\(\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\) [461]

Optimal result

Integrand size = 31, antiderivative size = 253 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {179 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{512 d}-\frac {179 a^2 \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt {a+a \sin (c+d x)}}+\frac {239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt {a+a \sin (c+d x)}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d} \]

[Out]

-179/512*a^(3/2)*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d-1/6*cot(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x
+c))^(3/2)/d-179/512*a^2*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+111/256*a^2*cot(d*x+c)*csc(d*x+c)/d/(a+a*sin(d*x+
c))^(1/2)+239/320*a^2*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c))^(1/2)+137/480*a^2*cot(d*x+c)*csc(d*x+c)^3/d/(
a+a*sin(d*x+c))^(1/2)-1/20*a*cot(d*x+c)*csc(d*x+c)^4*(a+a*sin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2960, 2841, 21, 2851, 2852, 212, 3123, 3054, 3059} \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {179 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{512 d}-\frac {179 a^2 \cot (c+d x)}{512 d \sqrt {a \sin (c+d x)+a}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a \sin (c+d x)+a}}+\frac {239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt {a \sin (c+d x)+a}}+\frac {111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{20 d} \]

[In]

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

[Out]

(-179*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(512*d) - (179*a^2*Cot[c + d*x])/(512*
d*Sqrt[a + a*Sin[c + d*x]]) + (111*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d*Sqrt[a + a*Sin[c + d*x]]) + (239*a^2*
Cot[c + d*x]*Csc[c + d*x]^2)/(320*d*Sqrt[a + a*Sin[c + d*x]]) + (137*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(480*d*S
qrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(20*d) - (Cot[c + d*x]*Csc
[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(6*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 2841

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c
+ a*d))), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1
)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], 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] && GtQ[m, 1] && LtQ[n, -1
] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 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 3054

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
*Sin[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 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) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {a^2 \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) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\int \csc ^6(c+d x) \left (\frac {3 a}{2}-\frac {17}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{6 a}-\frac {1}{2} a \int \frac {\csc ^2(c+d x) \left (-\frac {7 a}{2}-\frac {7}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {a^2 \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) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {137 a^2}{4}-\frac {149}{4} a^2 \sin (c+d x)\right ) \, dx}{30 a}+\frac {1}{4} (7 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {7 a^2 \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {1}{8} (7 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {1}{320} (717 a) \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {7 a^2 \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt {a+a \sin (c+d x)}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}-\frac {1}{128} (239 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {\left (7 a^2\right ) \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 {7 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {7 a^2 \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt {a+a \sin (c+d x)}}+\frac {239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt {a+a \sin (c+d x)}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}-\frac {1}{512} (717 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {7 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {179 a^2 \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt {a+a \sin (c+d x)}}+\frac {239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt {a+a \sin (c+d x)}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}-\frac {(717 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1024} \\ & = -\frac {7 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {179 a^2 \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt {a+a \sin (c+d x)}}+\frac {239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt {a+a \sin (c+d x)}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\left (717 a^2\right ) \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 {179 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{512 d}-\frac {179 a^2 \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}+\frac {111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt {a+a \sin (c+d x)}}+\frac {239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt {a+a \sin (c+d x)}}+\frac {137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{20 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.92 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.92 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {a \csc ^{19}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (25140 \cos \left (\frac {1}{2} (c+d x)\right )-71972 \cos \left (\frac {3}{2} (c+d x)\right )-42690 \cos \left (\frac {5}{2} (c+d x)\right )-5718 \cos \left (\frac {7}{2} (c+d x)\right )+18690 \cos \left (\frac {9}{2} (c+d x)\right )-5370 \cos \left (\frac {11}{2} (c+d x)\right )-26850 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+40275 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-16110 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2685 \cos (6 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+26850 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-40275 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+16110 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2685 \cos (6 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-25140 \sin \left (\frac {1}{2} (c+d x)\right )-71972 \sin \left (\frac {3}{2} (c+d x)\right )+42690 \sin \left (\frac {5}{2} (c+d x)\right )-5718 \sin \left (\frac {7}{2} (c+d x)\right )-18690 \sin \left (\frac {9}{2} (c+d x)\right )-5370 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{7680 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^6} \]

[In]

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

[Out]

(a*Csc[(c + d*x)/2]^19*Sqrt[a*(1 + Sin[c + d*x])]*(25140*Cos[(c + d*x)/2] - 71972*Cos[(3*(c + d*x))/2] - 42690
*Cos[(5*(c + d*x))/2] - 5718*Cos[(7*(c + d*x))/2] + 18690*Cos[(9*(c + d*x))/2] - 5370*Cos[(11*(c + d*x))/2] -
26850*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 40275*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c
+ d*x)/2]] - 16110*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2685*Cos[6*(c + d*x)]*Log[1
+ Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 26850*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 40275*Cos[2*(c
+ d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 16110*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(
c + d*x)/2]] - 2685*Cos[6*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 25140*Sin[(c + d*x)/2] - 7
1972*Sin[(3*(c + d*x))/2] + 42690*Sin[(5*(c + d*x))/2] - 5718*Sin[(7*(c + d*x))/2] - 18690*Sin[(9*(c + d*x))/2
] - 5370*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 = 0.12 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.78

method	result	size
default	\(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (2685 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{7} \left (\sin ^{6}\left (d x +c \right )\right )-2685 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {3}{2}}+10095 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {5}{2}}-7794 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {7}{2}}-10866 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {9}{2}}+15215 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {11}{2}}-2685 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {13}{2}}\right )}{7680 a^{\frac {11}{2}} \sin \left (d x +c \right )^{6} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(198\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/7680*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(2685*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^7*sin(d*x+
c)^6-2685*(-a*(sin(d*x+c)-1))^(11/2)*a^(3/2)+10095*(-a*(sin(d*x+c)-1))^(9/2)*a^(5/2)-7794*(-a*(sin(d*x+c)-1))^
(7/2)*a^(7/2)-10866*(-a*(sin(d*x+c)-1))^(5/2)*a^(9/2)+15215*(-a*(sin(d*x+c)-1))^(3/2)*a^(11/2)-2685*(-a*(sin(d
*x+c)-1))^(1/2)*a^(13/2))/a^(11/2)/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. 557 vs. \(2 (221) = 442\).

Time = 0.33 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.20 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {2685 \, {\left (a \cos \left (d x + c\right )^{7} + a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{5} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - 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 (2685 \, a \cos \left (d x + c\right )^{6} - 3330 \, a \cos \left (d x + c\right )^{5} - 5649 \, a \cos \left (d x + c\right )^{4} + 7188 \, a \cos \left (d x + c\right )^{3} + 6715 \, a \cos \left (d x + c\right )^{2} - 2578 \, a \cos \left (d x + c\right ) + {\left (2685 \, a \cos \left (d x + c\right )^{5} + 6015 \, a \cos \left (d x + c\right )^{4} + 366 \, a \cos \left (d x + c\right )^{3} - 6822 \, a \cos \left (d x + c\right )^{2} - 107 \, a \cos \left (d x + c\right ) + 2471 \, a\right )} \sin \left (d x + c\right ) - 2471 \, a\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))^(3/2),x, algorithm="fricas")

[Out]

1/30720*(2685*(a*cos(d*x + c)^7 + a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^5 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c
)^3 + 3*a*cos(d*x + c)^2 - a*cos(d*x + c) + (a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^2 - a)*s
in(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)*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*(2685*a*cos(d*x + c)^6 - 3330*a*cos(d*x + c)^5 - 5649*a*cos(d*x + c)^4 + 7188*a*cos(d*x
+ c)^3 + 6715*a*cos(d*x + c)^2 - 2578*a*cos(d*x + c) + (2685*a*cos(d*x + c)^5 + 6015*a*cos(d*x + c)^4 + 366*a*
cos(d*x + c)^3 - 6822*a*cos(d*x + c)^2 - 107*a*cos(d*x + c) + 2471*a)*sin(d*x + c) - 2471*a)*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. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**7*(a+a*sin(d*x+c))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{7} \,d x } \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(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)^7, x)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.10 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (2685 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (85920 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 161520 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 62352 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 43464 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30430 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2685 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}\right )} \sqrt {a}}{30720 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

-1/30720*sqrt(2)*(2685*sqrt(2)*a*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*(85920*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*
c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^11 - 161520*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1
/2*c)^9 + 62352*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 + 43464*a*sgn(cos(-1/4*
pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 30430*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4
*pi + 1/2*d*x + 1/2*c)^3 + 2685*a*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. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^7} \,d x \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^7,x)

[Out]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^7, x)