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

Optimal. Leaf size=253 \[ -\frac{179 a^2 \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{179 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{512 d}+\frac{137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt{a \sin (c+d x)+a}}+\frac{239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt{a \sin (c+d x)+a}}+\frac{111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{20 d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.918546, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2881, 2762, 21, 2772, 2773, 206, 3044, 2975, 2980} \[ -\frac{179 a^2 \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{179 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{512 d}+\frac{137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt{a \sin (c+d x)+a}}+\frac{239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt{a \sin (c+d x)+a}}+\frac{111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{20 d} \]

Antiderivative was successfully verified.

[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 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 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(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 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 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 2975

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*S
in[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 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) (a+a \sin (c+d x))^{3/2} \, dx &=\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 ) \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{7 a^{3/2} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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 ) \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{179 a^{3/2} \tanh ^{-1}\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]  time = 2.4293, size = 486, normalized size = 1.92 \[ \frac{a \csc ^{19}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-25140 \sin \left (\frac{1}{2} (c+d x)\right )-71972 \sin \left (\frac{3}{2} (c+d x)\right )+42690 \sin \left (\frac{5}{2} (c+d x)\right )-5718 \sin \left (\frac{7}{2} (c+d x)\right )-18690 \sin \left (\frac{9}{2} (c+d x)\right )-5370 \sin \left (\frac{11}{2} (c+d x)\right )+25140 \cos \left (\frac{1}{2} (c+d x)\right )-71972 \cos \left (\frac{3}{2} (c+d x)\right )-42690 \cos \left (\frac{5}{2} (c+d x)\right )-5718 \cos \left (\frac{7}{2} (c+d x)\right )+18690 \cos \left (\frac{9}{2} (c+d x)\right )-5370 \cos \left (\frac{11}{2} (c+d x)\right )+40275 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-16110 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+2685 \cos (6 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-26850 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-40275 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+16110 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-2685 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+26850 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{7680 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(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)

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Maple [A]  time = 1.198, 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 ( 2685\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{6}-2685\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{13/2}+15215\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{11/2}-10866\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{9/2}-7794\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{7/2}+10095\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{5/2}-2685\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{11/2}{a}^{3/2} \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)^7*(a+a*sin(d*x+c))^(3/2),x)

[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))^(1/2)*a^(13/2)+15215*(-a*(sin(d*x+c)-1))^(3/2)*a^(11/2)-10866*(-a*(sin(d*x+c)-1)
)^(5/2)*a^(9/2)-7794*(-a*(sin(d*x+c)-1))^(7/2)*a^(7/2)+10095*(-a*(sin(d*x+c)-1))^(9/2)*a^(5/2)-2685*(-a*(sin(d
*x+c)-1))^(11/2)*a^(3/2))/a^(11/2)/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{\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} \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))^(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)

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Fricas [B]  time = 1.30729, size = 1503, normalized size = 5.94 \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)^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)

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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))**(3/2),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))^(3/2),x, algorithm="giac")

[Out]

Timed out