3.563 \(\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=400 \[ -\frac{3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 a^{5/2} f (c-d)^5 (c+d)^{5/2}}-\frac{3 \left (c^2-10 c d+73 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{16 \sqrt{2} a^{5/2} f (c-d)^5}-\frac{(3 c-19 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2} \]

[Out]

(-3*(c^2 - 10*c*d + 73*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^
(5/2)*(c - d)^5*f) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]
*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e +
f*x])^(5/2)*(c + d*Sin[e + f*x])^2) - ((3*c - 19*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)
*(c + d*Sin[e + f*x])^2) - (d*(3*c^2 - 20*c*d - 31*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*S
in[e + f*x]]*(c + d*Sin[e + f*x])^2) - (3*d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^4*(
c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.51505, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2766, 2978, 2984, 2985, 2649, 206, 2773, 208} \[ -\frac{3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 a^{5/2} f (c-d)^5 (c+d)^{5/2}}-\frac{3 \left (c^2-10 c d+73 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{16 \sqrt{2} a^{5/2} f (c-d)^5}-\frac{(3 c-19 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]

[Out]

(-3*(c^2 - 10*c*d + 73*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^
(5/2)*(c - d)^5*f) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]
*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e +
f*x])^(5/2)*(c + d*Sin[e + f*x])^2) - ((3*c - 19*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)
*(c + d*Sin[e + f*x])^2) - (d*(3*c^2 - 20*c*d - 31*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*S
in[e + f*x]]*(c + d*Sin[e + f*x])^2) - (3*d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^4*(
c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*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]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac{\int \frac{-\frac{3}{2} a (c-4 d)-\frac{7}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx}{4 a^2 (c-d)}\\ &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac{(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac{\int \frac{\frac{1}{4} a^2 \left (3 c^2-15 c d+124 d^2\right )+\frac{5}{4} a^2 (3 c-19 d) d \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx}{8 a^4 (c-d)^2}\\ &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac{(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac{d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{\int \frac{-\frac{3}{2} a^3 \left (c^3-6 c^2 d+43 c d^2+42 d^3\right )-\frac{3}{2} a^3 d \left (3 c^2-20 c d-31 d^2\right ) \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx}{16 a^5 (c-d)^3 (c+d)}\\ &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac{(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac{d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac{\int \frac{\frac{3}{2} a^4 \left (c^4-7 c^3 d+47 c^2 d^2+99 c d^3+52 d^4\right )+\frac{3}{2} a^4 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{16 a^6 (c-d)^4 (c+d)^2}\\ &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac{(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac{d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac{\left (3 d^3 \left (21 c^2+30 c d+13 d^2\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{8 a^3 (c-d)^5 (c+d)^2}+\frac{\left (3 \left (c^2-10 c d+73 d^2\right )\right ) \int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx}{32 a^2 (c-d)^5}\\ &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac{(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac{d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac{\left (3 d^3 \left (21 c^2+30 c d+13 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a c+a d-d x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{4 a^2 (c-d)^5 (c+d)^2 f}-\frac{\left (3 \left (c^2-10 c d+73 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{16 a^2 (c-d)^5 f}\\ &=-\frac{3 \left (c^2-10 c d+73 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a+a \sin (e+f x)}}\right )}{16 \sqrt{2} a^{5/2} (c-d)^5 f}+\frac{3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac{(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac{d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac{3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}\\ \end{align*}

Mathematica [C]  time = 9.30076, size = 958, normalized size = 2.4 \[ \frac{3 \left (3 \cos \left (\frac{1}{2} (e+f x)\right ) d^4-3 \sin \left (\frac{1}{2} (e+f x)\right ) d^4+5 c \cos \left (\frac{1}{2} (e+f x)\right ) d^3-5 c \sin \left (\frac{1}{2} (e+f x)\right ) d^3\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{4 (c-d)^4 (c+d)^2 f (a (\sin (e+f x)+1))^{5/2} (c+d \sin (e+f x))}+\frac{\left (d^3 \cos \left (\frac{1}{2} (e+f x)\right )-d^3 \sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{2 (c-d)^3 (c+d) f (a (\sin (e+f x)+1))^{5/2} (c+d \sin (e+f x))^2}+\frac{(3+3 i) \left (c^2-10 d c+73 d^2\right ) \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \sec \left (\frac{1}{4} (e+f x)\right ) \left (\cos \left (\frac{1}{4} (e+f x)\right )-\sin \left (\frac{1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{\left (16 \sqrt [4]{-1} c^5-80 \sqrt [4]{-1} d c^4+160 \sqrt [4]{-1} d^2 c^3-160 \sqrt [4]{-1} d^3 c^2+80 \sqrt [4]{-1} d^4 c-16 \sqrt [4]{-1} d^5\right ) f (a (\sin (e+f x)+1))^{5/2}}+\frac{3 d^{5/2} \left (21 c^2+30 d c+13 d^2\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (\sqrt{d} \cos \left (\frac{1}{2} (e+f x)\right )-\sqrt{d} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{c+d}\right )\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{16 (c-d)^5 (c+d)^{5/2} f (a (\sin (e+f x)+1))^{5/2}}+\frac{3 d^{5/2} \left (21 c^2+30 d c+13 d^2\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (-\sqrt{d} \cos \left (\frac{1}{2} (e+f x)\right )+\sqrt{d} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{c+d}\right )\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{16 (d-c)^5 (c+d)^{5/2} f (a (\sin (e+f x)+1))^{5/2}}-\frac{3 (c-9 d) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}{16 (c-d)^4 f (a (\sin (e+f x)+1))^{5/2}}+\frac{3 \left (c \sin \left (\frac{1}{2} (e+f x)\right )-9 d \sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}{8 (c-d)^4 f (a (\sin (e+f x)+1))^{5/2}}-\frac{\left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}{4 (c-d)^3 f (a (\sin (e+f x)+1))^{5/2}}+\frac{\sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )}{2 (c-d)^3 f (a (\sin (e+f x)+1))^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]

[Out]

(Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(2*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(5/2)) - (Cos[(
e + f*x)/2] + Sin[(e + f*x)/2])^2/(4*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(5/2)) - (3*(c - 9*d)*(Cos[(e + f*x)/2
] + Sin[(e + f*x)/2])^4)/(16*(c - d)^4*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((3 + 3*I)*(c^2 - 10*c*d + 73*d^2)*Ar
cTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(
e + f*x)/2])^5)/((16*(-1)^(1/4)*c^5 - 80*(-1)^(1/4)*c^4*d + 160*(-1)^(1/4)*c^3*d^2 - 160*(-1)^(1/4)*c^2*d^3 +
80*(-1)^(1/4)*c*d^4 - 16*(-1)^(1/4)*d^5)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^
2)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - S
qrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(16*(c - d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[
e + f*x]))^(5/2)) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e
+ f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])^5)/(16*(-c + d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (3*(Cos[(e + f*x)/2] + Sin[(e + f*
x)/2])^3*(c*Sin[(e + f*x)/2] - 9*d*Sin[(e + f*x)/2]))/(8*(c - d)^4*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2])^5*(d^3*Cos[(e + f*x)/2] - d^3*Sin[(e + f*x)/2]))/(2*(c - d)^3*(c + d)*f*(a*(1 +
Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x])^2) + (3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(5*c*d^3*Cos[(e + f*
x)/2] + 3*d^4*Cos[(e + f*x)/2] - 5*c*d^3*Sin[(e + f*x)/2] - 3*d^4*Sin[(e + f*x)/2]))/(4*(c - d)^4*(c + d)^2*f*
(a*(1 + Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x]))

________________________________________________________________________________________

Maple [B]  time = 2.497, size = 3535, normalized size = 8.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x)

[Out]

-1/32*(-a*(-1+sin(f*x+e)))^(1/2)*(408*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)
/a^(1/2))*a^2*c^3*d^3+20*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^6+56*(-a*(-1+sin(f*x+e)))^(1/2
)*(a*(c+d)*d)^(1/2)*a^(3/2)*d^6-312*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)
^4*d^7-2448*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c^2*d^5-2064*arctanh(
(-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c*d^6-504*arctanh((-a*(-1+sin(f*x+e)))^(1
/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c^4*d^3-2736*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1
/2))*a^(5/2)*sin(f*x+e)^2*c^3*d^4-3696*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x
+e)^2*c^2*d^5-1968*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c*d^6+172*(-a*
(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*d^6-1008*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a
*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c^4*d^3-2448*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/
2)*sin(f*x+e)*c^3*d^4+69*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin
(f*x+e)^2*a^2*c^4*d^2+1032*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*s
in(f*x+e)^2*a^2*c^3*d^3+2013*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))
*sin(f*x+e)^2*a^2*c^2*d^4+1284*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2
))*sin(f*x+e)^2*a^2*c*d^5+3*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*
sin(f*x+e)^4*a^2*c^4*d^2-24*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*
sin(f*x+e)^4*a^2*c^3*d^3+162*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))
*sin(f*x+e)^4*a^2*c^2*d^4-624*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*d^7
-312*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*d^7-6*(-a*(-1+sin(f*x+e)))^(
3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^6-72*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*d^6-504*arctanh((-a
*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*c^4*d^3-720*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)
*d)^(1/2))*a^(5/2)*c^3*d^4-312*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*c^2*d^5+60*(-a*
(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^4*d^2-48*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/
2)*c^3*d^3+66*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^2*d^4-48*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c
+d)*d)^(1/2)*a^(1/2)*c*d^5+3*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))
*a^2*c^6-2064*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c^2*d^5-624*arctanh((
-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c*d^6+112*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d
)*d)^(1/2)*a^(3/2)*sin(f*x+e)*d^6-96*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^5*d-136*(-a*(-1+si
n(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^4*d^2-40*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^3
*d^3+60*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^2*d^4+136*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d
)^(1/2)*a^(3/2)*c*d^5-504*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^4*c^2*d^5
-720*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^4*c*d^6-1008*arctanh((-a*(-1+s
in(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c^3*d^4-126*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)
^(1/2)*a^(1/2)*sin(f*x+e)^2*d^6-144*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*d^6+48*(-a
*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^5*d+408*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(
f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^4*a^2*c*d^5+6*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x
+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^5*d-42*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e
)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^4*d^2-42*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e
)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^5*d+276*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))
^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^4*d^2+1140*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))
^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^3*d^3+1254*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))
^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^2*d^4+438*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^
(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c*d^5+276*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/
2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^3*d^3+1140*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(
1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c^2*d^4+1254*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))
^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*c*d^5-12*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(
1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^5*d+219*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1
/2)*2^(1/2)/a^(1/2))*a^2*c^2*d^4+219*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/
a^(1/2))*sin(f*x+e)^4*a^2*d^6-6*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)^2*c^4*d^2+48*(
-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)^2*c^3*d^3+180*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+
d)*d)^(1/2)*a^(1/2)*sin(f*x+e)^2*c^2*d^4-96*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)^2*
c*d^5+3*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*c^6
+219*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*d^6+20
*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*c^4*d^2-12*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c
+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^5*d+96*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^4*d
^2+120*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^3*d^3+144*(-a*(-1+sin(f*x+e)))^(3/2)*
(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^2*d^4-204*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+
e)*c*d^5+6*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^
6-232*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*c^3*d^3-192*(-a*(-1+sin(f*x+e)))^(1/2)
*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*c^2*d^4+232*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f
*x+e)^2*c*d^5+40*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c^5*d-192*(-a*(-1+sin(f*x+e))
)^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c^4*d^2-544*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*
sin(f*x+e)*c^3*d^3+80*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c^2*d^4+504*(-a*(-1+sin(
f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c*d^5-24*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin
(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^5*d+162*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2
)*2^(1/2)/a^(1/2))*a^2*c^4*d^2+438*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^
(1/2))*sin(f*x+e)^3*a^2*d^6)/a^(9/2)/(1+sin(f*x+e))/(a*(c+d)*d)^(1/2)/(c+d*sin(f*x+e))^2/(c+d)^2/(c-d)^5/cos(f
*x+e)/(a+a*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [B]  time = 24.1367, size = 13377, normalized size = 33.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[-1/64*(3*sqrt(2)*(4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*d^5 + 292*d^6 + (c^4*d
^2 - 8*c^3*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^5 + (2*c^5*d - 13*c^4*d^2 + 84*c^3*d^3 + 434*c^
2*d^4 + 554*c*d^5 + 219*d^6)*cos(f*x + e)^4 - (c^6 - 4*c^5*d + 25*c^4*d^2 + 328*c^3*d^3 + 779*c^2*d^4 + 700*c*
d^5 + 219*d^6)*cos(f*x + e)^3 - (3*c^6 - 14*c^5*d + 89*c^4*d^2 + 892*c^3*d^3 + 1957*c^2*d^4 + 1682*c*d^5 + 511
*d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c*d^5 + 73*d^6)*cos(f*x
+ e) + (4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*d^5 + 292*d^6 + (c^4*d^2 - 8*c^3
*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^4 - 2*(c^5*d - 7*c^4*d^2 + 46*c^3*d^3 + 190*c^2*d^4 + 209
*c*d^5 + 73*d^6)*cos(f*x + e)^3 - (c^6 - 2*c^5*d + 11*c^4*d^2 + 420*c^3*d^3 + 1159*c^2*d^4 + 1118*c*d^5 + 365*
d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c*d^5 + 73*d^6)*cos(f*x
+ e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e)
- sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x
+ e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 12*(84*a*c^4*d^2 + 288*a*c^3*d^3 + 376*a*c^2*d^4 + 224*a*c*d^5
+ 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^5 + (42*a*c^3*d^3 + 123*a*c^2*d^4 + 116*a*c*d
^5 + 39*a*d^6)*cos(f*x + e)^4 - (21*a*c^4*d^2 + 114*a*c^3*d^3 + 196*a*c^2*d^4 + 142*a*c*d^5 + 39*a*d^6)*cos(f*
x + e)^3 - (63*a*c^4*d^2 + 300*a*c^3*d^3 + 486*a*c^2*d^4 + 340*a*c*d^5 + 91*a*d^6)*cos(f*x + e)^2 + 2*(21*a*c^
4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*a*c*d^5 + 13*a*d^6)*cos(f*x + e) + (84*a*c^4*d^2 + 288*a*c^3*d^3 + 37
6*a*c^2*d^4 + 224*a*c*d^5 + 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^4 - 2*(21*a*c^3*d^3
+ 51*a*c^2*d^4 + 43*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^3 - (21*a*c^4*d^2 + 156*a*c^3*d^3 + 298*a*c^2*d^4 + 228*
a*c*d^5 + 65*a*d^6)*cos(f*x + e)^2 + 2*(21*a*c^4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*a*c*d^5 + 13*a*d^6)*co
s(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 -
2*c*d - d^2 - 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2
+ 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2
+ 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin
(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) +
(d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(4*c^6 - 8*c^5*d - 4*c^4*d^2
+ 16*c^3*d^3 - 4*c^2*d^4 - 8*c*d^5 + 4*d^6 - 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x
+ e)^4 - (6*c^5*d - 41*c^4*d^2 - 152*c^3*d^3 - 78*c^2*d^4 + 170*c*d^5 + 95*d^6)*cos(f*x + e)^3 + (3*c^6 - 16*
c^5*d - 31*c^4*d^2 - 84*c^3*d^3 - 23*c^2*d^4 + 100*c*d^5 + 51*d^6)*cos(f*x + e)^2 + (7*c^6 - 18*c^5*d - 79*c^4
*d^2 - 196*c^3*d^3 - 15*c^2*d^4 + 214*c*d^5 + 87*d^6)*cos(f*x + e) - (4*c^6 - 8*c^5*d - 4*c^4*d^2 + 16*c^3*d^3
- 4*c^2*d^4 - 8*c*d^5 + 4*d^6 + 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x + e)^3 - 2*(
3*c^5*d - 22*c^4*d^2 - 64*c^3*d^3 + 6*c^2*d^4 + 61*c*d^5 + 16*d^6)*cos(f*x + e)^2 - (3*c^6 - 10*c^5*d - 75*c^4
*d^2 - 212*c^3*d^3 - 11*c^2*d^4 + 222*c*d^5 + 83*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((
a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^
9)*f*cos(f*x + e)^5 + (2*a^3*c^8*d - 3*a^3*c^7*d^2 - 7*a^3*c^6*d^3 + 13*a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 17*a^3*c
^3*d^6 + 3*a^3*c^2*d^7 + 7*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^4 - (a^3*c^9 + a^3*c^8*d - 8*a^3*c^7*d^2 + 18
*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 16*a^3*c^3*d^6 + 8*a^3*c^2*d^7 + 5*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^3 - (3
*a^3*c^9 + a^3*c^8*d - 20*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 42*a^3*c^5*d^4 - 18*a^3*c^4*d^5 - 36*a^3*c^3*d^6 + 20*
a^3*c^2*d^7 + 11*a^3*c*d^8 - 7*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*
d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*
(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c
^2*d^7 + a^3*c*d^8 - a^3*d^9)*f + ((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6
- a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^4 - 2*(a^3*c^8*d - 2*a^3*c^7*d^2 - 2*a^3*c^6*d^3 + 6*a^3
*c^5*d^4 - 6*a^3*c^3*d^6 + 2*a^3*c^2*d^7 + 2*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^3 - (a^3*c^9 + 3*a^3*c^8*d -
12*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 30*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 28*a^3*c^3*d^6 + 12*a^3*c^2*d^7 + 9*a^3*c*d^
8 - 5*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a
^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*
a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*
d^9)*f)*sin(f*x + e)), -1/64*(3*sqrt(2)*(4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*
d^5 + 292*d^6 + (c^4*d^2 - 8*c^3*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^5 + (2*c^5*d - 13*c^4*d^2
+ 84*c^3*d^3 + 434*c^2*d^4 + 554*c*d^5 + 219*d^6)*cos(f*x + e)^4 - (c^6 - 4*c^5*d + 25*c^4*d^2 + 328*c^3*d^3
+ 779*c^2*d^4 + 700*c*d^5 + 219*d^6)*cos(f*x + e)^3 - (3*c^6 - 14*c^5*d + 89*c^4*d^2 + 892*c^3*d^3 + 1957*c^2*
d^4 + 1682*c*d^5 + 511*d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c
*d^5 + 73*d^6)*cos(f*x + e) + (4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*d^5 + 292*
d^6 + (c^4*d^2 - 8*c^3*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^4 - 2*(c^5*d - 7*c^4*d^2 + 46*c^3*d
^3 + 190*c^2*d^4 + 209*c*d^5 + 73*d^6)*cos(f*x + e)^3 - (c^6 - 2*c^5*d + 11*c^4*d^2 + 420*c^3*d^3 + 1159*c^2*d
^4 + 1118*c*d^5 + 365*d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c*
d^5 + 73*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*
sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos
(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 24*(84*a*c^4*d^2 + 288*a*c^3*d^3 + 376*a*
c^2*d^4 + 224*a*c*d^5 + 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^5 + (42*a*c^3*d^3 + 123
*a*c^2*d^4 + 116*a*c*d^5 + 39*a*d^6)*cos(f*x + e)^4 - (21*a*c^4*d^2 + 114*a*c^3*d^3 + 196*a*c^2*d^4 + 142*a*c*
d^5 + 39*a*d^6)*cos(f*x + e)^3 - (63*a*c^4*d^2 + 300*a*c^3*d^3 + 486*a*c^2*d^4 + 340*a*c*d^5 + 91*a*d^6)*cos(f
*x + e)^2 + 2*(21*a*c^4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*a*c*d^5 + 13*a*d^6)*cos(f*x + e) + (84*a*c^4*d^
2 + 288*a*c^3*d^3 + 376*a*c^2*d^4 + 224*a*c*d^5 + 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x +
e)^4 - 2*(21*a*c^3*d^3 + 51*a*c^2*d^4 + 43*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^3 - (21*a*c^4*d^2 + 156*a*c^3*d^3
+ 298*a*c^2*d^4 + 228*a*c*d^5 + 65*a*d^6)*cos(f*x + e)^2 + 2*(21*a*c^4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*
a*c*d^5 + 13*a*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*si
n(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) + 4*(4*c^6 - 8*c^5*d - 4*c^4*d^2 + 16*c^3*d^3 - 4
*c^2*d^4 - 8*c*d^5 + 4*d^6 - 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x + e)^4 - (6*c^5*
d - 41*c^4*d^2 - 152*c^3*d^3 - 78*c^2*d^4 + 170*c*d^5 + 95*d^6)*cos(f*x + e)^3 + (3*c^6 - 16*c^5*d - 31*c^4*d^
2 - 84*c^3*d^3 - 23*c^2*d^4 + 100*c*d^5 + 51*d^6)*cos(f*x + e)^2 + (7*c^6 - 18*c^5*d - 79*c^4*d^2 - 196*c^3*d^
3 - 15*c^2*d^4 + 214*c*d^5 + 87*d^6)*cos(f*x + e) - (4*c^6 - 8*c^5*d - 4*c^4*d^2 + 16*c^3*d^3 - 4*c^2*d^4 - 8*
c*d^5 + 4*d^6 + 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x + e)^3 - 2*(3*c^5*d - 22*c^4*
d^2 - 64*c^3*d^3 + 6*c^2*d^4 + 61*c*d^5 + 16*d^6)*cos(f*x + e)^2 - (3*c^6 - 10*c^5*d - 75*c^4*d^2 - 212*c^3*d^
3 - 11*c^2*d^4 + 222*c*d^5 + 83*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^7*d^2 - 3*a
^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)
^5 + (2*a^3*c^8*d - 3*a^3*c^7*d^2 - 7*a^3*c^6*d^3 + 13*a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 17*a^3*c^3*d^6 + 3*a^3*c^
2*d^7 + 7*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^4 - (a^3*c^9 + a^3*c^8*d - 8*a^3*c^7*d^2 + 18*a^3*c^5*d^4 - 6*
a^3*c^4*d^5 - 16*a^3*c^3*d^6 + 8*a^3*c^2*d^7 + 5*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^3 - (3*a^3*c^9 + a^3*c^
8*d - 20*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 42*a^3*c^5*d^4 - 18*a^3*c^4*d^5 - 36*a^3*c^3*d^6 + 20*a^3*c^2*d^7 + 11*
a^3*c*d^8 - 7*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d
^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^
8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^
8 - a^3*d^9)*f + ((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3
*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^4 - 2*(a^3*c^8*d - 2*a^3*c^7*d^2 - 2*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*
c^3*d^6 + 2*a^3*c^2*d^7 + 2*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^3 - (a^3*c^9 + 3*a^3*c^8*d - 12*a^3*c^7*d^2 -
4*a^3*c^6*d^3 + 30*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 28*a^3*c^3*d^6 + 12*a^3*c^2*d^7 + 9*a^3*c*d^8 - 5*a^3*d^9)*f*
cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^
3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a
^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f)*sin(f*x +
e))]

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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(1/(a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

Exception raised: TypeError