3.451 \(\int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=245 \[ -\frac{55 a \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{55 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{512 d}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a \sin (c+d x)+a}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a \sin (c+d x)+a}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a \sin (c+d x)+a}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-55*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(512*d) - (55*a*Cot[c + d*x])/(512*d*Sq
rt[a + a*Sin[c + d*x]]) - (55*a*Cot[c + d*x]*Csc[c + d*x])/(768*d*Sqrt[a + a*Sin[c + d*x]]) + (329*a*Cot[c + d
*x]*Csc[c + d*x]^2)/(960*d*Sqrt[a + a*Sin[c + d*x]]) + (47*a*Cot[c + d*x]*Csc[c + d*x]^3)/(160*d*Sqrt[a + a*Si
n[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(60*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^5*
Sqrt[a + a*Sin[c + d*x]])/(6*d)

________________________________________________________________________________________

Rubi [A]  time = 0.811591, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ -\frac{55 a \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{55 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{512 d}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a \sin (c+d x)+a}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a \sin (c+d x)+a}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a \sin (c+d x)+a}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-55*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(512*d) - (55*a*Cot[c + d*x])/(512*d*Sq
rt[a + a*Sin[c + d*x]]) - (55*a*Cot[c + d*x]*Csc[c + d*x])/(768*d*Sqrt[a + a*Sin[c + d*x]]) + (329*a*Cot[c + d
*x]*Csc[c + d*x]^2)/(960*d*Sqrt[a + a*Sin[c + d*x]]) + (47*a*Cot[c + d*x]*Csc[c + d*x]^3)/(160*d*Sqrt[a + a*Si
n[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(60*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^5*
Sqrt[a + a*Sin[c + d*x]])/(6*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 ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\int \csc ^7(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 (c+d x)}{2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}+\frac{3}{4} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\frac{\int \csc ^6(c+d x) \left (\frac{a}{2}-\frac{15}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{6 a}\\ &=-\frac{3 a \cot (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}+\frac{3}{8} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{47}{40} \int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 a \cot (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a+a \sin (c+d x)}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}-\frac{329}{320} \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{(3 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 )}{4 d}\\ &=-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}-\frac{3 a \cot (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a+a \sin (c+d x)}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a+a \sin (c+d x)}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}-\frac{329}{384} \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}-\frac{3 a \cot (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a+a \sin (c+d x)}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a+a \sin (c+d x)}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}-\frac{329}{512} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}-\frac{55 a \cot (c+d x)}{512 d \sqrt{a+a \sin (c+d x)}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a+a \sin (c+d x)}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a+a \sin (c+d x)}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}-\frac{329 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{1024}\\ &=-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}-\frac{55 a \cot (c+d x)}{512 d \sqrt{a+a \sin (c+d x)}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a+a \sin (c+d x)}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a+a \sin (c+d x)}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}+\frac{(329 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 )}{512 d}\\ &=-\frac{55 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{512 d}-\frac{55 a \cot (c+d x)}{512 d \sqrt{a+a \sin (c+d x)}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a+a \sin (c+d x)}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a+a \sin (c+d x)}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{6 d}\\ \end{align*}

Mathematica [A]  time = 7.57846, size = 485, normalized size = 1.98 \[ \frac{\csc ^{19}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (24540 \sin \left (\frac{1}{2} (c+d x)\right )-25684 \sin \left (\frac{3}{2} (c+d x)\right )+14490 \sin \left (\frac{5}{2} (c+d x)\right )-15006 \sin \left (\frac{7}{2} (c+d x)\right )+550 \sin \left (\frac{9}{2} (c+d x)\right )-1650 \sin \left (\frac{11}{2} (c+d x)\right )-24540 \cos \left (\frac{1}{2} (c+d x)\right )-25684 \cos \left (\frac{3}{2} (c+d x)\right )-14490 \cos \left (\frac{5}{2} (c+d x)\right )-15006 \cos \left (\frac{7}{2} (c+d x)\right )-550 \cos \left (\frac{9}{2} (c+d x)\right )-1650 \cos \left (\frac{11}{2} (c+d x)\right )+12375 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-4950 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+825 \cos (6 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-8250 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-12375 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+4950 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-825 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+8250 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{7680 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(Csc[(c + d*x)/2]^19*Sqrt[a*(1 + Sin[c + d*x])]*(-24540*Cos[(c + d*x)/2] - 25684*Cos[(3*(c + d*x))/2] - 14490*
Cos[(5*(c + d*x))/2] - 15006*Cos[(7*(c + d*x))/2] - 550*Cos[(9*(c + d*x))/2] - 1650*Cos[(11*(c + d*x))/2] - 82
50*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12375*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d
*x)/2]] - 4950*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 825*Cos[6*(c + d*x)]*Log[1 + Co
s[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8250*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 12375*Cos[2*(c + d*x)
]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4950*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x
)/2]] - 825*Cos[6*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24540*Sin[(c + d*x)/2] - 25684*Sin
[(3*(c + d*x))/2] + 14490*Sin[(5*(c + d*x))/2] - 15006*Sin[(7*(c + d*x))/2] + 550*Sin[(9*(c + d*x))/2] - 1650*
Sin[(11*(c + d*x))/2]))/(7680*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^6)

________________________________________________________________________________________

Maple [A]  time = 1.151, size = 198, normalized size = 0.8 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{7680\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 825\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{11/2}{a}^{5/2}-4675\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{7/2}-825\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}+7818\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{9/2}-1398\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{11/2}-4675\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{13/2}+825\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{15/2} \right ){a}^{-{\frac{15}{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)^7*(a+a*sin(d*x+c))^(1/2),x)

[Out]

1/7680*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(15/2)*(825*(-a*(sin(d*x+c)-1))^(11/2)*a^(5/2)-4675*(-a*(sin
(d*x+c)-1))^(9/2)*a^(7/2)-825*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^8*sin(d*x+c)^6+7818*(-a*(sin(d*x+c)
-1))^(7/2)*a^(9/2)-1398*(-a*(sin(d*x+c)-1))^(5/2)*a^(11/2)-4675*(-a*(sin(d*x+c)-1))^(3/2)*a^(13/2)+825*(-a*(si
n(d*x+c)-1))^(1/2)*a^(15/2))/sin(d*x+c)^6/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 )^{7}\,{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)^7*(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)^7, x)

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Fricas [B]  time = 1.2622, size = 1423, normalized size = 5.81 \begin{align*} \frac{825 \,{\left (\cos \left (d x + c\right )^{7} + \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1\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 (825 \, \cos \left (d x + c\right )^{6} + 550 \, \cos \left (d x + c\right )^{5} + 707 \, \cos \left (d x + c\right )^{4} + 1156 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} +{\left (825 \, \cos \left (d x + c\right )^{5} + 275 \, \cos \left (d x + c\right )^{4} + 982 \, \cos \left (d x + c\right )^{3} - 174 \, \cos \left (d x + c\right )^{2} - 399 \, \cos \left (d x + c\right ) + 27\right )} \sin \left (d x + c\right ) - 426 \, \cos \left (d x + c\right ) - 27\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{30720 \,{\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\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} - 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)^7*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/30720*(825*(cos(d*x + c)^7 + cos(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)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(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)*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)*s
in(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
*(825*cos(d*x + c)^6 + 550*cos(d*x + c)^5 + 707*cos(d*x + c)^4 + 1156*cos(d*x + c)^3 - 225*cos(d*x + c)^2 + (8
25*cos(d*x + c)^5 + 275*cos(d*x + c)^4 + 982*cos(d*x + c)^3 - 174*cos(d*x + c)^2 - 399*cos(d*x + c) + 27)*sin(
d*x + c) - 426*cos(d*x + c) - 27)*sqrt(a*sin(d*x + c) + a))/(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*cos(d*x + c)^6 -
3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - 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)**7*(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)^7*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out