3.489 \(\int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=229 \[ -\frac {363 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{64 a^{5/2} d}+\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d}+\frac {149 \cot (c+d x)}{64 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt {a \sin (c+d x)+a}} \]

[Out]

-363/64*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/a^(5/2)/d+4*arctanh(1/2*cos(d*x+c)*a^(1/2)*2^(1/2)/
(a+a*sin(d*x+c))^(1/2))/a^(5/2)/d*2^(1/2)+149/64*cot(d*x+c)/a^2/d/(a+a*sin(d*x+c))^(1/2)-107/96*cot(d*x+c)*csc
(d*x+c)/a^2/d/(a+a*sin(d*x+c))^(1/2)+17/24*cot(d*x+c)*csc(d*x+c)^2/a^2/d/(a+a*sin(d*x+c))^(1/2)-1/4*cot(d*x+c)
*csc(d*x+c)^3/a^2/d/(a+a*sin(d*x+c))^(1/2)

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Rubi [A]  time = 1.31, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2880, 2779, 2984, 2985, 2649, 206, 2773, 3044} \[ \frac {149 \cot (c+d x)}{64 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {363 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{64 a^{5/2} d}+\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt {a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-363*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*a^(5/2)*d) + (4*Sqrt[2]*ArcTanh[(Sqrt[a]*C
os[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (149*Cot[c + d*x])/(64*a^2*d*Sqrt[a + a*Sin[c
+ d*x]]) - (107*Cot[c + d*x]*Csc[c + d*x])/(96*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (17*Cot[c + d*x]*Csc[c + d*x]
^2)/(24*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]])

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 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 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 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 2880

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Dist[-2/(a*b*d), Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 2), x], x] + Dist[1/a^2
, Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}
, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]

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 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
])

Rubi steps

\begin {align*} \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {\int \frac {\csc ^5(c+d x) \left (1+\sin ^2(c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2}-\frac {2 \int \frac {\csc ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac {2 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^4(c+d x) \left (-\frac {a}{2}+\frac {15}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{4 a^3}+\frac {\int \frac {\csc ^3(c+d x) (a-5 a \sin (c+d x))}{\sqrt {a+a \sin (c+d x)}} \, dx}{3 a^3}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{6 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^3(c+d x) \left (\frac {91 a^2}{4}-\frac {5}{4} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{12 a^4}+\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {21 a^2}{2}+\frac {3}{2} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{6 a^4}\\ &=\frac {7 \cot (c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {111 a^3}{8}+\frac {273}{8} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{24 a^5}+\frac {\int \frac {\csc (c+d x) \left (\frac {27 a^3}{4}-\frac {21}{4} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{6 a^5}\\ &=\frac {149 \cot (c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc (c+d x) \left (\frac {657 a^4}{16}-\frac {111}{16} a^4 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{24 a^6}+\frac {9 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{8 a^3}-\frac {2 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac {149 \cot (c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {219 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{128 a^3}-\frac {2 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2}-\frac {9 \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 {4 \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 )}{a^2 d}\\ &=-\frac {9 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 a^{5/2} d}+\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{5/2} d}+\frac {149 \cot (c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {219 \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 a^2 d}+\frac {4 \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 )}{a^2 d}\\ &=-\frac {363 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 a^{5/2} d}+\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{5/2} d}+\frac {149 \cot (c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 5.12, size = 414, normalized size = 1.81 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (-\frac {16 \csc ^{12}\left (\frac {1}{2} (c+d x)\right ) \left (-6250 \sin \left (\frac {1}{2} (c+d x)\right )-4626 \sin \left (\frac {3}{2} (c+d x)\right )+1750 \sin \left (\frac {5}{2} (c+d x)\right )+894 \sin \left (\frac {7}{2} (c+d x)\right )+6250 \cos \left (\frac {1}{2} (c+d x)\right )-4626 \cos \left (\frac {3}{2} (c+d x)\right )-1750 \cos \left (\frac {5}{2} (c+d x)\right )+894 \cos \left (\frac {7}{2} (c+d x)\right )-4356 \cos (2 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+1089 \cos (4 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+3267 \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+4356 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-1089 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-3267 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{\left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^4}-(24576+24576 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (c+d x)\right )-1\right )\right )\right )}{3072 d (a (\sin (c+d x)+1))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*((-24576 - 24576*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + T
an[(c + d*x)/4])] - (16*Csc[(c + d*x)/2]^12*(6250*Cos[(c + d*x)/2] - 4626*Cos[(3*(c + d*x))/2] - 1750*Cos[(5*(
c + d*x))/2] + 894*Cos[(7*(c + d*x))/2] + 3267*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4356*Cos[2*(c +
d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 1089*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c +
d*x)/2]] - 3267*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4356*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2]] - 1089*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6250*Sin[(c + d*x)
/2] - 4626*Sin[(3*(c + d*x))/2] + 1750*Sin[(5*(c + d*x))/2] + 894*Sin[(7*(c + d*x))/2]))/(Csc[(c + d*x)/4]^2 -
Sec[(c + d*x)/4]^2)^4))/(3072*d*(a*(1 + Sin[c + d*x]))^(5/2))

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fricas [B]  time = 0.52, size = 643, normalized size = 2.81 \[ \frac {1089 \, {\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 ) + \frac {1536 \, \sqrt {2} {\left (a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - 4 \, {\left (447 \, \cos \left (d x + c\right )^{4} - 214 \, \cos \left (d x + c\right )^{3} - 1244 \, \cos \left (d x + c\right )^{2} + {\left (447 \, \cos \left (d x + c\right )^{3} + 661 \, \cos \left (d x + c\right )^{2} - 583 \, \cos \left (d x + c\right ) - 845\right )} \sin \left (d x + c\right ) + 262 \, \cos \left (d x + c\right ) + 845\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{768 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right ) + a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]

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))^(5/2),x, algorithm="fricas")

[Out]

1/768*(1089*(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)) + 1536*sqrt(2)*(a*cos(d*x + c)^5 + a*cos(d*x + c)^4 - 2*a*cos
(d*x + c)^3 - 2*a*cos(d*x + c)^2 + a*cos(d*x + c) + (a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*sin(d*x + c) +
a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c)
- sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x
+ c) - 2))/sqrt(a) - 4*(447*cos(d*x + c)^4 - 214*cos(d*x + c)^3 - 1244*cos(d*x + c)^2 + (447*cos(d*x + c)^3 +
661*cos(d*x + c)^2 - 583*cos(d*x + c) - 845)*sin(d*x + c) + 262*cos(d*x + c) + 845)*sqrt(a*sin(d*x + c) + a))/
(a^3*d*cos(d*x + c)^5 + a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x + c)^2 + a^3*d*cos(d*x
+ c) + a^3*d + (a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c))

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giac [B]  time = 1.29, size = 845, normalized size = 3.69 \[ \text {result too large to display} \]

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))^(5/2),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^3*sgn(tan(1/2*d*x + 1/2*c) + 1)) - 20
/(a^3*sgn(tan(1/2*d*x + 1/2*c) + 1)))*tan(1/2*d*x + 1/2*c) + 159/(a^3*sgn(tan(1/2*d*x + 1/2*c) + 1)))*tan(1/2*
d*x + 1/2*c) - 640/(a^3*sgn(tan(1/2*d*x + 1/2*c) + 1))) - (37026*sqrt(2)*sqrt(a)*arctan((sqrt(2)*sqrt(a) + sqr
t(a))/sqrt(-a)) - 73728*sqrt(2)*sqrt(a)*arctan(sqrt(a)/sqrt(-a)) - 18513*sqrt(2)*sqrt(-a)*log(sqrt(2)*sqrt(a)
+ sqrt(a)) + 52272*sqrt(a)*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 104448*sqrt(a)*arctan(sqrt(a)/sqrt(-
a)) - 26136*sqrt(-a)*log(sqrt(2)*sqrt(a) + sqrt(a)) - 29680*sqrt(2)*sqrt(-a) - 42100*sqrt(-a))*sgn(tan(1/2*d*x
+ 1/2*c) + 1)/(17*sqrt(2)*sqrt(-a)*a^(5/2) + 24*sqrt(-a)*a^(5/2)) - 3072*sqrt(2)*arctan(-1/2*sqrt(2)*(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))/(sqrt(-a)*a^2*sgn(tan(1/2*d*x
+ 1/2*c) + 1)) + 2178*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))/(s
qrt(-a)*a^2*sgn(tan(1/2*d*x + 1/2*c) + 1)) - 1089*log(abs(-sqrt(a)*tan(1/2*d*x + 1/2*c) + sqrt(a*tan(1/2*d*x +
1/2*c)^2 + a)))/(a^(5/2)*sgn(tan(1/2*d*x + 1/2*c) + 1)) + 2*(159*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1
/2*d*x + 1/2*c)^2 + a))^7 - 720*(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)
- 135*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^5*a + 1920*(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) - 135*(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 - 1840*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^(5/2) +
159*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))*a^3 + 640*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*a^2*sgn(tan(1/2*d*x + 1/2*c) + 1)))/d

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maple [A]  time = 1.40, size = 200, normalized size = 0.87 \[ \frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (321 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {13}{2}}-1049 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {11}{2}}+1127 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {9}{2}}-447 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {7}{2}}+768 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{7} \left (\sin ^{4}\left (d x +c \right )\right )-1089 a^{7} \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{4}\left (d x +c \right )\right )\right )}{192 a^{\frac {19}{2}} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]

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))^(5/2),x)

[Out]

1/192*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(321*(-a*(sin(d*x+c)-1))^(1/2)*a^(13/2)-1049*(-a*(sin(d*x+c)-1)
)^(3/2)*a^(11/2)+1127*(-a*(sin(d*x+c)-1))^(5/2)*a^(9/2)-447*(-a*(sin(d*x+c)-1))^(7/2)*a^(7/2)+768*2^(1/2)*arct
anh(1/2*(-a*(sin(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2))*a^7*sin(d*x+c)^4-1089*a^7*arctanh((-a*(sin(d*x+c)-1))^(1/2)
/a^(1/2))*sin(d*x+c)^4)/a^(19/2)/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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4}{{\sin \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4/(sin(c + d*x)^5*(a + a*sin(c + d*x))^(5/2)),x)

[Out]

int(cos(c + d*x)^4/(sin(c + d*x)^5*(a + a*sin(c + d*x))^(5/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**(5/2),x)

[Out]

Timed out

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