Integrand size = 25, antiderivative size = 150 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx=-\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d e^2 \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac {2}{9 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )} \]
2/3*sin(d*x+c)/a^2/d/e/(e*cos(d*x+c))^(1/2)-2/9/d/e/(a+a*sin(d*x+c))^2/(e* cos(d*x+c))^(1/2)-2/9/d/e/(a^2+a^2*sin(d*x+c))/(e*cos(d*x+c))^(1/2)-2/3*(c os(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)/a^2/d/e^2/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {13}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{2 \sqrt [4]{2} a^2 d e \sqrt {e \cos (c+d x)}} \]
(Hypergeometric2F1[-1/4, 13/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x ])^(1/4))/(2*2^(1/4)*a^2*d*e*Sqrt[e*Cos[c + d*x]])
Time = 0.66 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {5 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {1}{(e \cos (c+d x))^{3/2}}dx}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )}{5 a}-\frac {2}{5 d e (a \sin (c+d x)+a) \sqrt {e \cos (c+d x)}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}\) |
-2/(9*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) + (5*(-2/(5*d*e*Sqr t[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])) + (3*((-2*Sqrt[e*Cos[c + d*x]]*Ell ipticE[(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)
3.3.50.3.1 Defintions of rubi rules used
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. \(487\) vs. \(2(158)=316\).
Time = 7.23 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.25
| method | result | size |
| default | \(\frac {\frac {64 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{3}-\frac {32 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {128 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {64 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {304 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}-16 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {112 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}+\frac {16 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{3}-\frac {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}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3}-\frac {4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}}{\left (16 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a^{2} \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 d}\) | \(488\) |
2/9/(16*sin(1/2*d*x+1/2*c)^8-32*sin(1/2*d*x+1/2*c)^6+24*sin(1/2*d*x+1/2*c) ^4-8*sin(1/2*d*x+1/2*c)^2+1)/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c) ^2*e+e)^(1/2)/e*(96*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-48*EllipticE( cos(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)^(1/2)*sin(1/2*d*x+1/2*c)^8-192*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/ 2*c)^8+96*EllipticE(cos(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)^(1/2)*sin(1/2*d*x+1/2*c)^6+152*sin(1/2*d*x+1 /2*c)^6*cos(1/2*d*x+1/2*c)-72*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(c os(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c) ^4-56*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+24*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2) *sin(1/2*d*x+1/2*c)^2+12*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-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))-2*sin(1/2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx=\frac {3 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, \sqrt {e \cos \left (d x + c\right )} {\left (6 \, \cos \left (d x + c\right )^{2} + {\left (3 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 4\right )}}{9 \, {\left (a^{2} d e^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} d e^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d e^{2} \cos \left (d x + c\right )\right )}} \]
1/9*(3*(-I*sqrt(2)*cos(d*x + c)^3 + 2*I*sqrt(2)*cos(d*x + c)*sin(d*x + c) + 2*I*sqrt(2)*cos(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInv erse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(I*sqrt(2)*cos(d*x + c)^3 - 2*I*sqrt(2)*cos(d*x + c)*sin(d*x + c) - 2*I*sqrt(2)*cos(d*x + c))*sqrt(e )*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d *x + c))) + 2*sqrt(e*cos(d*x + c))*(6*cos(d*x + c)^2 + (3*cos(d*x + c)^2 - 5)*sin(d*x + c) - 4))/(a^2*d*e^2*cos(d*x + c)^3 - 2*a^2*d*e^2*cos(d*x + c )*sin(d*x + c) - 2*a^2*d*e^2*cos(d*x + c))
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]