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

Optimal. Leaf size=260 \[ -\frac{2 \sin ^5(c+d x) \cos (c+d x)}{11 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{46 \sin ^4(c+d x) \cos (c+d x)}{99 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{424 \sin ^3(c+d x) \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{200 \sin ^2(c+d x) \cos (c+d x)}{231 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1048 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 a^3 d}+\frac{4496 \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d} \]

[Out]

(-4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (4496*Cos[c + d*
x])/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (200*Cos[c + d*x]*Sin[c + d*x]^2)/(231*a^2*d*Sqrt[a + a*Sin[c + d*x
]]) - (424*Cos[c + d*x]*Sin[c + d*x]^3)/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (46*Cos[c + d*x]*Sin[c + d*x]^4
)/(99*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^5)/(11*a^2*d*Sqrt[a + a*Sin[c + d*x]]) -
(1048*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(693*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 1.35904, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2880, 2778, 2983, 2968, 3023, 2751, 2649, 206, 3046} \[ -\frac{2 \sin ^5(c+d x) \cos (c+d x)}{11 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{46 \sin ^4(c+d x) \cos (c+d x)}{99 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{424 \sin ^3(c+d x) \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{200 \sin ^2(c+d x) \cos (c+d x)}{231 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1048 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 a^3 d}+\frac{4496 \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (4496*Cos[c + d*
x])/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (200*Cos[c + d*x]*Sin[c + d*x]^2)/(231*a^2*d*Sqrt[a + a*Sin[c + d*x
]]) - (424*Cos[c + d*x]*Sin[c + d*x]^3)/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (46*Cos[c + d*x]*Sin[c + d*x]^4
)/(99*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^5)/(11*a^2*d*Sqrt[a + a*Sin[c + d*x]]) -
(1048*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(693*a^3*d)

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 2778

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp
[(-2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]]), x] - Dist[1/(b*(2*n
- 1)), Int[((c + d*Sin[e + f*x])^(n - 2)*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*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] && GtQ[n, 1] && IntegerQ[2*n]

Rule 2983

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*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(
m + n + 1)), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*b*c*(
m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*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] && GtQ[n, 0] && (I
ntegerQ[n] || EqQ[m + 1/2, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2751

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

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 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*
sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n
+ 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp
[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b
, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-
1)] && NeQ[m + n + 2, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\int \frac{\sin ^4(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{\sin ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{4 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \int \frac{\sin ^4(c+d x) \left (\frac{21 a}{2}-\frac{1}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{11 a^3}+\frac{2 \int \frac{\sin ^3(c+d x) (-8 a+a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{9 a^3}\\ &=-\frac{4 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{4 \int \frac{\sin ^3(c+d x) \left (-2 a^2+\frac{95}{2} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{99 a^4}+\frac{4 \int \frac{\sin ^2(c+d x) \left (3 a^2-\frac{57}{2} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{63 a^4}\\ &=\frac{76 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{8 \int \frac{\sin ^2(c+d x) \left (\frac{285 a^3}{2}-\frac{123}{4} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{693 a^5}+\frac{8 \int \frac{\sin (c+d x) \left (-57 a^3+\frac{87}{4} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{315 a^5}\\ &=\frac{200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{16 \int \frac{\sin (c+d x) \left (-\frac{123 a^4}{2}+\frac{2973}{8} a^4 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{3465 a^6}+\frac{8 \int \frac{-57 a^3 \sin (c+d x)+\frac{87}{4} a^3 \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{315 a^5}\\ &=\frac{200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{116 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{315 a^3 d}+\frac{16 \int \frac{-\frac{123}{2} a^4 \sin (c+d x)+\frac{2973}{8} a^4 \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{3465 a^6}+\frac{16 \int \frac{\frac{87 a^4}{8}-\frac{429}{4} a^4 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{945 a^6}\\ &=\frac{1144 \cos (c+d x)}{315 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{1048 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{693 a^3 d}+\frac{32 \int \frac{\frac{2973 a^5}{16}-\frac{3711}{8} a^5 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{10395 a^7}+\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{4496 \cos (c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{1048 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{693 a^3 d}+\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}-\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{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{4496 \cos (c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{1048 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{693 a^3 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{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{4496 \cos (c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{1048 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{693 a^3 d}\\ \end{align*}

Mathematica [C]  time = 1.05643, size = 224, normalized size = 0.86 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5 \left (-73458 \sin \left (\frac{1}{2} (c+d x)\right )-15246 \sin \left (\frac{3}{2} (c+d x)\right )+4851 \sin \left (\frac{5}{2} (c+d x)\right )+1485 \sin \left (\frac{7}{2} (c+d x)\right )-385 \sin \left (\frac{9}{2} (c+d x)\right )-63 \sin \left (\frac{11}{2} (c+d x)\right )+73458 \cos \left (\frac{1}{2} (c+d x)\right )-15246 \cos \left (\frac{3}{2} (c+d x)\right )-4851 \cos \left (\frac{5}{2} (c+d x)\right )+1485 \cos \left (\frac{7}{2} (c+d x)\right )+385 \cos \left (\frac{9}{2} (c+d x)\right )-63 \cos \left (\frac{11}{2} (c+d x)\right )+(88704+88704 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{11088 d (a (\sin (c+d x)+1))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*((88704 + 88704*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Ta
n[(c + d*x)/4])] + 73458*Cos[(c + d*x)/2] - 15246*Cos[(3*(c + d*x))/2] - 4851*Cos[(5*(c + d*x))/2] + 1485*Cos[
(7*(c + d*x))/2] + 385*Cos[(9*(c + d*x))/2] - 63*Cos[(11*(c + d*x))/2] - 73458*Sin[(c + d*x)/2] - 15246*Sin[(3
*(c + d*x))/2] + 4851*Sin[(5*(c + d*x))/2] + 1485*Sin[(7*(c + d*x))/2] - 385*Sin[(9*(c + d*x))/2] - 63*Sin[(11
*(c + d*x))/2]))/(11088*d*(a*(1 + Sin[c + d*x]))^(5/2))

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Maple [A]  time = 1.031, size = 166, normalized size = 0.6 \begin{align*} -{\frac{2+2\,\sin \left ( dx+c \right ) }{693\,{a}^{8}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 1386\,{a}^{11/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) -63\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{11/2}+154\,a \left ( a-a\sin \left ( dx+c \right ) \right ) ^{9/2}-198\,{a}^{2} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{7/2}-231\,{a}^{4} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}-1386\,{a}^{5}\sqrt{a-a\sin \left ( dx+c \right ) } \right ){\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*sin(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x)

[Out]

-2/693*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(1386*a^(11/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1
/2)/a^(1/2))-63*(a-a*sin(d*x+c))^(11/2)+154*a*(a-a*sin(d*x+c))^(9/2)-198*a^2*(a-a*sin(d*x+c))^(7/2)-231*a^4*(a
-a*sin(d*x+c))^(3/2)-1386*a^5*(a-a*sin(d*x+c))^(1/2))/a^8/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 \frac{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.19554, size = 846, normalized size = 3.25 \begin{align*} \frac{2 \,{\left (\frac{693 \, \sqrt{2}{\left (a \cos \left (d x + c\right ) + a \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}} -{\left (63 \, \cos \left (d x + c\right )^{6} - 161 \, \cos \left (d x + c\right )^{5} - 562 \, \cos \left (d x + c\right )^{4} + 622 \, \cos \left (d x + c\right )^{3} + 1759 \, \cos \left (d x + c\right )^{2} +{\left (63 \, \cos \left (d x + c\right )^{5} + 224 \, \cos \left (d x + c\right )^{4} - 338 \, \cos \left (d x + c\right )^{3} - 960 \, \cos \left (d x + c\right )^{2} + 799 \, \cos \left (d x + c\right ) + 2984\right )} \sin \left (d x + c\right ) - 2185 \, \cos \left (d x + c\right ) - 2984\right )} \sqrt{a \sin \left (d x + c\right ) + a}\right )}}{693 \,{\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/693*(693*sqrt(2)*(a*cos(d*x + c) + 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) - (63*cos(d*x + c)^6 - 161*cos(d*x + c
)^5 - 562*cos(d*x + c)^4 + 622*cos(d*x + c)^3 + 1759*cos(d*x + c)^2 + (63*cos(d*x + c)^5 + 224*cos(d*x + c)^4
- 338*cos(d*x + c)^3 - 960*cos(d*x + c)^2 + 799*cos(d*x + c) + 2984)*sin(d*x + c) - 2185*cos(d*x + c) - 2984)*
sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c) + a^3*d*sin(d*x + c) + a^3*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*sin(d*x+c)**4/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.67354, size = 637, normalized size = 2.45 \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*sin(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/709632*(5677056*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)) - (((((((((((431*sgn(tan(1/2*d*x + 1/2*c)
+ 1)*tan(1/2*d*x + 1/2*c)/a^15 - 693*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) + 2717*sgn(tan(
1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) - 3927*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*
c) + 7326*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) - 8778*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*
tan(1/2*d*x + 1/2*c) + 8778*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) - 7326*sgn(tan(1/2*d*x +
1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) + 3927*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) - 2717*
sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x + 1/2*c) + 693*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)*tan(1/2*d*x
+ 1/2*c) - 431*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^15)/(a*tan(1/2*d*x + 1/2*c)^2 + a)^(11/2) - 2*sqrt(2)*(2838528
*a^(39/2)*arctan(sqrt(a)/sqrt(-a)) + 373*sqrt(-a)*a)*sgn(tan(1/2*d*x + 1/2*c) + 1)/(sqrt(-a)*a^(43/2)))/d