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

Optimal. Leaf size=205 \[ -\frac{9 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*Sqrt[a]*d) - (9*Cot[c + d*x])/(128*d*Sqrt[a
+ a*Sin[c + d*x]]) - (3*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (29*Cot[c + d*x]*Csc[c +
d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (Co
t[c + d*x]*Csc[c + d*x]^4)/(5*d*Sqrt[a + a*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.14305, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2881, 2779, 2985, 2649, 206, 2773, 3044, 2984} \[ -\frac{9 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}} \]

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]

(-9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*Sqrt[a]*d) - (9*Cot[c + d*x])/(128*d*Sqrt[a
+ a*Sin[c + d*x]]) - (3*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (29*Cot[c + d*x]*Csc[c +
d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (Co
t[c + d*x]*Csc[c + d*x]^4)/(5*d*Sqrt[a + a*Sin[c + d*x]])

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 2779

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> -Sim
p[(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[1/
(2*b*(n + 1)*(c^2 - d^2)), Int[((c + d*Sin[e + f*x])^(n + 1)*Simp[a*d - 2*b*c*(n + 1) + b*d*(2*n + 3)*Sin[e +
f*x], x])/Sqrt[a + b*Sin[e + f*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] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2985

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[(
B*c - A*d)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*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]

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^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 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 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 2984

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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^(n + 1))/(f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin
[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e +
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])

Rubi steps

\begin{align*} \int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\int \frac{\csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx+\int \frac{\csc ^6(c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^5(c+d x) \left (-\frac{a}{2}-\frac{11}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{5 a}-\frac{\int \frac{\csc (c+d x) (a-a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{2 a}\\ &=-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{\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)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^4(c+d x) \left (-\frac{87 a^2}{4}-\frac{7}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{20 a^2}-\frac{\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{2 a}+\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}+\frac{\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)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^3(c+d x) \left (\frac{45 a^3}{8}-\frac{435}{8} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{60 a^3}+\frac{\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{2 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}+\frac{\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)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^2(c+d x) \left (-\frac{1785 a^4}{16}+\frac{135}{16} a^4 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{120 a^4}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{9 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}+\frac{\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)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc (c+d x) \left (\frac{2055 a^5}{32}-\frac{1785}{32} a^5 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{120 a^5}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{9 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}+\frac{\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)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{137 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{256 a}-\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{9 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}+\frac{\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)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{137 \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{2 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 \sqrt{a} d}-\frac{9 \cot (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a+a \sin (c+d x)}}+\frac{\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)}{5 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.996352, size = 410, normalized size = 2. \[ -\frac{\csc ^{15}\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (-820 \sin \left (\frac{1}{2} (c+d x)\right )+1600 \sin \left (\frac{3}{2} (c+d x)\right )-1616 \sin \left (\frac{5}{2} (c+d x)\right )-30 \sin \left (\frac{7}{2} (c+d x)\right )-90 \sin \left (\frac{9}{2} (c+d x)\right )+820 \cos \left (\frac{1}{2} (c+d x)\right )+1600 \cos \left (\frac{3}{2} (c+d x)\right )+1616 \cos \left (\frac{5}{2} (c+d x)\right )-30 \cos \left (\frac{7}{2} (c+d x)\right )+90 \cos \left (\frac{9}{2} (c+d x)\right )+450 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-450 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-225 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+225 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+45 \sin (5 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-45 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{640 d \sqrt{a (\sin (c+d x)+1)} \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^5} \]

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]^15*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(820*Cos[(c + d*x)/2] + 1600*Cos[(3*(c + d*x))/2]
+ 1616*Cos[(5*(c + d*x))/2] - 30*Cos[(7*(c + d*x))/2] + 90*Cos[(9*(c + d*x))/2] - 820*Sin[(c + d*x)/2] + 450*L
og[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 450*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Si
n[c + d*x] + 1600*Sin[(3*(c + d*x))/2] - 1616*Sin[(5*(c + d*x))/2] - 225*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d
*x)/2]]*Sin[3*(c + d*x)] + 225*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 30*Sin[(7*(c +
d*x))/2] - 90*Sin[(9*(c + d*x))/2] + 45*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 45*Log
[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(640*d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^
5*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]  time = 1.147, size = 180, normalized size = 0.9 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{640\, \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 ( 45\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{5/2}-210\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{7/2}+45\,{\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 ) ^{5}+128\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{9/2}+210\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{11/2}-45\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{13/2} \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)^6/(a+a*sin(d*x+c))^(1/2),x)

[Out]

-1/640*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(15/2)*(45*(-a*(sin(d*x+c)-1))^(9/2)*a^(5/2)-210*(-a*(sin(d*
x+c)-1))^(7/2)*a^(7/2)+45*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^7*sin(d*x+c)^5+128*(-a*(sin(d*x+c)-1))^
(5/2)*a^(9/2)+210*(-a*(sin(d*x+c)-1))^(3/2)*a^(11/2)-45*(-a*(sin(d*x+c)-1))^(1/2)*a^(13/2))/sin(d*x+c)^5/cos(d
*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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Maxima [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, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 1.30192, size = 1273, normalized size = 6.21 \begin{align*} \frac{45 \,{\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 (45 \, \cos \left (d x + c\right )^{5} + 15 \, \cos \left (d x + c\right )^{4} + 142 \, \cos \left (d x + c\right )^{3} + 186 \, \cos \left (d x + c\right )^{2} -{\left (45 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{3} + 172 \, \cos \left (d x + c\right )^{2} - 14 \, \cos \left (d x + c\right ) - 73\right )} \sin \left (d x + c\right ) - 59 \, \cos \left (d x + c\right ) - 73\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{2560 \,{\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d -{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )\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]

1/2560*(45*(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)*sq
rt(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*(45*cos(d*x + c)^5 + 15*cos(d*x + c)^4
+ 142*cos(d*x + c)^3 + 186*cos(d*x + c)^2 - (45*cos(d*x + c)^4 + 30*cos(d*x + c)^3 + 172*cos(d*x + c)^2 - 14*c
os(d*x + c) - 73)*sin(d*x + c) - 59*cos(d*x + c) - 73)*sqrt(a*sin(d*x + c) + a))/(a*d*cos(d*x + c)^6 - 3*a*d*c
os(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d - (a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 -
2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)*sin(d*x + c))

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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)**6/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.4837, size = 1083, normalized size = 5.28 \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)^6/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/1280*(sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a)*((2*((4*tan(1/2*d*x + 1/2*c)/(a*sgn(tan(1/2*d*x + 1/2*c) + 1)) - 5/
(a*sgn(tan(1/2*d*x + 1/2*c) + 1)))*tan(1/2*d*x + 1/2*c) - 12/(a*sgn(tan(1/2*d*x + 1/2*c) + 1)))*tan(1/2*d*x +
1/2*c) + 35/(a*sgn(tan(1/2*d*x + 1/2*c) + 1)))*tan(1/2*d*x + 1/2*c) - 32/(a*sgn(tan(1/2*d*x + 1/2*c) + 1))) -
(2610*sqrt(2)*sqrt(a)*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 1305*sqrt(2)*sqrt(-a)*log(sqrt(2)*sqrt(a)
+ sqrt(a)) + 3690*sqrt(a)*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 1845*sqrt(-a)*log(sqrt(2)*sqrt(a) +
sqrt(a)) - 5058*sqrt(2)*sqrt(-a) - 7156*sqrt(-a))*sgn(tan(1/2*d*x + 1/2*c) + 1)/(29*sqrt(2)*sqrt(-a)*sqrt(a) +
41*sqrt(-a)*sqrt(a)) + 90*arctan(-(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))/sqrt(-a
))/(sqrt(-a)*sgn(tan(1/2*d*x + 1/2*c) + 1)) - 45*log(abs(-sqrt(a)*tan(1/2*d*x + 1/2*c) + sqrt(a*tan(1/2*d*x +
1/2*c)^2 + a)))/(sqrt(a)*sgn(tan(1/2*d*x + 1/2*c) + 1)) + 2*(35*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2
*d*x + 1/2*c)^2 + a))^9 - 80*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^8*sqrt(a) - 1
10*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^7*a + 240*(sqrt(a)*tan(1/2*d*x + 1/2*c)
- sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*a^(3/2) - 80*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2
*c)^2 + a))^4*a^(5/2) + 110*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^3*a^3 + 80*(sq
rt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^(7/2) - 35*(sqrt(a)*tan(1/2*d*x + 1/2*c)
- sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))*a^4 - 32*a^(9/2))/(((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x +
1/2*c)^2 + a))^2 - a)^5*sgn(tan(1/2*d*x + 1/2*c) + 1)))/d