3.450 \(\int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=209 \[ -\frac{31 a \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{31 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-31*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d) - (31*a*Cot[c + d*x])/(128*d*Sq
rt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*x]*Csc[c + d*x])/(192*d*Sqrt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*
x]*Csc[c + d*x]^2)/(240*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c +
d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.690556, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{31 a \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{31 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-31*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d) - (31*a*Cot[c + d*x])/(128*d*Sq
rt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*x]*Csc[c + d*x])/(192*d*Sqrt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*
x]*Csc[c + d*x]^2)/(240*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c +
d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(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 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 ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\int \csc ^6(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)}{d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}+\frac{1}{2} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\frac{\int \csc ^5(c+d x) \left (\frac{a}{2}-\frac{13}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{5 a}\\ &=-\frac{a \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{80} \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx-\frac{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 )}{d}\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{a \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{96} \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{a \cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{128} \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{31 a \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}-\frac{97}{256} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{31 a \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}+\frac{(97 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 )}{128 d}\\ &=-\frac{31 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 d}-\frac{31 a \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a+a \sin (c+d x)}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a+a \sin (c+d x)}}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}\\ \end{align*}

Mathematica [A]  time = 4.37091, size = 403, normalized size = 1.93 \[ -\frac{\csc ^{16}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-10180 \sin \left (\frac{1}{2} (c+d x)\right )-2240 \sin \left (\frac{3}{2} (c+d x)\right )+1392 \sin \left (\frac{5}{2} (c+d x)\right )+4810 \sin \left (\frac{7}{2} (c+d x)\right )-930 \sin \left (\frac{9}{2} (c+d x)\right )+10180 \cos \left (\frac{1}{2} (c+d x)\right )-2240 \cos \left (\frac{3}{2} (c+d x)\right )-1392 \cos \left (\frac{5}{2} (c+d x)\right )+4810 \cos \left (\frac{7}{2} (c+d x)\right )+930 \cos \left (\frac{9}{2} (c+d x)\right )+4650 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-4650 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-2325 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+2325 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+465 \sin (5 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-465 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{1920 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

-(Csc[(c + d*x)/2]^16*Sqrt[a*(1 + Sin[c + d*x])]*(10180*Cos[(c + d*x)/2] - 2240*Cos[(3*(c + d*x))/2] - 1392*Co
s[(5*(c + d*x))/2] + 4810*Cos[(7*(c + d*x))/2] + 930*Cos[(9*(c + d*x))/2] - 10180*Sin[(c + d*x)/2] + 4650*Log[
1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 4650*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[
c + d*x] - 2240*Sin[(3*(c + d*x))/2] + 1392*Sin[(5*(c + d*x))/2] - 2325*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*
x)/2]]*Sin[3*(c + d*x)] + 2325*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 4810*Sin[(7*(c
+ d*x))/2] - 930*Sin[(9*(c + d*x))/2] + 465*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 46
5*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(1920*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x
)/4]^2 - Sec[(c + d*x)/4]^2)^5)

________________________________________________________________________________________

Maple [A]  time = 1.2, size = 180, normalized size = 0.9 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{1920\, \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 ( 465\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{11/2}-2170\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{9/2}+896\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{7/2}+890\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{5/2}-465\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{3/2}-465\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{5} \right ){a}^{-{\frac{11}{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))^(1/2),x)

[Out]

1/1920*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(465*(-a*(sin(d*x+c)-1))^(1/2)*a^(11/2)-2170*(-a*(sin(d*x+c)-1
))^(3/2)*a^(9/2)+896*(-a*(sin(d*x+c)-1))^(5/2)*a^(7/2)+890*(-a*(sin(d*x+c)-1))^(7/2)*a^(5/2)-465*(-a*(sin(d*x+
c)-1))^(9/2)*a^(3/2)-465*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^6*sin(d*x+c)^5)/a^(11/2)/sin(d*x+c)^5/co
s(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

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 )^{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))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.30777, size = 1261, normalized size = 6.03 \begin{align*} \frac{465 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \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 (465 \, \cos \left (d x + c\right )^{5} + 1435 \, \cos \left (d x + c\right )^{4} - 154 \, \cos \left (d x + c\right )^{3} - 1662 \, \cos \left (d x + c\right )^{2} -{\left (465 \, \cos \left (d x + c\right )^{4} - 970 \, \cos \left (d x + c\right )^{3} - 1124 \, \cos \left (d x + c\right )^{2} + 538 \, \cos \left (d x + c\right ) + 611\right )} \sin \left (d x + c\right ) + 73 \, \cos \left (d x + c\right ) + 611\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{7680 \,{\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))^(1/2),x, algorithm="fricas")

[Out]

1/7680*(465*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d
*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sin(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)*s
qrt(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*(465*cos(d*x + c)^5 + 1435*cos(d*x + c
)^4 - 154*cos(d*x + c)^3 - 1662*cos(d*x + c)^2 - (465*cos(d*x + c)^4 - 970*cos(d*x + c)^3 - 1124*cos(d*x + c)^
2 + 538*cos(d*x + c) + 611)*sin(d*x + c) + 73*cos(d*x + c) + 611)*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*co
s(d*x + c)^2 + d*cos(d*x + c) + d)*sin(d*x + c) - d)

________________________________________________________________________________________

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))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

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)^6*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError