∫ 1_____________ (a+a sin(e+fx))5∕2(c+d sin(e+fx))3 dx

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

[Out]

3/4*d^(5/2)*(21*c^2+30*c*d+13*d^2)*arctanh(cos(f*x+e)*a^(1/2)*d^(1/2)/(c+d)^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(5

/2)/(c-d)^5/(c+d)^(5/2)/f-1/4*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^2-1/16*(3*c-19*d)*cos

(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^2-3/32*(c^2-10*c*d+73*d^2)*arctanh(1/2*cos(f*x+e)*

a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(5/2)/(c-d)^5/f*2^(1/2)-1/16*d*(3*c^2-20*c*d-31*d^2)*cos(f*x+e)/a^2/

(c-d)^3/(c+d)/f/(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(1/2)-3/16*d*(c+3*d)*(c^2-10*c*d-7*d^2)*cos(f*x+e)/a^2/(c-

d)^4/(c+d)^2/f/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)

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Rubi [A] time = 1.52, antiderivative size = 400, normalized size of antiderivative = 1.00, 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 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 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]

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 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 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 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]

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*}

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Mathematica [C] time = 9.36, size = 958, normalized size = 2.40 \[ \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]))

________________________________________________________________________________________

fricas [B] time = 3.54, size = 5999, normalized size = 15.00 \[ \text {result too large to display} \]

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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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab

le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (4*p

i/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unab

le to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*p