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

Optimal. Leaf size=313 \[ -\frac{\left (3 c^2-22 c d+115 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{16 \sqrt{2} a^{5/2} f (c-d)^4}+\frac{d^{5/2} (7 c+5 d) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac{d (c-7 d) (3 c+5 d) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{3 (c-5 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))} \]

[Out]

-((3*c^2 - 22*c*d + 115*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)^4*f) + (d^(5/2)*(7*c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[
e + f*x]])])/(a^(5/2)*(c - d)^4*(c + d)^(3/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d
*Sin[e + f*x])) - (3*(c - 5*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])
) - ((c - 7*d)*d*(3*c + 5*d)*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]))

________________________________________________________________________________________

Rubi [A]  time = 1.09969, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{\left (3 c^2-22 c d+115 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{16 \sqrt{2} a^{5/2} f (c-d)^4}+\frac{d^{5/2} (7 c+5 d) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac{d (c-7 d) (3 c+5 d) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{3 (c-5 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*c^2 - 22*c*d + 115*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)^4*f) + (d^(5/2)*(7*c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[
e + f*x]])])/(a^(5/2)*(c - d)^4*(c + d)^(3/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d
*Sin[e + f*x])) - (3*(c - 5*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])
) - ((c - 7*d)*d*(3*c + 5*d)*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]))

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))^2} \, dx &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\frac{\int \frac{-\frac{1}{2} a (3 c-10 d)-\frac{5}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2} \, 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))}-\frac{3 (c-5 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))}+\frac{\int \frac{\frac{1}{4} a^2 \left (3 c^2-13 c d+70 d^2\right )+\frac{9}{4} a^2 (c-5 d) d \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, 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))}-\frac{3 (c-5 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))}-\frac{(c-7 d) d (3 c+5 d) \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))}-\frac{\int \frac{-\frac{1}{4} a^3 \left (3 c^3-16 c^2 d+77 c d^2+80 d^3\right )-\frac{1}{4} a^3 (c-7 d) d (3 c+5 d) \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{8 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))}-\frac{3 (c-5 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))}-\frac{(c-7 d) d (3 c+5 d) \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))}-\frac{\left (d^3 (7 c+5 d)\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 a^3 (c-d)^4 (c+d)}+\frac{\left (3 c^2-22 c d+115 d^2\right ) \int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx}{32 a^2 (c-d)^4}\\ &=-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\frac{3 (c-5 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))}-\frac{(c-7 d) d (3 c+5 d) \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))}+\frac{\left (d^3 (7 c+5 d)\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 )}{a^2 (c-d)^4 (c+d) f}-\frac{\left (3 c^2-22 c d+115 d^2\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)^4 f}\\ &=-\frac{\left (3 c^2-22 c d+115 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)^4 f}+\frac{d^{5/2} (7 c+5 d) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a+a \sin (e+f x)}}\right )}{a^{5/2} (c-d)^4 (c+d)^{3/2} f}-\frac{\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\frac{3 (c-5 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))}-\frac{(c-7 d) d (3 c+5 d) \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))}\\ \end{align*}

Mathematica [C]  time = 5.76415, size = 570, normalized size = 1.82 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left ((1+i) (-1)^{3/4} \left (3 c^2-22 c d+115 d^2\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (e+f x)\right )-1\right )\right )+\frac{16 d^3 (c-d) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4}{(c+d) (c+d \sin (e+f x))}+\frac{4 d^{5/2} (7 c+5 d) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \left (2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (\sqrt{c+d}-\sqrt{d} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{d} \cos \left (\frac{1}{2} (e+f x)\right )\right )\right )-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{3/2}}-\frac{4 d^{5/2} (7 c+5 d) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \left (2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (\sqrt{c+d}+\sqrt{d} \sin \left (\frac{1}{2} (e+f x)\right )-\sqrt{d} \cos \left (\frac{1}{2} (e+f x)\right )\right )\right )-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{3/2}}+8 (c-d)^2 \sin \left (\frac{1}{2} (e+f x)\right )+(c-d) (19 d-3 c) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+2 (3 c-19 d) (c-d) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-4 (c-d)^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{16 f (c-d)^4 (a (\sin (e+f x)+1))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(c - d)^2*Sin[(e + f*x)/2] - 4*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2]) + 2*(3*c - 19*d)*(c - d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (c - d)*(-3*c +
19*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(3*c^2 - 22*c*d + 115*d^2)*ArcTanh[(1/2 +
I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (4*d^(5/2)*(7*c + 5*d)*(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])^4)/(c + d)^(3/2) - (4*d^(5/2)*(7*c + 5*d)*(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])^4)/(c + d)^(3/2) + (16*(c - d)*d^3*(Cos[(e + f*x)/2] - Sin
[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/((c + d)*(c + d*Sin[e + f*x]))))/(16*(c - d)^4*f*(a*(1
+ Sin[e + f*x]))^(5/2))

________________________________________________________________________________________

Maple [B]  time = 1.894, size = 1972, normalized size = 6.3 \begin{align*} \text{result 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))^2,x)

[Out]

-1/32*(115*(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^4+55*(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^2+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)^3*a^2*c^3*d
-19*(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^2
+301*(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^3-
35*(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+167*
(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^2+323*(
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^3+93*(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^3-13*(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^3*d-160*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^5-320*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^5-160*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^
(5/2)*sin(f*x+e)*d^5-224*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*c^2*d^3-160*arctanh((
-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*c*d^4+20*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a
^(3/2)*c^4+32*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*d^4-6*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d
)^(1/2)*a^(1/2)*c^4-32*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*c*d^3+20*(-a*(-1+sin(
f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c^3*d-84*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3
/2)*sin(f*x+e)*c^2*d^2-84*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*c*d^3+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^4-6*(-a*(-1+sin(f*x+e
)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^3*d+38*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*s
in(f*x+e)*c^2*d^2+6*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c*d^3+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^4+115*(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*d^4-19*(a*(c+d)*d)^(1/2)*2^(1/2)*arc
tanh(1/2*(-a*(-1+sin(f*x+e)))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^3*d+93*(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^2+115*(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*d^3+230*(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^4+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^4-224*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^4-38*(-a*(
-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*d^4+52*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)
*a^(3/2)*c*d^3-608*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^4+32*(-a*(
-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)^2*d^4-448*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^3-544*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^4+148*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*d^4-84*(-a*(-1+sin(f*x+e)
))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^3*d-20*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^2*d^2-224*a
rctanh((-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^3+38*(-a*(-1+sin(f*x+e)))^(3
/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^3*d+6*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c^2*d^2-38*(-a*(-1+
sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*c*d^3)*(-a*(-1+sin(f*x+e)))^(1/2)/a^(9/2)/(1+sin(f*x+e))/(a*(c+d)
*d)^(1/2)/(c+d*sin(f*x+e))/(c+d)/(c-d)^4/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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

[Out]

Timed out

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Fricas [B]  time = 12.7231, size = 8342, normalized size = 26.65 \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))^2,x, algorithm="fricas")

[Out]

[1/64*(sqrt(2)*((3*c^3*d - 19*c^2*d^2 + 93*c*d^3 + 115*d^4)*cos(f*x + e)^4 + 12*c^4 - 64*c^3*d + 296*c^2*d^2 +
832*c*d^3 + 460*d^4 - (3*c^4 - 13*c^3*d + 55*c^2*d^2 + 301*c*d^3 + 230*d^4)*cos(f*x + e)^3 - (9*c^4 - 42*c^3*
d + 184*c^2*d^2 + 810*c*d^3 + 575*d^4)*cos(f*x + e)^2 + 2*(3*c^4 - 16*c^3*d + 74*c^2*d^2 + 208*c*d^3 + 115*d^4
)*cos(f*x + e) + (12*c^4 - 64*c^3*d + 296*c^2*d^2 + 832*c*d^3 + 460*d^4 - (3*c^3*d - 19*c^2*d^2 + 93*c*d^3 + 1
15*d^4)*cos(f*x + e)^3 - (3*c^4 - 10*c^3*d + 36*c^2*d^2 + 394*c*d^3 + 345*d^4)*cos(f*x + e)^2 + 2*(3*c^4 - 16*
c^3*d + 74*c^2*d^2 + 208*c*d^3 + 115*d^4)*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)) + 16*(28*a*c^2*
d^2 + 48*a*c*d^3 + 20*a*d^4 + (7*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^4 - (7*a*c^2*d^2 + 19*a*c*d^3 + 10*a*d^4)*cos
(f*x + e)^3 - (21*a*c^2*d^2 + 50*a*c*d^3 + 25*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^2 + 12*a*c*d^3 + 5*a*d^4)*c
os(f*x + e) + (28*a*c^2*d^2 + 48*a*c*d^3 + 20*a*d^4 - (7*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^3 - (7*a*c^2*d^2 + 26
*a*c*d^3 + 15*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^2 + 12*a*c*d^3 + 5*a*d^4)*cos(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)*c
os(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*co
s(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) - 4*(4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19*c^2*d^2
- 19*c*d^3 + 35*d^4)*cos(f*x + e)^3 + (3*c^4 - 15*c^3*d - 7*c^2*d^2 - c*d^3 + 20*d^4)*cos(f*x + e)^2 + (7*c^4
- 20*c^3*d - 26*c^2*d^2 - 12*c*d^3 + 51*d^4)*cos(f*x + e) - (4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19
*c^2*d^2 - 19*c*d^3 + 35*d^4)*cos(f*x + e)^2 - (3*c^4 - 12*c^3*d - 26*c^2*d^2 - 20*c*d^3 + 55*d^4)*cos(f*x + e
))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3
*c*d^5 + a^3*d^6)*f*cos(f*x + e)^4 - (a^3*c^6 - a^3*c^5*d - 4*a^3*c^4*d^2 + 6*a^3*c^3*d^3 + a^3*c^2*d^4 - 5*a^
3*c*d^5 + 2*a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*d^
4 - 12*a^3*c*d^5 + 5*a^3*d^6)*f*cos(f*x + e)^2 + 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*
c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) + 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3
*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f - ((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^
5 + a^3*d^6)*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a^3
*d^6)*f*cos(f*x + e)^2 - 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 +
a^3*d^6)*f*cos(f*x + e) - 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 +
a^3*d^6)*f)*sin(f*x + e)), 1/64*(sqrt(2)*((3*c^3*d - 19*c^2*d^2 + 93*c*d^3 + 115*d^4)*cos(f*x + e)^4 + 12*c^4
- 64*c^3*d + 296*c^2*d^2 + 832*c*d^3 + 460*d^4 - (3*c^4 - 13*c^3*d + 55*c^2*d^2 + 301*c*d^3 + 230*d^4)*cos(f*
x + e)^3 - (9*c^4 - 42*c^3*d + 184*c^2*d^2 + 810*c*d^3 + 575*d^4)*cos(f*x + e)^2 + 2*(3*c^4 - 16*c^3*d + 74*c^
2*d^2 + 208*c*d^3 + 115*d^4)*cos(f*x + e) + (12*c^4 - 64*c^3*d + 296*c^2*d^2 + 832*c*d^3 + 460*d^4 - (3*c^3*d
- 19*c^2*d^2 + 93*c*d^3 + 115*d^4)*cos(f*x + e)^3 - (3*c^4 - 10*c^3*d + 36*c^2*d^2 + 394*c*d^3 + 345*d^4)*cos(
f*x + e)^2 + 2*(3*c^4 - 16*c^3*d + 74*c^2*d^2 + 208*c*d^3 + 115*d^4)*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)) + 32*(28*a*c^2*d^2 + 48*a*c*d^3 + 20*a*d^4 + (7*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^4 - (7*a*c^2*d^2 +
19*a*c*d^3 + 10*a*d^4)*cos(f*x + e)^3 - (21*a*c^2*d^2 + 50*a*c*d^3 + 25*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^
2 + 12*a*c*d^3 + 5*a*d^4)*cos(f*x + e) + (28*a*c^2*d^2 + 48*a*c*d^3 + 20*a*d^4 - (7*a*c*d^3 + 5*a*d^4)*cos(f*x
+ e)^3 - (7*a*c^2*d^2 + 26*a*c*d^3 + 15*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^2 + 12*a*c*d^3 + 5*a*d^4)*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*sin(f*x + e) - c - 2*d)*sqrt
(-d/(a*c + a*d))/(d*cos(f*x + e))) - 4*(4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19*c^2*d^2 - 19*c*d^3 +
35*d^4)*cos(f*x + e)^3 + (3*c^4 - 15*c^3*d - 7*c^2*d^2 - c*d^3 + 20*d^4)*cos(f*x + e)^2 + (7*c^4 - 20*c^3*d -
26*c^2*d^2 - 12*c*d^3 + 51*d^4)*cos(f*x + e) - (4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19*c^2*d^2 - 1
9*c*d^3 + 35*d^4)*cos(f*x + e)^2 - (3*c^4 - 12*c^3*d - 26*c^2*d^2 - 20*c*d^3 + 55*d^4)*cos(f*x + e))*sin(f*x +
e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3
*d^6)*f*cos(f*x + e)^4 - (a^3*c^6 - a^3*c^5*d - 4*a^3*c^4*d^2 + 6*a^3*c^3*d^3 + a^3*c^2*d^4 - 5*a^3*c*d^5 + 2*
a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*d^4 - 12*a^3*c
*d^5 + 5*a^3*d^6)*f*cos(f*x + e)^2 + 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*
a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) + 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2
*a^3*c*d^5 + a^3*d^6)*f - ((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)
*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a^3*d^6)*f*cos(
f*x + e)^2 - 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*c
os(f*x + e) - 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*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))**2,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))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError