Integrand size = 25, antiderivative size = 216 \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2 a b \left (a^2+14 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 \left (a^4-12 a^2 b^2-4 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}} \]
2/3*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^3/d/e/(e*cos(d*x+c))^(3/2)+2/3*(a^4- 12*a^2*b^2-4*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elliptic F(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/e^2/(e*cos(d*x+c))^(1/2)+ 2/3*a*b*(a^2+14*b^2)*(e*cos(d*x+c))^(1/2)/d/e^3+2/3*b*(a^2+2*b^2)*(a+b*sin (d*x+c))*(e*cos(d*x+c))^(1/2)/d/e^3+2/3*a*b*(a+b*sin(d*x+c))^2*(e*cos(d*x+ c))^(1/2)/d/e^3
Time = 1.77 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx=\frac {16 a^3 b+40 a b^3+24 a b^3 \cos (2 (c+d x))+4 \left (a^4-12 a^2 b^2-4 b^4\right ) \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+4 a^4 \sin (c+d x)+24 a^2 b^2 \sin (c+d x)+5 b^4 \sin (c+d x)+b^4 \sin (3 (c+d x))}{6 d e (e \cos (c+d x))^{3/2}} \]
(16*a^3*b + 40*a*b^3 + 24*a*b^3*Cos[2*(c + d*x)] + 4*(a^4 - 12*a^2*b^2 - 4 *b^4)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 4*a^4*Sin[c + d*x] + 24*a^2*b^2*Sin[c + d*x] + 5*b^4*Sin[c + d*x] + b^4*Sin[3*(c + d*x)])/(6*d* e*(e*Cos[c + d*x])^(3/2))
Time = 1.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3170, 27, 3042, 3341, 27, 3042, 3341, 27, 3042, 3148, 3042, 3121, 3042, 3120}
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.
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}-\frac {2 \int -\frac {(a+b \sin (c+d x))^2 \left (a^2-5 b \sin (c+d x) a-6 b^2\right )}{2 \sqrt {e \cos (c+d x)}}dx}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x))^2 \left (a^2-5 b \sin (c+d x) a-6 b^2\right )}{\sqrt {e \cos (c+d x)}}dx}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x))^2 \left (a^2-5 b \sin (c+d x) a-6 b^2\right )}{\sqrt {e \cos (c+d x)}}dx}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {5 (a+b \sin (c+d x)) \left (a \left (a^2-10 b^2\right )-3 b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {e \cos (c+d x)}}dx+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x)) \left (a \left (a^2-10 b^2\right )-3 b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{\sqrt {e \cos (c+d x)}}dx+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x)) \left (a \left (a^2-10 b^2\right )-3 b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{\sqrt {e \cos (c+d x)}}dx+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {\frac {2}{3} \int \frac {3 \left (a^4-12 b^2 a^2-b \left (a^2+14 b^2\right ) \sin (c+d x) a-4 b^4\right )}{2 \sqrt {e \cos (c+d x)}}dx+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^4-12 b^2 a^2-b \left (a^2+14 b^2\right ) \sin (c+d x) a-4 b^4}{\sqrt {e \cos (c+d x)}}dx+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a^4-12 b^2 a^2-b \left (a^2+14 b^2\right ) \sin (c+d x) a-4 b^4}{\sqrt {e \cos (c+d x)}}dx+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {\left (a^4-12 a^2 b^2-4 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 a b \left (a^2+14 b^2\right ) \sqrt {e \cos (c+d x)}}{d e}+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^4-12 a^2 b^2-4 b^4\right ) \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a b \left (a^2+14 b^2\right ) \sqrt {e \cos (c+d x)}}{d e}+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {\frac {\left (a^4-12 a^2 b^2-4 b^4\right ) \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{\sqrt {e \cos (c+d x)}}+\frac {2 a b \left (a^2+14 b^2\right ) \sqrt {e \cos (c+d x)}}{d e}+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^4-12 a^2 b^2-4 b^4\right ) \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {e \cos (c+d x)}}+\frac {2 a b \left (a^2+14 b^2\right ) \sqrt {e \cos (c+d x)}}{d e}+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 a b \left (a^2+14 b^2\right ) \sqrt {e \cos (c+d x)}}{d e}+\frac {2 b \left (a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{d e}+\frac {2 \left (a^4-12 a^2 b^2-4 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{d e}}{3 e^2}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}\) |
(2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(3*d*e*(e*Cos[c + d*x])^(3 /2)) + ((2*a*b*(a^2 + 14*b^2)*Sqrt[e*Cos[c + d*x]])/(d*e) + (2*(a^4 - 12*a ^2*b^2 - 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Co s[c + d*x]]) + (2*b*(a^2 + 2*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x] ))/(d*e) + (2*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(d*e))/(3*e ^2)
3.6.70.3.1 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[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, 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] && S implerQ[c + d*x, a + b*x])
Leaf count of result is larger than twice the leaf count of optimal. \(574\) vs. \(2(220)=440\).
Time = 8.52 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.66
| method | result | size |
| default | \(-\frac {2 \left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+2 \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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}-24 \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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}-8 \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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+48 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}+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^{2} b^{2}+4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-\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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{4}+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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b^{2}+4 \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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{4}-48 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b +16 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{3}\right )}{3 \left (2 \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^{2} d}\) | \(575\) |
| parts | \(\text {Expression too large to display}\) | \(840\) |
-2/3/(2*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^2*(8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*b^4-8*cos(1/2* d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*b^4+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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d* x+1/2*c)^2*a^4-24*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2*a^2*b^2-8 *(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(c os(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2*b^4+48*sin(1/2*d*x+1/2*c)^ 5*a*b^3+2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^4+12*cos(1/2*d*x+1/2*c )*sin(1/2*d*x+1/2*c)^2*a^2*b^2+4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b ^4-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF (cos(1/2*d*x+1/2*c),2^(1/2))*a^4+12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/ 2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^2+4*(s in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos( 1/2*d*x+1/2*c),2^(1/2))*b^4-48*sin(1/2*d*x+1/2*c)^3*a*b^3+4*sin(1/2*d*x+1/ 2*c)*a^3*b+16*sin(1/2*d*x+1/2*c)*a*b^3)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (-i \, a^{4} + 12 i \, a^{2} b^{2} + 4 i \, b^{4}\right )} \sqrt {e} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (i \, a^{4} - 12 i \, a^{2} b^{2} - 4 i \, b^{4}\right )} \sqrt {e} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (12 \, a b^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} b + 4 \, a b^{3} + {\left (b^{4} \cos \left (d x + c\right )^{2} + a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{3 \, d e^{3} \cos \left (d x + c\right )^{2}} \]
1/3*(sqrt(2)*(-I*a^4 + 12*I*a^2*b^2 + 4*I*b^4)*sqrt(e)*cos(d*x + c)^2*weie rstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + sqrt(2)*(I*a^4 - 1 2*I*a^2*b^2 - 4*I*b^4)*sqrt(e)*cos(d*x + c)^2*weierstrassPInverse(-4, 0, c os(d*x + c) - I*sin(d*x + c)) + 2*(12*a*b^3*cos(d*x + c)^2 + 4*a^3*b + 4*a *b^3 + (b^4*cos(d*x + c)^2 + a^4 + 6*a^2*b^2 + b^4)*sin(d*x + c))*sqrt(e*c os(d*x + c)))/(d*e^3*cos(d*x + c)^2)
Timed out. \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]