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

Optimal. Leaf size=215 \[ \frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{165 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{40 d} \]

[Out]

(-165*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d) + (91*a^2*Cot[c + d*x])/(128*d
*Sqrt[a + a*Sin[c + d*x]]) + (73*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (31*a^2*Cot[
c + d*x]*Csc[c + d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c +
d*x]])/(40*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.807951, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2881, 2762, 21, 2773, 206, 3044, 2975, 2980, 2772} \[ \frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{165 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{40 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-165*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d) + (91*a^2*Cot[c + d*x])/(128*d
*Sqrt[a + a*Sin[c + d*x]]) + (73*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (31*a^2*Cot[
c + d*x]*Csc[c + d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c +
d*x]])/(40*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(5*d)

Rule 2881

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 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(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] + Dist[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]))

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 2773

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3044

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*Sin[
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
])

Rule 2975

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*S
in[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 2980

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 2772

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]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^6(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{\int \csc ^5(c+d x) \left (\frac{3 a}{2}-\frac{15}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{5 a}-a \int \frac{\csc (c+d x) \left (-\frac{3 a}{2}-\frac{3}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{\int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \left (-\frac{93 a^2}{4}-\frac{105}{4} a^2 \sin (c+d x)\right ) \, dx}{20 a}+\frac{1}{2} (3 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{1}{32} (73 a) \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{\left (3 a^2\right ) \operatorname{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{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{a^2 \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{1}{128} (219 a) \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac{1}{256} (219 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{\left (219 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 d}\\ &=-\frac{165 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 d}+\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}\\ \end{align*}

Mathematica [A]  time = 1.70755, size = 404, normalized size = 1.88 \[ -\frac{a \csc ^{16}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-1380 \sin \left (\frac{1}{2} (c+d x)\right )+320 \sin \left (\frac{3}{2} (c+d x)\right )-1296 \sin \left (\frac{5}{2} (c+d x)\right )+2010 \sin \left (\frac{7}{2} (c+d x)\right )+910 \sin \left (\frac{9}{2} (c+d x)\right )+1380 \cos \left (\frac{1}{2} (c+d x)\right )+320 \cos \left (\frac{3}{2} (c+d x)\right )+1296 \cos \left (\frac{5}{2} (c+d x)\right )+2010 \cos \left (\frac{7}{2} (c+d x)\right )-910 \cos \left (\frac{9}{2} (c+d x)\right )+8250 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-8250 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-4125 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+4125 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+825 \sin (5 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-825 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{640 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Csc[(c + d*x)/2]^16*Sqrt[a*(1 + Sin[c + d*x])]*(1380*Cos[(c + d*x)/2] + 320*Cos[(3*(c + d*x))/2] + 1296*Co
s[(5*(c + d*x))/2] + 2010*Cos[(7*(c + d*x))/2] - 910*Cos[(9*(c + d*x))/2] - 1380*Sin[(c + d*x)/2] + 8250*Log[1
+ Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 8250*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c
+ d*x] + 320*Sin[(3*(c + d*x))/2] - 1296*Sin[(5*(c + d*x))/2] - 4125*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)
/2]]*Sin[3*(c + d*x)] + 4125*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2010*Sin[(7*(c +
d*x))/2] + 910*Sin[(9*(c + d*x))/2] + 825*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 825*
Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(640*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4
]^2 - Sec[(c + d*x)/4]^2)^5)

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Maple [A]  time = 1.161, size = 180, normalized size = 0.8 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{640\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( -825\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{5}+455\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}\sqrt{a}-2550\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{3/2}+4992\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{5/2}-3850\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{7/2}+825\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{9/2} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x)

[Out]

1/640*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(7/2)*(-825*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^5*si
n(d*x+c)^5+455*(-a*(sin(d*x+c)-1))^(9/2)*a^(1/2)-2550*(-a*(sin(d*x+c)-1))^(7/2)*a^(3/2)+4992*(-a*(sin(d*x+c)-1
))^(5/2)*a^(5/2)-3850*(-a*(sin(d*x+c)-1))^(3/2)*a^(7/2)+825*(-a*(sin(d*x+c)-1))^(1/2)*a^(9/2))/sin(d*x+c)^5/co
s(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(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)^6, x)

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Fricas [B]  time = 1.27496, size = 1307, normalized size = 6.08 \begin{align*} \frac{825 \,{\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} -{\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 ) + 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 (455 \, a \cos \left (d x + c\right )^{5} - 275 \, a \cos \left (d x + c\right )^{4} - 982 \, a \cos \left (d x + c\right )^{3} + 174 \, a \cos \left (d x + c\right )^{2} + 399 \, a \cos \left (d x + c\right ) -{\left (455 \, a \cos \left (d x + c\right )^{4} + 730 \, a \cos \left (d x + c\right )^{3} - 252 \, a \cos \left (d x + c\right )^{2} - 426 \, a \cos \left (d x + c\right ) - 27 \, a\right )} \sin \left (d x + c\right ) - 27 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{2560 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} -{\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 ) + d\right )} \sin \left (d x + c\right ) - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/2560*(825*(a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^2 - (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)*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*(455*a*cos(d*x + c)^5
- 275*a*cos(d*x + c)^4 - 982*a*cos(d*x + c)^3 + 174*a*cos(d*x + c)^2 + 399*a*cos(d*x + c) - (455*a*cos(d*x + c
)^4 + 730*a*cos(d*x + c)^3 - 252*a*cos(d*x + c)^2 - 426*a*cos(d*x + c) - 27*a)*sin(d*x + c) - 27*a)*sqrt(a*sin
(d*x + c) + a))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - (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)*sin(d*x + c) - d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**6*(a+a*sin(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out