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

Optimal. Leaf size=197 \[ -\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{29 a^2 \cot (c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d} \]

[Out]

(37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (8*a^2*Cos[c + d*x])/(3*d*Sqrt[a
+ a*Sin[c + d*x]]) + (29*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin
[c + d*x]])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^
2*(a + a*Sin[c + d*x])^(3/2))/(3*d)

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Rubi [A]  time = 0.499766, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {2718, 2647, 2646, 3044, 2975, 2980, 2773, 206} \[ -\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{29 a^2 \cot (c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (8*a^2*Cos[c + d*x])/(3*d*Sqrt[a
+ a*Sin[c + d*x]]) + (29*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin
[c + d*x]])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^
2*(a + a*Sin[c + d*x])^(3/2))/(3*d)

Rule 2718

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]
&& EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] &&  !LtQ[m, -1]

Rule 2647

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] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2646

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] && EqQ[a^2 - b^2, 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 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])

Rubi steps

\begin{align*} \int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{2 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}+\frac{\int \csc ^3(c+d x) \left (\frac{3 a}{2}-\frac{11}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{3 a}+\frac{1}{3} (4 a) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{8 a^2 \cos (c+d x)}{3 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}+\frac{\int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \left (-\frac{29 a^2}{4}-\frac{41}{4} a^2 \sin (c+d x)\right ) \, dx}{6 a}\\ &=-\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}-\frac{1}{16} (37 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\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}+\frac{\left (37 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 )}{8 d}\\ &=\frac{37 a^{3/2} \tanh ^{-1}\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}\\ \end{align*}

Mathematica [A]  time = 1.26978, size = 334, normalized size = 1.7 \[ -\frac{a \csc ^{10}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (276 \sin \left (\frac{1}{2} (c+d x)\right )+326 \sin \left (\frac{3}{2} (c+d x)\right )-78 \sin \left (\frac{5}{2} (c+d x)\right )-72 \sin \left (\frac{7}{2} (c+d x)\right )-8 \sin \left (\frac{9}{2} (c+d x)\right )-276 \cos \left (\frac{1}{2} (c+d x)\right )+326 \cos \left (\frac{3}{2} (c+d x)\right )+78 \cos \left (\frac{5}{2} (c+d x)\right )-72 \cos \left (\frac{7}{2} (c+d x)\right )+8 \cos \left (\frac{9}{2} (c+d x)\right )-333 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+333 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+111 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-111 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{24 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 + Cos[(
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*Si
n[(9*(c + d*x))/2]))/(24*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)

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Maple [A]  time = 1.03, size = 196, normalized size = 1. \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{24\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 96\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{5/2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}-16\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{3/2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}-111\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}+15\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{5/2}+8\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{3/2}-15\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}\sqrt{a} \right ){a}^{-{\frac{3}{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)^4*(a+a*sin(d*x+c))^(3/2),x)

[Out]

-1/24*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(96*(-a*(sin(d*x+c)-1))^(1/2)*a^(5/2)*sin(d*x+c)^3-16*(-a*(sin(
d*x+c)-1))^(3/2)*a^(3/2)*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))^(1/2)*a^(5/2)+8*(-a*(sin(d*x+c)-1))^(3/2)*a^(3/2)-15*(-a*(sin(d*x+c)-1))^(5/2)*a^(1/2))/a^(3/2)
/sin(d*x+c)^3/cos(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 )^{4}\,{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)^4*(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)^4, x)

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Fricas [B]  time = 1.2388, size = 1123, normalized size = 5.7 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(111*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 - (a*cos(d*x + c)^3 + 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*(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)*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)**4*(a+a*sin(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError