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}\, \sin \left (d x +c \right )^{4} a^{2}+438 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right )^{3} a^{2}-312 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \sin \left (d x +c \right )^{4} a^{2}+172 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}} \sin \left (d x +c \right )^{2}-126 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}\, \sin \left (d x +c \right )^{2}+219 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right )^{2} a^{2}-624 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \sin \left (d x +c \right )^{3} a^{2}+112 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}} \sin \left (d x +c \right )-144 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}\, \sin \left (d x +c \right )-312 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \sin \left (d x +c \right )^{2} a^{2}+56 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}}-72 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{32 a^{\frac {9}{2}} \sin \left (d x +c \right )^{2} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(404\)

Input:

int(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/32*(219*arctanh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*s 
in(d*x+c)^4*a^2+438*2^(1/2)*arctanh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2)/ 
a^(1/2))*sin(d*x+c)^3*a^2-312*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*s 
in(d*x+c)^4*a^2+172*(-a*(sin(d*x+c)-1))^(1/2)*a^(3/2)*sin(d*x+c)^2-126*(-a 
*(sin(d*x+c)-1))^(3/2)*a^(1/2)*sin(d*x+c)^2+219*2^(1/2)*arctanh(1/2*(-a*(s 
in(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(d*x+c)^2*a^2-624*arctanh((-a*(sin 
(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+c)^3*a^2+112*(-a*(sin(d*x+c)-1))^(1/2)* 
a^(3/2)*sin(d*x+c)-144*(-a*(sin(d*x+c)-1))^(3/2)*a^(1/2)*sin(d*x+c)-312*ar 
ctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+c)^2*a^2+56*(-a*(sin(d*x+ 
c)-1))^(1/2)*a^(3/2)-72*(-a*(sin(d*x+c)-1))^(3/2)*a^(1/2))*(-a*(sin(d*x+c) 
-1))^(1/2)/a^(9/2)/sin(d*x+c)^2/(1+sin(d*x+c))/cos(d*x+c)/(a+a*sin(d*x+c)) 
^(1/2)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (189) = 378\).

Time = 0.12 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.19 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

1/64*(219*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 - 3*cos(d*x + c)^3 - 
7*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 5*cos(d*x + c)^2 + 
 2*cos(d*x + c) + 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*sqrt(a)*log(-(a*co 
s(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - 
sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) 
 + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 
 2)) + 156*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 - 3*cos(d*x + c)^3 - 7*cos(d 
*x + c)^2 + (cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 5*cos(d*x + c)^2 + 2*cos( 
d*x + c) + 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*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*(63*cos(d*x + c)^4 + 95*cos(d*x + c)^3 - 51*cos(d*x + c)^2 + 
 (63*cos(d*x + c)^3 - 32*cos(d*x + c)^2 - 83*cos(d*x + c) + 4)*sin(d*x + c 
) - 87*cos(d*x + c) - 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^5 + 
 3*a^3*d*cos(d*x + c)^4 - 3*a^3*d*cos(d*x + c)^3 - 7*a^3*d*cos(d*x + c)^2 
+ 2*a^3*d*cos(d*x + c) + 4*a^3*d + (a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x 
 + c)^3 - 5*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d)*sin(d*x 
 + c))
 

Sympy [F]

\[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(csc(d*x+c)**3/(a+a*sin(d*x+c))**(5/2),x)
 

Output:

Integral(csc(c + d*x)**3/(a*(sin(c + d*x) + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate(csc(d*x + c)^3/(a*sin(d*x + c) + a)^(5/2), x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.33 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\frac {219 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {219 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {312 \, \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {312 \, \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (252 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 568 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 399 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 85 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{64 \, d} \] Input:

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

-1/64*(219*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(5/2)*sgn(co 
s(-1/4*pi + 1/2*d*x + 1/2*c))) - 219*sqrt(2)*log(-sin(-1/4*pi + 1/2*d*x + 
1/2*c) + 1)/(a^(5/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 312*log(abs(1/ 
2*sqrt(2) + sin(-1/4*pi + 1/2*d*x + 1/2*c)))/(a^(5/2)*sgn(cos(-1/4*pi + 1/ 
2*d*x + 1/2*c))) + 312*log(abs(-1/2*sqrt(2) + sin(-1/4*pi + 1/2*d*x + 1/2* 
c)))/(a^(5/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 2*sqrt(2)*(252*sqrt(a 
)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 568*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1 
/2*c)^5 + 399*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 85*sqrt(a)*sin(-1 
/4*pi + 1/2*d*x + 1/2*c))/((2*sin(-1/4*pi + 1/2*d*x + 1/2*c)^4 - 3*sin(-1/ 
4*pi + 1/2*d*x + 1/2*c)^2 + 1)^2*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))) 
/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:

int(1/(sin(c + d*x)^3*(a + a*sin(c + d*x))^(5/2)),x)
 

Output:

int(1/(sin(c + d*x)^3*(a + a*sin(c + d*x))^(5/2)), x)
 

Reduce [F]

\[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \csc \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )+1}d x \right )}{a^{3}} \] Input:

int(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x)
 

Output:

(sqrt(a)*int((sqrt(sin(c + d*x) + 1)*csc(c + d*x)**3)/(sin(c + d*x)**3 + 3 
*sin(c + d*x)**2 + 3*sin(c + d*x) + 1),x))/a**3