\(\int \frac {\csc ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\) [67]

Optimal result

Integrand size = 23, antiderivative size = 146 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {7 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 \sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {\cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a+a \sin (c+d x)}} \] Output:

-7/4*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(1/2)/d+2^(1/2)* 
arctanh(1/2*a^(1/2)*cos(d*x+c)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))/a^(1/2)/d+1 
/4*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-1/2*cot(d*x+c)*csc(d*x+c)/d/(a+a*si 
n(d*x+c))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.51 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.10 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (-8-(64+64 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right )+4 \cot \left (\frac {1}{4} (c+d x)\right )-\csc ^2\left (\frac {1}{4} (c+d x)\right )-28 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+28 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{4} (c+d x)\right )+\frac {2}{\left (\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}-\frac {8 \sin \left (\frac {1}{4} (c+d x)\right )}{\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )}-\frac {2}{\left (\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}+\frac {8 \sin \left (\frac {1}{4} (c+d x)\right )}{\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )}+4 \tan \left (\frac {1}{4} (c+d x)\right )\right )}{32 d \sqrt {a (1+\sin (c+d x))}} \] Input:

Integrate[Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-8 - (64 + 64*I)*(-1)^(3/4)*ArcTan 
h[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 4*Cot[(c + d*x)/4] - C 
sc[(c + d*x)/4]^2 - 28*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 28*L 
og[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2/(Cos[ 
(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (8*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4 
] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (8*Sin 
[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 4*Tan[(c + d*x)/4]) 
)/(32*d*Sqrt[a*(1 + Sin[c + d*x])])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3258, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3463

\(\displaystyle -\frac {\frac {\int -\frac {\csc (c+d x) \left (7 a^2-a^2 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\csc (c+d x) \left (7 a^2-a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3464

\(\displaystyle -\frac {-\frac {7 a \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-8 a^2 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {7 a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-8 a^2 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3128

\(\displaystyle -\frac {-\frac {\frac {16 a^2 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+7 a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 3252

\(\displaystyle -\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {14 a^2 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\frac {8 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {14 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\)

Input:

Int[Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

-1/2*(Cot[c + d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (-1/2*((-1 
4*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (8 
*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c 
+ d*x]])])/d)/a - (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/(4*a)
 

Defintions of rubi rules used

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 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 3258

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_. 
) + (f_.)*(x_)]], x_Symbol] :> Simp[(-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] - Simp[1/(2* 
b*(n + 1)*(c^2 - d^2))   Int[(c + d*Sin[e + f*x])^(n + 1)*(Simp[a*d - 2*b*c 
*(n + 1) + b*d*(2*n + 3)*Sin[e + f*x], x]/Sqrt[a + b*Sin[e + f*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] && LtQ[n, -1] && IntegerQ[2*n]
 

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 [A] (verified)

Time = 0.34 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-7 a^{5} \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \sin \left (d x +c \right )^{2}-\left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {7}{2}}+4 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{5} \sin \left (d x +c \right )^{2}-\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {9}{2}}\right )}{4 a^{\frac {11}{2}} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(164\)

Input:

int(csc(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(11/2)*(-7*a^5*arctanh((-a* 
(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+c)^2-(-a*(sin(d*x+c)-1))^(3/2)*a^(7 
/2)+4*2^(1/2)*arctanh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2))*a^5*s 
in(d*x+c)^2-(-a*(sin(d*x+c)-1))^(1/2)*a^(9/2))/sin(d*x+c)^2/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. 492 vs. \(2 (121) = 242\).

Time = 0.11 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.37 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {7 \, {\left (\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 )} \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 ) + \frac {8 \, \sqrt {2} {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - 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}}{16 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d + {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )\right )}} \] Input:

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

1/16*(7*(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)*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)) + 8*sqrt(2)*(a*c 
os(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) + (a*cos(d*x + c)^2 - 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)/sqr 
t(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*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*s 
in(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a))/(a*d*cos(d*x + 
 c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d + (a*d*cos(d*x + c)^2 
- a*d)*sin(d*x + c))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.52 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\frac {4 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {7 \, \log \left (\frac {{\left | -16 \, \sqrt {2} - 32 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 16 \, \sqrt {2} - 32 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{8 \, d} \] Input:

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

-1/8*(4*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(sqrt(a)*sgn(cos(- 
1/4*pi + 1/2*d*x + 1/2*c))) - 4*sqrt(2)*log(-sin(-1/4*pi + 1/2*d*x + 1/2*c 
) + 1)/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 7*log(abs(-16*sqrt( 
2) - 32*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(16*sqrt(2) - 32*sin(-1/4*pi + 
1/2*d*x + 1/2*c)))/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 2*sqrt( 
2)*(2*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 + 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)^2*sqrt(a)*sgn(cos(-1/4*pi + 1/2*d 
*x + 1/2*c))))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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