\(\int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\) [458]

Optimal result

Integrand size = 23, antiderivative size = 197 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {37 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {29 a^2 \cot (c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {a \cot (c+d x) \csc (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d} \] Output:

37/8*a^(3/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/d-8/3*a^2* 
cos(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+29/24*a^2*cot(d*x+c)/d/(a+a*sin(d*x+c) 
)^(1/2)-2/3*a*cos(d*x+c)*(a+a*sin(d*x+c))^(1/2)/d-1/4*a*cot(d*x+c)*csc(d*x 
+c)*(a+a*sin(d*x+c))^(1/2)/d-1/3*cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^ 
(3/2)/d
 

Mathematica [A] (verified)

Time = 2.09 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.70 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-276 \cos \left (\frac {1}{2} (c+d x)\right )+326 \cos \left (\frac {3}{2} (c+d x)\right )+78 \cos \left (\frac {5}{2} (c+d x)\right )-72 \cos \left (\frac {7}{2} (c+d x)\right )+8 \cos \left (\frac {9}{2} (c+d x)\right )+276 \sin \left (\frac {1}{2} (c+d x)\right )-333 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+333 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+326 \sin \left (\frac {3}{2} (c+d x)\right )-78 \sin \left (\frac {5}{2} (c+d x)\right )+111 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-111 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-72 \sin \left (\frac {7}{2} (c+d x)\right )-8 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{24 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3} \] Input:

Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]
 

Output:

-1/24*(a*Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(-276*Cos[(c + d*x 
)/2] + 326*Cos[(3*(c + d*x))/2] + 78*Cos[(5*(c + d*x))/2] - 72*Cos[(7*(c + 
 d*x))/2] + 8*Cos[(9*(c + d*x))/2] + 276*Sin[(c + d*x)/2] - 333*Log[1 + Co 
s[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 333*Log[1 - Cos[(c + d*x 
)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 326*Sin[(3*(c + d*x))/2] - 78*Sin[ 
(5*(c + d*x))/2] + 111*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3* 
(c + d*x)] - 111*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d 
*x)] - 72*Sin[(7*(c + d*x))/2] - 8*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + 
 d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 3197, 3042, 3126, 3042, 3125, 3523, 27, 3042, 3454, 27, 3042, 3459, 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.

\(\displaystyle \int \cot ^4(c+d x) (a \sin (c+d x)+a)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (c+d x)+a)^{3/2}}{\tan (c+d x)^4}dx\)

\(\Big \downarrow \) 3197

\(\displaystyle \int (\sin (c+d x) a+a)^{3/2}dx+\int \csc ^4(c+d x) (\sin (c+d x) a+a)^{3/2} \left (1-2 \sin ^2(c+d x)\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (\sin (c+d x) a+a)^{3/2}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^4}dx\)

\(\Big \downarrow \) 3126

\(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^4}dx+\frac {4}{3} a \int \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{3} a \int \sqrt {\sin (c+d x) a+a}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^4}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\)

\(\Big \downarrow \) 3125

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

\(\Big \downarrow \) 3523

\(\displaystyle \frac {\int \frac {1}{2} \csc ^3(c+d x) (3 a-11 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{3 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \csc ^3(c+d x) (3 a-11 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(3 a-11 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)^3}dx}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {1}{2} \int -\frac {1}{2} \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a} \left (41 \sin (c+d x) a^2+29 a^2\right )dx-\frac {3 a^2 \cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a} \left (41 \sin (c+d x) a^2+29 a^2\right )dx-\frac {3 a^2 \cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {1}{4} \int \frac {\sqrt {\sin (c+d x) a+a} \left (41 \sin (c+d x) a^2+29 a^2\right )}{\sin (c+d x)^2}dx-\frac {3 a^2 \cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 3459

\(\displaystyle \frac {\frac {1}{4} \left (\frac {29 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {111}{2} a^2 \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx\right )-\frac {3 a^2 \cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{4} \left (\frac {29 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {111}{2} a^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx\right )-\frac {3 a^2 \cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {1}{4} \left (\frac {111 a^3 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\frac {29 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {3 a^2 \cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}}{6 a}-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}+\frac {\frac {1}{4} \left (\frac {111 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}+\frac {29 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {3 a^2 \cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}}{6 a}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}\)

Input:

Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]
 

Output:

(-8*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*S 
qrt[a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c 
 + d*x])^(3/2))/(3*d) + ((-3*a^2*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[ 
c + d*x]])/(2*d) + ((111*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a 
*Sin[c + d*x]]])/d + (29*a^3*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/4 
)/(6*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 3125

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos 
[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[a^2 - b^2, 0]
 

rule 3126

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos 
[c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) 
 Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ 
a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
 

rule 3197

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, 
x_Symbol] :> Int[(a + b*Sin[e + f*x])^m, x] + Int[(a + b*Sin[e + f*x])^m*(( 
1 - 2*Sin[e + f*x]^2)/Sin[e + f*x]^4), x] /; FreeQ[{a, b, e, f, m}, x] && E 
qQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] &&  !LtQ[m, -1]
 

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 3454

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 
])
 

rule 3459

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]
 

rule 3523

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + 
 f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - 
d^2))   Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a 
*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* 
(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, 
 e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - 
d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (16 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} \sin \left (d x +c \right )^{3}-96 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {5}{2}} \sin \left (d x +c \right )^{3}+111 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{3} \sin \left (d x +c \right )^{3}+15 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} \sqrt {a}-8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-15 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {5}{2}}\right )}{24 a^{\frac {3}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(196\)

Input:

int(cot(d*x+c)^4*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/24*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(3/2)*(16*(-a*(sin(d*x+c)- 
1))^(3/2)*a^(3/2)*sin(d*x+c)^3-96*(-a*(sin(d*x+c)-1))^(1/2)*a^(5/2)*sin(d* 
x+c)^3+111*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^3*sin(d*x+c)^3+15* 
(-a*(sin(d*x+c)-1))^(5/2)*a^(1/2)-8*(-a*(sin(d*x+c)-1))^(3/2)*a^(3/2)-15*( 
-a*(sin(d*x+c)-1))^(1/2)*a^(5/2))/sin(d*x+c)^3/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. 424 vs. \(2 (169) = 338\).

Time = 0.10 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.15 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {111 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (16 \, a \cos \left (d x + c\right )^{5} - 64 \, a \cos \left (d x + c\right )^{4} - 17 \, a \cos \left (d x + c\right )^{3} + 165 \, a \cos \left (d x + c\right )^{2} + 9 \, a \cos \left (d x + c\right ) - {\left (16 \, a \cos \left (d x + c\right )^{4} + 80 \, a \cos \left (d x + c\right )^{3} + 63 \, a \cos \left (d x + c\right )^{2} - 102 \, a \cos \left (d x + c\right ) - 93 \, a\right )} \sin \left (d x + c\right ) - 93 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \] Input:

integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

1/96*(111*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 - (a*cos(d*x + c)^3 + a*c 
os(d*x + c)^2 - a*cos(d*x + c) - a)*sin(d*x + c) + a)*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)*si 
n(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*co 
s(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*(16*a*cos(d*x + c)^5 - 64*a*cos(d*x + c)^4 - 17*a*cos(d 
*x + c)^3 + 165*a*cos(d*x + c)^2 + 9*a*cos(d*x + c) - (16*a*cos(d*x + c)^4 
 + 80*a*cos(d*x + c)^3 + 63*a*cos(d*x + c)^2 - 102*a*cos(d*x + c) - 93*a)* 
sin(d*x + c) - 93*a)*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)*s 
in(d*x + c) + d)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**4*(a+a*sin(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate((a*sin(d*x + c) + a)^(3/2)*cot(d*x + c)^4, x)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.25 \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (128 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 111 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 384 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {4 \, {\left (60 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{96 \, d} \] Input:

integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

-1/96*sqrt(2)*(128*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2 
*d*x + 1/2*c)^3 - 111*sqrt(2)*a*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)) - 384*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin( 
-1/4*pi + 1/2*d*x + 1/2*c) - 4*(60*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*s 
in(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 16*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) 
*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 15*a*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
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:

int(cot(c + d*x)^4*(a + a*sin(c + d*x))^(3/2),x)
 

Output:

int(cot(c + d*x)^4*(a + a*sin(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \sin \left (d x +c \right )d x +\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4}d x \right ) \] Input:

int(cot(d*x+c)^4*(a+a*sin(d*x+c))^(3/2),x)
 

Output:

sqrt(a)*a*(int(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**4*sin(c + d*x),x) + in 
t(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**4,x))