3.84 \(\int \frac{\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=224 \[ \frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{39 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 a^{5/2} d}+\frac{219 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}} \]

[Out]

(-39*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(5/2)*d) + (219*ArcTanh[(Sqrt[a]*Cos[c + d
*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(4*d*(a + a*Sin
[c + d*x])^(5/2)) + (19*Cot[c + d*x]*Csc[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) + (63*Cot[c + d*x])/(16
*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (31*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.660126, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2766, 2978, 2984, 2985, 2649, 206, 2773} \[ \frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{39 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 a^{5/2} d}+\frac{219 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-39*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(5/2)*d) + (219*ArcTanh[(Sqrt[a]*Cos[c + d
*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(4*d*(a + a*Sin
[c + d*x])^(5/2)) + (19*Cot[c + d*x]*Csc[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) + (63*Cot[c + d*x])/(16
*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (31*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]])

Rule 2766

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dis
t[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*
(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d,
0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] &&  !GtQ[n, 0] && (IntegersQ[2*m, 2*n] || (IntegerQ
[m] && EqQ[c, 0]))

Rule 2978

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

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]

Rubi steps

\begin{align*} \int \frac{\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{\int \frac{\csc ^3(c+d x) \left (6 a-\frac{7}{2} a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{\int \frac{\csc ^3(c+d x) \left (31 a^2-\frac{95}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{8 a^4}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^2(c+d x) \left (-63 a^3+\frac{93}{2} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{16 a^5}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc (c+d x) \left (78 a^4-\frac{63}{2} a^4 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{16 a^6}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{39 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{8 a^3}-\frac{219 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{32 a^2}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{39 \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 a^2 d}+\frac{219 \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 )}{16 a^2 d}\\ &=-\frac{39 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 a^{5/2} d}+\frac{219 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 1.07375, size = 680, normalized size = 3.04 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (-16 \sin \left (\frac{1}{2} (c+d x)\right )-\frac{40 \sin \left (\frac{1}{4} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}{\cos \left (\frac{1}{4} (c+d x)\right )-\sin \left (\frac{1}{4} (c+d x)\right )}+\frac{40 \sin \left (\frac{1}{4} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}{\sin \left (\frac{1}{4} (c+d x)\right )+\cos \left (\frac{1}{4} (c+d x)\right )}+\frac{2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}{\left (\cos \left (\frac{1}{4} (c+d x)\right )-\sin \left (\frac{1}{4} (c+d x)\right )\right )^2}-\frac{2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}{\left (\sin \left (\frac{1}{4} (c+d x)\right )+\cos \left (\frac{1}{4} (c+d x)\right )\right )^2}-40 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4+54 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3-108 \sin \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 )^2+8 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-156 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+156 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+20 \tan \left (\frac{1}{4} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4+20 \cot \left (\frac{1}{4} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4-\csc ^2\left (\frac{1}{4} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4+\sec ^2\left (\frac{1}{4} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4-(438+438 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{32 d (a (\sin (c+d x)+1))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-16*Sin[(c + d*x)/2] + 8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 108*S
in[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 54*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 40*(Cos
[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - (438 + 438*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d
*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 20*Cot[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4
- Csc[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 156*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2
]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 156*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2
] + Sin[(c + d*x)/2])^4 + Sec[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (2*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (40*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^4)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - (2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d
*x)/4] + Sin[(c + d*x)/4])^2 + (40*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4]
+ Sin[(c + d*x)/4]) + 20*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Tan[(c + d*x)/4]))/(32*d*(a*(1 + Sin[c + d*x
]))^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.853, size = 404, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( 32+32\,\sin \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) d} \left ( -219\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{2}+312\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{2}-438\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}{a}^{2}+126\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}\sqrt{a} \left ( \sin \left ( dx+c \right ) \right ) ^{2}+624\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}{a}^{2}-219\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}+144\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}\sqrt{a}\sin \left ( dx+c \right ) -172\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{3/2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}+312\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}+72\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}\sqrt{a}-112\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{3/2}\sin \left ( dx+c \right ) -56\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{3/2} \right ) \sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{-{\frac{9}{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(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x)

[Out]

-1/32/a^(9/2)*(-219*2^(1/2)*arctanh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(d*x+c)^4*a^2+312*arctan
h((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+c)^4*a^2-438*2^(1/2)*arctanh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2
)/a^(1/2))*sin(d*x+c)^3*a^2+126*(-a*(sin(d*x+c)-1))^(3/2)*a^(1/2)*sin(d*x+c)^2+624*arctanh((-a*(sin(d*x+c)-1))
^(1/2)/a^(1/2))*sin(d*x+c)^3*a^2-219*2^(1/2)*arctanh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(d*x+c)
^2*a^2+144*(-a*(sin(d*x+c)-1))^(3/2)*a^(1/2)*sin(d*x+c)-172*(-a*(sin(d*x+c)-1))^(1/2)*a^(3/2)*sin(d*x+c)^2+312
*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+c)^2*a^2+72*(-a*(sin(d*x+c)-1))^(3/2)*a^(1/2)-112*(-a*(sin
(d*x+c)-1))^(1/2)*a^(3/2)*sin(d*x+c)-56*(-a*(sin(d*x+c)-1))^(1/2)*a^(3/2))*(-a*(sin(d*x+c)-1))^(1/2)/(1+sin(d*
x+c))/sin(d*x+c)^2/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

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(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [B]  time = 2.07802, size = 1909, normalized size = 8.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/64*(219*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 - 3*cos(d*x + c)^3 - 7*cos(d*x + c)^2 + (cos(d*x + c)^4 -
2*cos(d*x + c)^3 - 5*cos(d*x + c)^2 + 2*cos(d*x + c) + 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*sqrt(a)*log(-(a*
cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x +
c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x +
c) - 2)) + 156*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 - 3*cos(d*x + c)^3 - 7*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*
cos(d*x + c)^3 - 5*cos(d*x + c)^2 + 2*cos(d*x + c) + 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*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)*sq
rt(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) - cos(d*x + c) - 1)) - 4*(63*cos(d*x +
c)^4 + 95*cos(d*x + c)^3 - 51*cos(d*x + c)^2 + (63*cos(d*x + c)^3 - 32*cos(d*x + c)^2 - 83*cos(d*x + c) + 4)*s
in(d*x + c) - 87*cos(d*x + c) - 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 -
3*a^3*d*cos(d*x + c)^3 - 7*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d + (a^3*d*cos(d*x + c)^4 - 2*a
^3*d*cos(d*x + c)^3 - 5*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d)*sin(d*x + c))

________________________________________________________________________________________

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(csc(d*x+c)**3/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

sage2