3.452 \(\int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=281 \[ -\frac{61 a \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{61 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{1024 d}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a \sin (c+d x)+a}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a \sin (c+d x)+a}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-61*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d) - (61*a*Cot[c + d*x])/(1024*d*
Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c + d*x]*Csc[c + d*x])/(1536*d*Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c +
d*x]*Csc[c + d*x]^2)/(1920*d*Sqrt[a + a*Sin[c + d*x]]) + (579*a*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*d*Sqrt[a +
a*Sin[c + d*x]]) + (193*a*Cot[c + d*x]*Csc[c + d*x]^4)/(840*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc
[c + d*x]^5)/(84*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]])/(7*d)

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Rubi [A]  time = 0.947658, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2881, 2772, 2773, 206, 3044, 2980} \[ -\frac{61 a \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{61 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{1024 d}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a \sin (c+d x)+a}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a \sin (c+d x)+a}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-61*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d) - (61*a*Cot[c + d*x])/(1024*d*
Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c + d*x]*Csc[c + d*x])/(1536*d*Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c +
d*x]*Csc[c + d*x]^2)/(1920*d*Sqrt[a + a*Sin[c + d*x]]) + (579*a*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*d*Sqrt[a +
a*Sin[c + d*x]]) + (193*a*Cot[c + d*x]*Csc[c + d*x]^4)/(840*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc
[c + d*x]^5)/(84*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]])/(7*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 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]

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 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]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\int \csc ^8(c+d x) \sqrt{a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}+\frac{5}{6} \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\frac{\int \csc ^7(c+d x) \left (\frac{a}{2}-\frac{17}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{7 a}\\ &=-\frac{5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}+\frac{5}{8} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{193}{168} \int \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{5 a \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}+\frac{5}{16} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{579}{560} \int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{5 a \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}-\frac{579}{640} \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{(5 a) \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{5 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}-\frac{5 a \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a+a \sin (c+d x)}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}-\frac{193}{256} \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{5 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}-\frac{5 a \cot (c+d x)}{8 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a+a \sin (c+d x)}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}-\frac{579 \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{1024}\\ &=-\frac{5 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}-\frac{61 a \cot (c+d x)}{1024 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a+a \sin (c+d x)}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}-\frac{579 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{2048}\\ &=-\frac{5 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 d}-\frac{61 a \cot (c+d x)}{1024 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a+a \sin (c+d x)}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}+\frac{(579 a) \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 )}{1024 d}\\ &=-\frac{61 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{1024 d}-\frac{61 a \cot (c+d x)}{1024 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a+a \sin (c+d x)}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a+a \sin (c+d x)}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a+a \sin (c+d x)}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)}}{7 d}\\ \end{align*}

Mathematica [A]  time = 2.05157, size = 191, normalized size = 0.68 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (-102480 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+102480 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+\csc ^7(c+d x) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (49128 \sin (c+d x)-179636 \sin (3 (c+d x))-8540 \sin (5 (c+d x))-244533 \cos (2 (c+d x))-52094 \cos (4 (c+d x))+6405 \cos (6 (c+d x))-201298)\right )}{3440640 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(Sqrt[a*(1 + Sin[c + d*x])]*(-102480*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 102480*Log[1 - Cos[(c + d*
x)/2] + Sin[(c + d*x)/2]] + Csc[c + d*x]^7*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(-201298 - 244533*Cos[2*(c +
d*x)] - 52094*Cos[4*(c + d*x)] + 6405*Cos[6*(c + d*x)] + 49128*Sin[c + d*x] - 179636*Sin[3*(c + d*x)] - 8540*S
in[5*(c + d*x)])))/(3440640*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

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Maple [A]  time = 1.28, size = 216, normalized size = 0.8 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{107520\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 6405\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{13/2}{a}^{7/2}-42700\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{11/2}{a}^{9/2}+120841\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{11/2}+6405\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{10} \left ( \sin \left ( dx+c \right ) \right ) ^{7}-156672\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{13/2}+51191\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{15/2}+42700\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{17/2}-6405\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{19/2} \right ){a}^{-{\frac{19}{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)^8*(a+a*sin(d*x+c))^(1/2),x)

[Out]

-1/107520*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(19/2)*(6405*(-a*(sin(d*x+c)-1))^(13/2)*a^(7/2)-42700*(-a
*(sin(d*x+c)-1))^(11/2)*a^(9/2)+120841*(-a*(sin(d*x+c)-1))^(9/2)*a^(11/2)+6405*arctanh((-a*(sin(d*x+c)-1))^(1/
2)/a^(1/2))*a^10*sin(d*x+c)^7-156672*(-a*(sin(d*x+c)-1))^(7/2)*a^(13/2)+51191*(-a*(sin(d*x+c)-1))^(5/2)*a^(15/
2)+42700*(-a*(sin(d*x+c)-1))^(3/2)*a^(17/2)-6405*(-a*(sin(d*x+c)-1))^(1/2)*a^(19/2))/sin(d*x+c)^7/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 \sqrt{a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{8}\,{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)^8*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*cos(d*x + c)^4*csc(d*x + c)^8, x)

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Fricas [B]  time = 1.22736, size = 1565, normalized size = 5.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/430080*(6405*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 - (cos(d*x + c)^7 + co
s(d*x + c)^6 - 3*cos(d*x + c)^5 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^3 + 3*cos(d*x + c)^2 - cos(d*x + c) - 1)*s
in(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)*s
in(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) - c
os(d*x + c) - 1)) + 4*(6405*cos(d*x + c)^7 + 2135*cos(d*x + c)^6 - 22631*cos(d*x + c)^5 - 37613*cos(d*x + c)^4
+ 1343*cos(d*x + c)^3 + 27477*cos(d*x + c)^2 - (6405*cos(d*x + c)^6 + 4270*cos(d*x + c)^5 - 18361*cos(d*x + c
)^4 + 19252*cos(d*x + c)^3 + 20595*cos(d*x + c)^2 - 6882*cos(d*x + c) - 7359)*sin(d*x + c) - 477*cos(d*x + c)
- 7359)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x +
c)^2 - (d*cos(d*x + c)^7 + d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^5 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 + 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)**8*(a+a*sin(d*x+c))**(1/2),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)^8*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out