\(\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx\) [459]

Optimal result

Integrand size = 29, antiderivative size = 205 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {21 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {149 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d} \]

[Out]

21/64*a^(3/2)*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d-1/4*cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c)
)^(3/2)/d-2*a^2*cos(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+149/64*a^2*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+19/32*a^2*c
ot(d*x+c)*csc(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-1/8*a*cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2960, 2842, 21, 2852, 212, 3123, 3054, 3059, 2851} \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {21 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{64 d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}+\frac {149 a^2 \cot (c+d x)}{64 d \sqrt {a \sin (c+d x)+a}}+\frac {19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{8 d} \]

[In]

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

[Out]

(21*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) - (2*a^2*Cos[c + d*x])/(d*Sqrt[a
+ a*Sin[c + d*x]]) + (149*a^2*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (19*a^2*Cot[c + d*x]*Csc[c + d*x
])/(32*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(8*d) - (Cot[c +
d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(4*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/(
d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d
*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*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] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m,
2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2851

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]
+ Dist[(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]

Rule 2852

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[-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 2960

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Dist[1/d^4, Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^
n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&
!IGtQ[m, 0]

Rule 3054

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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]*(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] - Dist[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 3059

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] + Dist[(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 3123

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(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*Si
n[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[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])

Rubi steps \begin{align*} \text {integral}& = \int \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+2 \int \frac {\csc (c+d x) \left (\frac {a^2}{2}+\frac {1}{2} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx+\frac {\int \csc ^4(c+d x) \left (\frac {3 a}{2}-\frac {13}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{4 a} \\ & = -\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+\frac {\int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {57 a^2}{4}-\frac {69}{4} a^2 \sin (c+d x)\right ) \, dx}{12 a}+a \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}-\frac {1}{64} (149 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d} \\ & = -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {149 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}-\frac {1}{128} (149 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {149 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+\frac {\left (149 a^2\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d} \\ & = \frac {21 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {2 a^2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {149 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.64 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.91 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \csc ^{13}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (1486 \cos \left (\frac {1}{2} (c+d x)\right )-1030 \cos \left (\frac {3}{2} (c+d x)\right )-754 \cos \left (\frac {5}{2} (c+d x)\right )+426 \cos \left (\frac {7}{2} (c+d x)\right )+128 \cos \left (\frac {9}{2} (c+d x)\right )-63 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+84 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-21 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+63 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-84 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+21 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-1486 \sin \left (\frac {1}{2} (c+d x)\right )-1030 \sin \left (\frac {3}{2} (c+d x)\right )+754 \sin \left (\frac {5}{2} (c+d x)\right )+426 \sin \left (\frac {7}{2} (c+d x)\right )-128 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{64 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^4} \]

[In]

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

[Out]

-1/64*(a*Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(1486*Cos[(c + d*x)/2] - 1030*Cos[(3*(c + d*x))/2] - 7
54*Cos[(5*(c + d*x))/2] + 426*Cos[(7*(c + d*x))/2] + 128*Cos[(9*(c + d*x))/2] - 63*Log[1 + Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]] + 84*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 21*Cos[4*(c + d*x)]*Log
[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 63*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 84*Cos[2*(c + d*
x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 21*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x
)/2]] - 1486*Sin[(c + d*x)/2] - 1030*Sin[(3*(c + d*x))/2] + 754*Sin[(5*(c + d*x))/2] + 426*Sin[(7*(c + d*x))/2
] - 128*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)^4)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-128 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sin ^{4}\left (d x +c \right )\right ) a^{\frac {7}{2}}+21 \left (\sin ^{4}\left (d x +c \right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4}-149 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} \sqrt {a}+461 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}-435 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}+107 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}\right )}{64 a^{\frac {5}{2}} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(188\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/64*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(-128*(-a*(sin(d*x+c)-1))^(1/2)*sin(d*x+c)^4*a^(7/2)+21*sin(d*x+
c)^4*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^4-149*(-a*(sin(d*x+c)-1))^(7/2)*a^(1/2)+461*(-a*(sin(d*x+c)-
1))^(5/2)*a^(3/2)-435*(-a*(sin(d*x+c)-1))^(3/2)*a^(5/2)+107*(-a*(sin(d*x+c)-1))^(1/2)*a^(7/2))/a^(5/2)/sin(d*x
+c)^4/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. 460 vs. \(2 (179) = 358\).

Time = 0.30 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.24 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {21 \, {\left (a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (128 \, a \cos \left (d x + c\right )^{5} + 277 \, a \cos \left (d x + c\right )^{4} - 242 \, a \cos \left (d x + c\right )^{3} - 500 \, a \cos \left (d x + c\right )^{2} + 130 \, a \cos \left (d x + c\right ) - {\left (128 \, a \cos \left (d x + c\right )^{4} - 149 \, a \cos \left (d x + c\right )^{3} - 391 \, a \cos \left (d x + c\right )^{2} + 109 \, a \cos \left (d x + c\right ) + 239 \, a\right )} \sin \left (d x + c\right ) + 239 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{256 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right ) + d\right )}} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/256*(21*(a*cos(d*x + c)^5 + a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 - 2*a*cos(d*x + c)^2 + a*cos(d*x + c) + (a
*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + 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)*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*(128*a*cos(d*x + c)^5 + 277*a*cos(d*x + c)^4
- 242*a*cos(d*x + c)^3 - 500*a*cos(d*x + c)^2 + 130*a*cos(d*x + c) - (128*a*cos(d*x + c)^4 - 149*a*cos(d*x + c
)^3 - 391*a*cos(d*x + c)^2 + 109*a*cos(d*x + c) + 239*a)*sin(d*x + c) + 239*a)*sqrt(a*sin(d*x + c) + a))/(d*co
s(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)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**5*(a+a*sin(d*x+c))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{5} \,d x } \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^4*csc(d*x + c)^5, x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.20 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (21 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 512 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {4 \, {\left (1192 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1844 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 870 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 107 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}\right )} \sqrt {a}}{256 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

1/256*sqrt(2)*(21*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)) + 512*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*(1192*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 18
44*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 870*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 - 107*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)^4)*sqrt(a)/d

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^5} \,d x \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^5,x)

[Out]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^5, x)