Integrand size = 25, antiderivative size = 187 \[ \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac {6 a \left (a^2-2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}} \]
2/5*b*(3*a^2-4*b^2)*(e*cos(d*x+c))^(3/2)/d/e^5+2/5*(b+a*sin(d*x+c))*(a+b*s in(d*x+c))^2/d/e/(e*cos(d*x+c))^(5/2)-2/5*(a+b*sin(d*x+c))*(a*b-(3*a^2-4*b ^2)*sin(d*x+c))/d/e^3/(e*cos(d*x+c))^(1/2)-6/5*a*(a^2-2*b^2)*(cos(1/2*d*x+ 1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*( e*cos(d*x+c))^(1/2)/d/e^4/cos(d*x+c)^(1/2)
Time = 1.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 \left (-5 b^3-3 a \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \left (3 a^2+b^2\right ) \sec ^2(c+d x)+3 a^3 \sin (c+d x)-6 a b^2 \sin (c+d x)+a \left (a^2+3 b^2\right ) \sec (c+d x) \tan (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}} \]
(2*(-5*b^3 - 3*a*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2 ] + b*(3*a^2 + b^2)*Sec[c + d*x]^2 + 3*a^3*Sin[c + d*x] - 6*a*b^2*Sin[c + d*x] + a*(a^2 + 3*b^2)*Sec[c + d*x]*Tan[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]])
Time = 0.86 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3170, 27, 3042, 3340, 27, 3042, 3148, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 \int -\frac {(a+b \sin (c+d x)) \left (3 a^2-b \sin (c+d x) a-4 b^2\right )}{2 (e \cos (c+d x))^{3/2}}dx}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x)) \left (3 a^2-b \sin (c+d x) a-4 b^2\right )}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x)) \left (3 a^2-b \sin (c+d x) a-4 b^2\right )}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3340 |
| \(\displaystyle \frac {-\frac {2 \int \frac {3}{2} \sqrt {e \cos (c+d x)} \left (a \left (a^2-2 b^2\right )+b \left (3 a^2-4 b^2\right ) \sin (c+d x)\right )dx}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \int \sqrt {e \cos (c+d x)} \left (a \left (a^2-2 b^2\right )+b \left (3 a^2-4 b^2\right ) \sin (c+d x)\right )dx}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \int \sqrt {e \cos (c+d x)} \left (a \left (a^2-2 b^2\right )+b \left (3 a^2-4 b^2\right ) \sin (c+d x)\right )dx}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {-\frac {3 \left (a \left (a^2-2 b^2\right ) \int \sqrt {e \cos (c+d x)}dx-\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (a \left (a^2-2 b^2\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {a \left (a^2-2 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}-\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {a \left (a^2-2 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}-\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {2 a \left (a^2-2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )}{e^2}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}}}{5 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}\) |
(2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(5*d*e*(e*Cos[c + d*x])^(5 /2)) + ((-3*((-2*b*(3*a^2 - 4*b^2)*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a* (a^2 - 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[ c + d*x]])))/e^2 - (2*(a + b*Sin[c + d*x])*(a*b - (3*a^2 - 4*b^2)*Sin[c + d*x]))/(d*e*Sqrt[e*Cos[c + d*x]]))/(5*e^2)
3.6.62.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x ])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g }, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2* p] || IntegerQ[m])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(g* Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Si n[e + f*x])^(m - 1)*Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ [m, 0] && LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])
Leaf count of result is larger than twice the leaf count of optimal. \(617\) vs. \(2(195)=390\).
Time = 8.39 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.30
| method | result | size |
| default | \(\frac {\frac {48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{5}-\frac {96 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{5}-\frac {24 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{5}+\frac {48 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{5}-\frac {48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{5}+\frac {96 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{5}+\frac {24 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{5}-\frac {48 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{5}-8 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+\frac {16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{5}-\frac {12 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{5}-\frac {6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}}{5}+\frac {12 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}}{5}+8 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+\frac {6 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b}{5}-\frac {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{5}}{\left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d}\) | \(618\) |
| parts | \(\text {Expression too large to display}\) | \(799\) |
2/5/(4*sin(1/2*d*x+1/2*c)^4-4*sin(1/2*d*x+1/2*c)^2+1)/sin(1/2*d*x+1/2*c)/( -2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^3*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+ 1/2*c)^6*a^3-48*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*a*b^2-12*EllipticE (cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1 /2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4*a^3+24*EllipticE(cos(1/2*d*x+1/2*c), 2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin (1/2*d*x+1/2*c)^4*a*b^2-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^3+48* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a*b^2+12*EllipticE(cos(1/2*d*x+1/2 *c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2) *sin(1/2*d*x+1/2*c)^2*a^3-24*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^ 2*a*b^2-20*sin(1/2*d*x+1/2*c)^5*b^3+8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c )^2*a^3-6*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a*b^2-3*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c) ,2^(1/2))*a^3+6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1 /2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+20*sin(1/2*d*x+1/2*c)^3*b^ 3+3*sin(1/2*d*x+1/2*c)*a^2*b-4*sin(1/2*d*x+1/2*c)*b^3)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (i \, a^{3} - 2 i \, a b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-i \, a^{3} + 2 i \, a b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (5 \, b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3} - {\left (a^{3} + 3 \, a b^{2} + 3 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{5 \, d e^{4} \cos \left (d x + c\right )^{3}} \]
-1/5*(3*sqrt(2)*(I*a^3 - 2*I*a*b^2)*sqrt(e)*cos(d*x + c)^3*weierstrassZeta (-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*sqr t(2)*(-I*a^3 + 2*I*a*b^2)*sqrt(e)*cos(d*x + c)^3*weierstrassZeta(-4, 0, we ierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(5*b^3*cos(d* x + c)^2 - 3*a^2*b - b^3 - (a^3 + 3*a*b^2 + 3*(a^3 - 2*a*b^2)*cos(d*x + c) ^2)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/(d*e^4*cos(d*x + c)^3)
Timed out. \[ \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]