Integrand size = 27, antiderivative size = 154 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {32 (a+a \sin (c+d x))^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}} \]
-12/5*(a+a*sin(d*x+c))^(3/2)/a/d/e/(e*cos(d*x+c))^(7/2)+16/5*(a+a*sin(d*x+ c))^(5/2)/a^2/d/e/(e*cos(d*x+c))^(7/2)-32/35*(a+a*sin(d*x+c))^(7/2)/a^3/d/ e/(e*cos(d*x+c))^(7/2)-2/5*(a+a*sin(d*x+c))^(1/2)/d/e/(e*cos(d*x+c))^(7/2)
Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=\frac {2 \sqrt {e \cos (c+d x)} \sec ^4(c+d x) \sqrt {a (1+\sin (c+d x))} (-5-4 \cos (2 (c+d x))+10 \sin (c+d x)+4 \sin (3 (c+d x)))}{35 d e^5} \]
(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*Sqrt[a*(1 + Sin[c + d*x])]*(-5 - 4* Cos[2*(c + d*x)] + 10*Sin[c + d*x] + 4*Sin[3*(c + d*x)]))/(35*d*e^5)
Time = 0.72 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3042, 3151, 3042, 3151, 3042, 3151, 3042, 3150}
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.
\(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{(e \cos (c+d x))^{9/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{(e \cos (c+d x))^{9/2}}dx\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {6 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{9/2}}dx}{5 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {6 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{9/2}}dx}{5 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {6 \left (\frac {4 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{9/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\right )}{5 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {6 \left (\frac {4 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{9/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\right )}{5 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{9/2}}dx}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\right )}{5 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{9/2}}dx}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\right )}{5 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\right )}{5 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}\) |
(-2*Sqrt[a + a*Sin[c + d*x]])/(5*d*e*(e*Cos[c + d*x])^(7/2)) + (6*((-2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(7/2)) + (4*((2*(a + a*Sin[ c + d*x])^(5/2))/(3*d*e*(e*Cos[c + d*x])^(7/2)) - (4*(a + a*Sin[c + d*x])^ (7/2))/(21*a*d*e*(e*Cos[c + d*x])^(7/2))))/a))/(5*a)
3.3.79.3.1 Defintions of rubi rules used
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*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
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*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[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] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Time = 2.74 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {2 \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \left (16 \tan \left (d x +c \right )-8 \sec \left (d x +c \right )+6 \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )-\left (\sec ^{3}\left (d x +c \right )\right )\right )}{35 d \sqrt {e \cos \left (d x +c \right )}\, e^{4}}\) | \(74\) |
2/35/d*(a*(1+sin(d*x+c)))^(1/2)/(e*cos(d*x+c))^(1/2)/e^4*(16*tan(d*x+c)-8* sec(d*x+c)+6*tan(d*x+c)*sec(d*x+c)^2-sec(d*x+c)^3)
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35 \, d e^{5} \cos \left (d x + c\right )^{4}} \]
-2/35*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 - 2*(8*cos(d*x + c)^2 + 3)*si n(d*x + c) + 1)*sqrt(a*sin(d*x + c) + a)/(d*e^5*cos(d*x + c)^4)
Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (130) = 260\).
Time = 0.32 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \, {\left (9 \, \sqrt {a} \sqrt {e} - \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {9 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{35 \, {\left (e^{5} + \frac {4 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, e^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {e^{5} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]
-2/35*(9*sqrt(a)*sqrt(e) - 44*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 14*sqrt(a)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 84*sqrt(a)* sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 84*sqrt(a)*sqrt(e)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 14*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 44*sqrt(a)*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 9*sqr t(a)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^4/((e^5 + 4*e^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6* e^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*e^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + e^5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin(d*x + c)/(cos( d*x + c) + 1) + 1)^(7/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2))
Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \]
Time = 7.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {16\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (23\,\cos \left (c+d\,x\right )+11\,\cos \left (3\,c+3\,d\,x\right )+2\,\cos \left (5\,c+5\,d\,x\right )-16\,\sin \left (2\,c+2\,d\,x\right )-11\,\sin \left (4\,c+4\,d\,x\right )-2\,\sin \left (6\,c+6\,d\,x\right )\right )}{35\,d\,e^4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (15\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+\cos \left (6\,c+6\,d\,x\right )+10\right )} \]