Integrand size = 27, antiderivative size = 152 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx=-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac {4 (a+a \sin (c+d x))^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac {16 (a+a \sin (c+d x))^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac {32 (a+a \sin (c+d x))^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}} \]
-2/3*(a+a*sin(d*x+c))^(3/2)/d/e/(e*cos(d*x+c))^(9/2)+4*(a+a*sin(d*x+c))^(5 /2)/a/d/e/(e*cos(d*x+c))^(9/2)-16/5*(a+a*sin(d*x+c))^(7/2)/a^2/d/e/(e*cos( d*x+c))^(9/2)+32/45*(a+a*sin(d*x+c))^(9/2)/a^3/d/e/(e*cos(d*x+c))^(9/2)
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.49 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx=\frac {2 \sqrt {e \cos (c+d x)} \sec ^5(c+d x) (a (1+\sin (c+d x)))^{3/2} (7+12 \cos (2 (c+d x))+6 \sin (c+d x)-4 \sin (3 (c+d x)))}{45 d e^6} \]
(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^5*(a*(1 + Sin[c + d*x]))^(3/2)*(7 + 1 2*Cos[2*(c + d*x)] + 6*Sin[c + d*x] - 4*Sin[3*(c + d*x)]))/(45*d*e^6)
Time = 0.72 (sec) , antiderivative size = 158, 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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^{3/2}}{(e \cos (c+d x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {2 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{11/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{11/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {2 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {4 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{11/2}}dx}{a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {4 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{11/2}}dx}{a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {2 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{5 d e (e \cos (c+d x))^{9/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{9/2}}{(e \cos (c+d x))^{11/2}}dx}{5 a}\right )}{a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{5 d e (e \cos (c+d x))^{9/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{9/2}}{(e \cos (c+d x))^{11/2}}dx}{5 a}\right )}{a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle \frac {2 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{5 d e (e \cos (c+d x))^{9/2}}-\frac {4 (a \sin (c+d x)+a)^{9/2}}{45 a d e (e \cos (c+d x))^{9/2}}\right )}{a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}\) |
(-2*(a + a*Sin[c + d*x])^(3/2))/(3*d*e*(e*Cos[c + d*x])^(9/2)) + (2*((2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(9/2)) - (4*((2*(a + a*Sin [c + d*x])^(7/2))/(5*d*e*(e*Cos[c + d*x])^(9/2)) - (4*(a + a*Sin[c + d*x]) ^(9/2))/(45*a*d*e*(e*Cos[c + d*x])^(9/2))))/a))/a
3.3.88.3.1 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 = 2.67 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(\frac {2 a \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \left (16+8 \sec \left (d x +c \right ) \tan \left (d x +c \right )-2 \left (\sec ^{2}\left (d x +c \right )\right )+5 \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )+5 \left (\sec ^{4}\left (d x +c \right )\right )\right )}{45 d \sqrt {e \cos \left (d x +c \right )}\, e^{5}}\) | \(84\) |
2/45/d*a*(a*(1+sin(d*x+c)))^(1/2)/(e*cos(d*x+c))^(1/2)/e^5*(16+8*sec(d*x+c )*tan(d*x+c)-2*sec(d*x+c)^2+5*tan(d*x+c)*sec(d*x+c)^3+5*sec(d*x+c)^4)
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx=-\frac {2 \, {\left (24 \, a \cos \left (d x + c\right )^{2} - 2 \, {\left (8 \, a \cos \left (d x + c\right )^{2} - 5 \, a\right )} \sin \left (d x + c\right ) - 5 \, a\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45 \, {\left (d e^{6} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - d e^{6} \cos \left (d x + c\right )^{3}\right )}} \]
-2/45*(24*a*cos(d*x + c)^2 - 2*(8*a*cos(d*x + c)^2 - 5*a)*sin(d*x + c) - 5 *a)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*e^6*cos(d*x + c)^3*si n(d*x + c) - d*e^6*cos(d*x + c)^3)
Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (130) = 260\).
Time = 0.33 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.35 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx=\frac {2 \, {\left (19 \, a^{\frac {3}{2}} \sqrt {e} - \frac {12 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {58 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {116 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {116 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {58 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {12 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {19 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{45 \, {\left (e^{6} + \frac {4 \, e^{6} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, e^{6} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, e^{6} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {e^{6} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}}} \]
2/45*(19*a^(3/2)*sqrt(e) - 12*a^(3/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 58*a^(3/2)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 116*a^(3/2) *sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 116*a^(3/2)*sqrt(e)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 58*a^(3/2)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 12*a^(3/2)*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 19* a^(3/2)*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^6 + 4*e^6*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*e^6*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*e^6*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + e^6*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin(d*x + c)/(c os(d*x + c) + 1) + 1)^(5/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2))
Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx=\text {Timed out} \]
Time = 11.69 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.72 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx=\frac {14\,a\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+12\,a\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+24\,a\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-8\,a\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {45\,d\,e^5\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}+\frac {45\,d\,e^5\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}-\frac {45\,d\,e^5\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {45\,d\,e^5\,\sin \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \]
(14*a*(a + a*sin(c + d*x))^(1/2) + 12*a*sin(c + d*x)*(a + a*sin(c + d*x))^ (1/2) + 24*a*cos(2*c + 2*d*x)*(a + a*sin(c + d*x))^(1/2) - 8*a*sin(3*c + 3 *d*x)*(a + a*sin(c + d*x))^(1/2))/((45*d*e^5*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/2 + (45*d*e^5*cos(2*c + 2*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/2 - (45*d*e^5*sin(3* c + 3*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/ 4 - (45*d*e^5*sin(c + d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x *1i))/2)^(1/2))/4)