\(\int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [451]

Optimal result

Integrand size = 31, antiderivative size = 245 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {55 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{512 d}-\frac {55 a \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a+a \sin (c+d x)}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d} \]

[Out]

-55/512*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))*a^(1/2)/d-55/512*a*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1
/2)-55/768*a*cot(d*x+c)*csc(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+329/960*a*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c
))^(1/2)+47/160*a*cot(d*x+c)*csc(d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)-1/60*a*cot(d*x+c)*csc(d*x+c)^4/d/(a+a*sin(d
*x+c))^(1/2)-1/6*cot(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2960, 2851, 2852, 212, 3123, 3059} \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {55 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{512 d}-\frac {55 a \cot (c+d x)}{512 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a \sin (c+d x)+a}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a \sin (c+d x)+a}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a \sin (c+d x)+a}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a \sin (c+d x)+a}} \]

[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 212

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

Rule 2851

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 2852

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 2960

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 3059

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 3123

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*Si
n[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])

Rubi steps \begin{align*} \text {integral}& = \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) \text {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} \text {arctanh}\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} \text {arctanh}\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} \text {arctanh}\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} \text {arctanh}\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) \text {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} \text {arctanh}\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] (verified)

Time = 5.81 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.98 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\csc ^{19}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-24540 \cos \left (\frac {1}{2} (c+d x)\right )-25684 \cos \left (\frac {3}{2} (c+d x)\right )-14490 \cos \left (\frac {5}{2} (c+d x)\right )-15006 \cos \left (\frac {7}{2} (c+d x)\right )-550 \cos \left (\frac {9}{2} (c+d x)\right )-1650 \cos \left (\frac {11}{2} (c+d x)\right )-8250 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12375 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-4950 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+825 \cos (6 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8250 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-12375 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4950 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-825 \cos (6 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24540 \sin \left (\frac {1}{2} (c+d x)\right )-25684 \sin \left (\frac {3}{2} (c+d x)\right )+14490 \sin \left (\frac {5}{2} (c+d x)\right )-15006 \sin \left (\frac {7}{2} (c+d x)\right )+550 \sin \left (\frac {9}{2} (c+d x)\right )-1650 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{7680 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^6} \]

[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] (verified)

Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (825 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {5}{2}}-4675 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {7}{2}}-825 \left (\sin ^{6}\left (d x +c \right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{8}+7818 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {9}{2}}-1398 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {11}{2}}-4675 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {13}{2}}+825 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {15}{2}}\right )}{7680 a^{\frac {15}{2}} \sin \left (d x +c \right )^{6} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(198\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[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*sin(d*x+c)^6*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^8+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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (213) = 426\).

Time = 0.29 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.14 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\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 )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**7*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{7} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.11 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (825 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (26400 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 74800 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 62544 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5592 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9350 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 825 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}\right )} \sqrt {a}}{30720 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-1/30720*sqrt(2)*(825*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(-1/
4*pi + 1/2*d*x + 1/2*c)))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) + 4*(26400*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*s
in(-1/4*pi + 1/2*d*x + 1/2*c)^11 - 74800*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9
+ 62544*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 5592*sgn(cos(-1/4*pi + 1/2*d*x
+ 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 9350*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x +
1/2*c)^3 + 825*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c))/(2*sin(-1/4*pi + 1/2*d*x +
1/2*c)^2 - 1)^6)*sqrt(a)/d

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^7} \,d x \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^7,x)

[Out]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^7, x)