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

Optimal. Leaf size=170 \[ -\frac{11 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{11 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*Sqrt[a]*d) - (11*Cot[c + d*x])/(64*d*Sqrt[a
+ a*Sin[c + d*x]]) + (53*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d
*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*d*Sqrt[a + a*Sin[c + d*x]])

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Rubi [A]  time = 0.886759, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2881, 2780, 2649, 206, 2773, 3044, 2984, 2985} \[ -\frac{11 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{11 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*Sqrt[a]*d) - (11*Cot[c + d*x])/(64*d*Sqrt[a
+ a*Sin[c + d*x]]) + (53*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d
*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*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 2780

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

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]

Rubi steps

\begin{align*} \int \frac{\cot ^4(c+d x) \csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\int \frac{\csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx+\int \frac{\csc ^5(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) \csc ^3(c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^4(c+d x) \left (-\frac{a}{2}-\frac{9}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{4 a}+\frac{\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{a}-\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^3(c+d x) \left (-\frac{53 a^2}{4}-\frac{5}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{12 a^2}-\frac{2 \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{2 \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{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^2(c+d x) \left (\frac{33 a^3}{8}-\frac{159}{8} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{24 a^3}\\ &=-\frac{2 \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{11 \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc (c+d x) \left (-\frac{351 a^4}{16}+\frac{33}{16} a^4 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{24 a^4}\\ &=-\frac{2 \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{11 \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{117 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{128 a}+\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{2 \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{11 \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{117 \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 )}{64 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{11 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 \sqrt{a} d}-\frac{11 \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [B]  time = 0.88838, size = 374, normalized size = 2.2 \[ \frac{\csc ^{12}\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (-214 \sin \left (\frac{1}{2} (c+d x)\right )-558 \sin \left (\frac{3}{2} (c+d x)\right )+490 \sin \left (\frac{5}{2} (c+d x)\right )+66 \sin \left (\frac{7}{2} (c+d x)\right )+214 \cos \left (\frac{1}{2} (c+d x)\right )-558 \cos \left (\frac{3}{2} (c+d x)\right )-490 \cos \left (\frac{5}{2} (c+d x)\right )+66 \cos \left (\frac{7}{2} (c+d x)\right )+132 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-33 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-99 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-132 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+33 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+99 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{192 d \sqrt{a (\sin (c+d x)+1)} \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[(c + d*x)/2]^12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(214*Cos[(c + d*x)/2] - 558*Cos[(3*(c + d*x))/2] -
490*Cos[(5*(c + d*x))/2] + 66*Cos[(7*(c + d*x))/2] - 99*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 132*Cos
[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 33*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Si
n[(c + d*x)/2]] + 99*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 132*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)
/2] + Sin[(c + d*x)/2]] + 33*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 214*Sin[(c + d*x)
/2] - 558*Sin[(3*(c + d*x))/2] + 490*Sin[(5*(c + d*x))/2] + 66*Sin[(7*(c + d*x))/2]))/(192*d*(Csc[(c + d*x)/4]
^2 - Sec[(c + d*x)/4]^2)^4*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]  time = 1.016, size = 162, normalized size = 1. \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{192\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 33\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{3/2}-33\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{4}+7\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{5/2}-121\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{7/2}+33\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{9/2} \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)^5/(a+a*sin(d*x+c))^(1/2),x)

[Out]

1/192*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(11/2)*(33*(-a*(sin(d*x+c)-1))^(7/2)*a^(3/2)-33*arctanh((-a*(
sin(d*x+c)-1))^(1/2)/a^(1/2))*a^5*sin(d*x+c)^4+7*(-a*(sin(d*x+c)-1))^(5/2)*a^(5/2)-121*(-a*(sin(d*x+c)-1))^(3/
2)*a^(7/2)+33*(-a*(sin(d*x+c)-1))^(1/2)*a^(9/2))/sin(d*x+c)^4/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)^5/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 1.1929, size = 1161, normalized size = 6.83 \begin{align*} \frac{33 \,{\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} +{\left (\cos \left (d x + c\right )^{4} - 2 \, \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 (33 \, \cos \left (d x + c\right )^{4} - 106 \, \cos \left (d x + c\right )^{3} - 164 \, \cos \left (d x + c\right )^{2} +{\left (33 \, \cos \left (d x + c\right )^{3} + 139 \, \cos \left (d x + c\right )^{2} - 25 \, \cos \left (d x + c\right ) - 83\right )} \sin \left (d x + c\right ) + 58 \, \cos \left (d x + c\right ) + 83\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{768 \,{\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 +{\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + 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)^5/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/768*(33*(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)^4 - 2*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)*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*(33*cos(d*x + c)^4 - 106*cos(d*x + c)^3 - 164*cos(d*x + c)^
2 + (33*cos(d*x + c)^3 + 139*cos(d*x + c)^2 - 25*cos(d*x + c) - 83)*sin(d*x + c) + 58*cos(d*x + c) + 83)*sqrt(
a*sin(d*x + c) + a))/(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 + (a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + 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)**5/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.41396, size = 994, normalized size = 5.85 \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)^5/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/384*(sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a)*((2*(3*tan(1/2*d*x + 1/2*c)/(a*sgn(tan(1/2*d*x + 1/2*c) + 1)) - 4/(a
*sgn(tan(1/2*d*x + 1/2*c) + 1)))*tan(1/2*d*x + 1/2*c) - 33/(a*sgn(tan(1/2*d*x + 1/2*c) + 1)))*tan(1/2*d*x + 1/
2*c) + 64/(a*sgn(tan(1/2*d*x + 1/2*c) + 1))) - (792*sqrt(2)*a^(3/2)*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a
)) - 396*sqrt(2)*sqrt(-a)*a*log(sqrt(2)*sqrt(a) + sqrt(a)) + 1122*a^(3/2)*arctan((sqrt(2)*sqrt(a) + sqrt(a))/s
qrt(-a)) - 561*sqrt(-a)*a*log(sqrt(2)*sqrt(a) + sqrt(a)) + 2054*sqrt(2)*sqrt(-a)*a + 2896*sqrt(-a)*a)*sgn(tan(
1/2*d*x + 1/2*c) + 1)/(12*sqrt(2)*sqrt(-a)*a^(3/2) + 17*sqrt(-a)*a^(3/2)) + 66*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)) - 33*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*(33*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^7 - 48*(sqrt(a)*tan(1/2*d*x + 1/2*c)
- sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*sqrt(a) - 57*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2
*c)^2 + a))^5*a + 192*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^4*a^(3/2) - 57*(sqrt
(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^3*a^2 - 208*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqr
t(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^(5/2) + 33*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2
+ a))*a^3 + 64*a^(7/2))/(((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^4*sgn(tan
(1/2*d*x + 1/2*c) + 1)))/d