Integrand size = 29, antiderivative size = 270 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {3 \left (c^2-6 c d+25 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^{7/2} f}-\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}-\frac {(3 c-13 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}-\frac {(c-7 d) d (3 c+7 d) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \] Output:
-3/32*(c^2-6*c*d+25*d^2)*arctanh(1/2*a^(1/2)*(c-d)^(1/2)*cos(f*x+e)*2^(1/2 )/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))*2^(1/2)/a^(5/2)/(c-d)^(7/ 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))^(1/2)- 1/16*(3*c-13*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x +e))^(1/2)-1/16*(c-7*d)*d*(3*c+7*d)*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(a+a*si n(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)
Time = 7.88 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-14 c^3+25 c^2 d+56 c d^2+113 d^3+d \left (3 c^2-14 c d-49 d^2\right ) \cos (2 (e+f x))+\left (-6 c^3+14 c^2 d+62 c d^2+170 d^3\right ) \sin (e+f x)\right )}{(c+d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {3 \left (c^2-6 c d+25 d^2\right ) \left (\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2+2 \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}}\right )}{32 (c-d)^3 f (a (1+\sin (e+f x)))^{5/2} \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(((Cos[(e + f*x)/2] - Sin[(e + f* x)/2])*(-14*c^3 + 25*c^2*d + 56*c*d^2 + 113*d^3 + d*(3*c^2 - 14*c*d - 49*d ^2)*Cos[2*(e + f*x)] + (-6*c^3 + 14*c^2*d + 62*c*d^2 + 170*d^3)*Sin[e + f* x]))/((c + d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + (3*(c^2 - 6*c*d + 25*d^2)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]) )/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f *x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f* x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqr t[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f* x)/2]))))/(32*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]])
Time = 1.39 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 3245, 27, 3042, 3457, 27, 3042, 3463, 27, 3042, 3261, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle -\frac {\int -\frac {3 a (c-3 d)+4 a d \sin (e+f x)}{2 (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^{3/2}}dx}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a (c-3 d)+4 a d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^{3/2}}dx}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a (c-3 d)+4 a d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^{3/2}}dx}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (3 c^2-12 d c+49 d^2\right ) a^2+2 (3 c-13 d) d \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}}dx}{2 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-12 d c+49 d^2\right ) a^2+2 (3 c-13 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}}dx}{4 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-12 d c+49 d^2\right ) a^2+2 (3 c-13 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}}dx}{4 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {3 a^3 (c+d) \left (c^2-6 d c+25 d^2\right )}{2 \sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-14 c d-49 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{4 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 (c+d) \left (c^2-6 c d+25 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d \left (3 c^2-14 c d-49 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{4 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 (c+d) \left (c^2-6 c d+25 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d \left (3 c^2-14 c d-49 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{4 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \frac {\frac {-\frac {6 a^3 (c+d) \left (c^2-6 c d+25 d^2\right ) \int \frac {1}{2 a^2-\frac {a^3 (c-d) \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-14 c d-49 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{4 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {-\frac {3 \sqrt {2} a^{3/2} (c+d) \left (c^2-6 c d+25 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f \sqrt {c-d} \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-14 c d-49 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{4 a^2 (c-d)}-\frac {a (3 c-13 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt {c+d \sin (e+f x)}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt {c+d \sin (e+f x)}}\) |
Input:
Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]
Output:
-1/4*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]]) + (-1/2*(a*(3*c - 13*d)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f* x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) + ((-3*Sqrt[2]*a^(3/2)*(c + d)*(c^2 - 6*c*d + 25*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[c - d]*(c^2 - d^2)*f) - (2*a^2*d*(3*c^2 - 14*c*d - 49*d^2)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))/(4*a^2*(c - d)))/(8*a^2*(c - d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[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] + Simp[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] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[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] + Simp[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])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(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] + Simp[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] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Timed out.
\[\int \frac {1}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
Input:
int(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x)
Output:
int(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 1375 vs. \(2 (235) = 470\).
Time = 0.74 (sec) , antiderivative size = 2984, normalized size of antiderivative = 11.05 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="fr icas")
Output:
[-1/128*(3*((c^3*d - 5*c^2*d^2 + 19*c*d^3 + 25*d^4)*cos(f*x + e)^4 + 4*c^4 - 16*c^3*d + 56*c^2*d^2 + 176*c*d^3 + 100*d^4 - (c^4 - 3*c^3*d + 9*c^2*d^ 2 + 63*c*d^3 + 50*d^4)*cos(f*x + e)^3 - (3*c^4 - 10*c^3*d + 32*c^2*d^2 + 1 70*c*d^3 + 125*d^4)*cos(f*x + e)^2 + 2*(c^4 - 4*c^3*d + 14*c^2*d^2 + 44*c* d^3 + 25*d^4)*cos(f*x + e) + (4*c^4 - 16*c^3*d + 56*c^2*d^2 + 176*c*d^3 + 100*d^4 - (c^3*d - 5*c^2*d^2 + 19*c*d^3 + 25*d^4)*cos(f*x + e)^3 - (c^4 - 2*c^3*d + 4*c^2*d^2 + 82*c*d^3 + 75*d^4)*cos(f*x + e)^2 + 2*(c^4 - 4*c^3*d + 14*c^2*d^2 + 44*c*d^3 + 25*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a *c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 + 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4 *c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e ) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f *x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/( cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)* sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*(4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 17*c^2*d^2 - 35*c*d^3 + 49*d^4)*cos(f*x + e)^3 + (3*c^4 - 13 *c^3*d - 7*c^2*d^2 - 19*c*d^3 + 36*d^4)*cos(f*x + e)^2 + (7*c^4 - 18*c^3*d - 24*c^2*d^2 - 46*c*d^3 + 81*d^4)*cos(f*x + e) - (4*c^4 - 8*c^3*d + 8*...
Timed out. \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="ma xima")
Output:
integrate(1/((a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^(3/2)), x)
\[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="gi ac")
Output:
integrate(1/((a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^(3/2)), x)
Timed out. \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int(1/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2)),x)
Output:
int(1/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2)), x)
\[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{5} d^{2}+2 \sin \left (f x +e \right )^{4} c d +3 \sin \left (f x +e \right )^{4} d^{2}+\sin \left (f x +e \right )^{3} c^{2}+6 \sin \left (f x +e \right )^{3} c d +3 \sin \left (f x +e \right )^{3} d^{2}+3 \sin \left (f x +e \right )^{2} c^{2}+6 \sin \left (f x +e \right )^{2} c d +\sin \left (f x +e \right )^{2} d^{2}+3 \sin \left (f x +e \right ) c^{2}+2 \sin \left (f x +e \right ) c d +c^{2}}d x \right )}{a^{3}} \] Input:
int(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x)
Output:
(sqrt(a)*int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x) + 1))/(sin(e + f* x)**5*d**2 + 2*sin(e + f*x)**4*c*d + 3*sin(e + f*x)**4*d**2 + sin(e + f*x) **3*c**2 + 6*sin(e + f*x)**3*c*d + 3*sin(e + f*x)**3*d**2 + 3*sin(e + f*x) **2*c**2 + 6*sin(e + f*x)**2*c*d + sin(e + f*x)**2*d**2 + 3*sin(e + f*x)*c **2 + 2*sin(e + f*x)*c*d + c**2),x))/a**3