Integrand size = 25, antiderivative size = 225 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=-\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )} \] Output:
-42/221*(e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^4/d/e ^2/cos(d*x+c)^(1/2)+42/221*sin(d*x+c)/a^4/d/e/(e*cos(d*x+c))^(1/2)-2/17/d/ e/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^4-18/221/a/d/e/(e*cos(d*x+c))^(1/2 )/(a+a*sin(d*x+c))^3-14/221/d/e/(e*cos(d*x+c))^(1/2)/(a^2+a^2*sin(d*x+c))^ 2-14/221/d/e/(e*cos(d*x+c))^(1/2)/(a^4+a^4*sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.29 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {21}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{8 \sqrt [4]{2} a^4 d e \sqrt {e \cos (c+d x)}} \] Input:
Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4),x]
Output:
(Hypergeometric2F1[-1/4, 21/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x ])^(1/4))/(8*2^(1/4)*a^4*d*e*Sqrt[e*Cos[c + d*x]])
Time = 1.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3160, 3042, 3160, 3042, 3160, 3042, 3162, 3042, 3116, 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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^4 (e \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^4 (e \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {9 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^3}dx}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^3}dx}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^2}dx}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^2}dx}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{(e \cos (c+d x))^{3/2}}dx}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}\right )}{17 a}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}}\) |
Input:
Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4),x]
Output:
-2/(17*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4) + (9*(-2/(13*d*e*S qrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) + (7*(-2/(9*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) + (5*(-2/(5*d*e*Sqrt[e*Cos[c + d*x]]*(a + a *Sin[c + d*x])) + (3*((-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/ (d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]]))) /(5*a)))/(9*a)))/(13*a)))/(17*a)
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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 _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*(2*m + p + 1))), x] + Simp[(m + p + 1)/(a*(2*m + p + 1)) Int[ (g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] & & IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[b*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*S in[e + f*x]))), x] + Simp[p/(a*(p - 1)) Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && Intege rQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(877\) vs. \(2(201)=402\).
Time = 7.85 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.90
Input:
int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
2/221/(256*sin(1/2*d*x+1/2*c)^16-1024*sin(1/2*d*x+1/2*c)^14+1792*sin(1/2*d *x+1/2*c)^12-1792*sin(1/2*d*x+1/2*c)^10+1120*sin(1/2*d*x+1/2*c)^8-448*sin( 1/2*d*x+1/2*c)^6+112*sin(1/2*d*x+1/2*c)^4-16*sin(1/2*d*x+1/2*c)^2+1)/a^4/s in(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e*(10752*sin(1/2*d*x +1/2*c)^18*cos(1/2*d*x+1/2*c)-5376*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1 /2*c)^16-43008*sin(1/2*d*x+1/2*c)^16*cos(1/2*d*x+1/2*c)+21504*(2*sin(1/2*d *x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/ 2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^14+76160*sin(1/2*d*x+1/2*c)^14*cos(1/2*d* x+1/2*c)-37632*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^12-77952*cos(1 /2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+37632*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2) *(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/ 2*d*x+1/2*c)^10+50560*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-23520*(2*si n(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2 *d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^8-21376*cos(1/2*d*x+1/2*c)*sin(1/2 *d*x+1/2*c)^8+9408*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2) ^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^6+5656*sin (1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-2352*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...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\frac {2 \, {\left (21 \, \sqrt {\frac {1}{2}} {\left (-i \, \cos \left (d x + c\right )^{5} + 8 i \, \cos \left (d x + c\right )^{3} + 4 \, {\left (i \, \cos \left (d x + c\right )^{3} - 2 i \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 8 i \, \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {\frac {1}{2}} {\left (i \, \cos \left (d x + c\right )^{5} - 8 i \, \cos \left (d x + c\right )^{3} + 4 \, {\left (-i \, \cos \left (d x + c\right )^{3} + 2 i \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 8 i \, \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (84 \, \cos \left (d x + c\right )^{4} - 224 \, \cos \left (d x + c\right )^{2} + {\left (21 \, \cos \left (d x + c\right )^{4} - 161 \, \cos \left (d x + c\right )^{2} + 117\right )} \sin \left (d x + c\right ) + 104\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{221 \, {\left (a^{4} d e^{2} \cos \left (d x + c\right )^{5} - 8 \, a^{4} d e^{2} \cos \left (d x + c\right )^{3} + 8 \, a^{4} d e^{2} \cos \left (d x + c\right ) - 4 \, {\left (a^{4} d e^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{4} d e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
2/221*(21*sqrt(1/2)*(-I*cos(d*x + c)^5 + 8*I*cos(d*x + c)^3 + 4*(I*cos(d*x + c)^3 - 2*I*cos(d*x + c))*sin(d*x + c) - 8*I*cos(d*x + c))*sqrt(e)*weier strassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c) )) + 21*sqrt(1/2)*(I*cos(d*x + c)^5 - 8*I*cos(d*x + c)^3 + 4*(-I*cos(d*x + c)^3 + 2*I*cos(d*x + c))*sin(d*x + c) + 8*I*cos(d*x + c))*sqrt(e)*weierst rassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + (84*cos(d*x + c)^4 - 224*cos(d*x + c)^2 + (21*cos(d*x + c)^4 - 161*cos( d*x + c)^2 + 117)*sin(d*x + c) + 104)*sqrt(e*cos(d*x + c)))/(a^4*d*e^2*cos (d*x + c)^5 - 8*a^4*d*e^2*cos(d*x + c)^3 + 8*a^4*d*e^2*cos(d*x + c) - 4*(a ^4*d*e^2*cos(d*x + c)^3 - 2*a^4*d*e^2*cos(d*x + c))*sin(d*x + c))
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**4,x)
Output:
Timed out
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
integrate(1/((e*cos(d*x + c))^(3/2)*(a*sin(d*x + c) + a)^4), x)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \] Input:
int(1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^4),x)
Output:
int(1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^4), x)
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+\cos \left (d x +c \right )^{2}}d x \right )}{a^{4} e^{2}} \] Input:
int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x)
Output:
(sqrt(e)*int(sqrt(cos(c + d*x))/(cos(c + d*x)**2*sin(c + d*x)**4 + 4*cos(c + d*x)**2*sin(c + d*x)**3 + 6*cos(c + d*x)**2*sin(c + d*x)**2 + 4*cos(c + d*x)**2*sin(c + d*x) + cos(c + d*x)**2),x))/(a**4*e**2)