Integrand size = 27, antiderivative size = 154 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {16 \sqrt {a+a \sin (c+d x)}}{15 a^2 d e (e \cos (c+d x))^{3/2}}+\frac {32 (a+a \sin (c+d x))^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}} \]
-2/9/d/e/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(3/2)+32/45*(a+a*sin(d*x+c) )^(3/2)/a^3/d/e/(e*cos(d*x+c))^(3/2)-4/15/a/d/e/(e*cos(d*x+c))^(3/2)/(a+a* sin(d*x+c))^(1/2)-16/15*(a+a*sin(d*x+c))^(1/2)/a^2/d/e/(e*cos(d*x+c))^(3/2 )
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 (7+12 \cos (2 (c+d x))-6 \sin (c+d x)+4 \sin (3 (c+d x)))}{45 d e (e \cos (c+d x))^{3/2} (a (1+\sin (c+d x)))^{3/2}} \]
(-2*(7 + 12*Cos[2*(c + d*x)] - 6*Sin[c + d*x] + 4*Sin[3*(c + d*x)]))/(45*d *e*(e*Cos[c + d*x])^(3/2)*(a*(1 + Sin[c + d*x]))^(3/2))
Time = 0.72 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {2 \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {\sin (c+d x) a+a}}dx}{3 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {\sin (c+d x) a+a}}dx}{3 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {2 \left (\frac {4 \int \frac {\sqrt {\sin (c+d x) a+a}}{(e \cos (c+d x))^{5/2}}dx}{5 a}-\frac {2}{5 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}\right )}{3 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 \int \frac {\sqrt {\sin (c+d x) a+a}}{(e \cos (c+d x))^{5/2}}dx}{5 a}-\frac {2}{5 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}\right )}{3 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {2 \left (\frac {4 \left (\frac {2 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{5/2}}dx}{a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}}\right )}{5 a}-\frac {2}{5 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}\right )}{3 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 \left (\frac {2 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{5/2}}dx}{a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}}\right )}{5 a}-\frac {2}{5 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}\right )}{3 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle \frac {2 \left (\frac {4 \left (\frac {4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}}\right )}{5 a}-\frac {2}{5 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}\right )}{3 a}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}\) |
-2/(9*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2)) + (2*(-2/(5*d *e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) + (4*((-2*Sqrt[a + a*S in[c + d*x]])/(d*e*(e*Cos[c + d*x])^(3/2)) + (4*(a + a*Sin[c + d*x])^(3/2) )/(3*a*d*e*(e*Cos[c + d*x])^(3/2))))/(5*a)))/(3*a)
3.4.11.3.1 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 = 83, normalized size of antiderivative = 0.54
| method | result | size |
| default | \(-\frac {2 \left (16 \cos \left (d x +c \right ) \sin \left (d x +c \right )+24 \cos \left (d x +c \right )-10 \tan \left (d x +c \right )-5 \sec \left (d x +c \right )\right )}{45 d a \,e^{2} \left (1+\sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {e \cos \left (d x +c \right )}}\) | \(83\) |
-2/45/d/a/e^2/(1+sin(d*x+c))/(a*(1+sin(d*x+c)))^(1/2)/(e*cos(d*x+c))^(1/2) *(16*cos(d*x+c)*sin(d*x+c)+24*cos(d*x+c)-10*tan(d*x+c)-5*sec(d*x+c))
Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (24 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 5\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45 \, {\left (a^{2} d e^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{2} d e^{3} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} d e^{3} \cos \left (d x + c\right )^{2}\right )}} \]
2/45*sqrt(e*cos(d*x + c))*(24*cos(d*x + c)^2 + 2*(8*cos(d*x + c)^2 - 5)*si n(d*x + c) - 5)*sqrt(a*sin(d*x + c) + a)/(a^2*d*e^3*cos(d*x + c)^4 - 2*a^2 *d*e^3*cos(d*x + c)^2*sin(d*x + c) - 2*a^2*d*e^3*cos(d*x + c)^2)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (130) = 260\).
Time = 0.34 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (19 \, \sqrt {a} \sqrt {e} + \frac {12 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {58 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {116 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {116 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {58 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {12 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {19 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{45 \, {\left (a^{2} e^{3} + \frac {4 \, a^{2} e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{2} e^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} e^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \]
-2/45*(19*sqrt(a)*sqrt(e) + 12*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 58*sqrt(a)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 116*sqrt(a )*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 116*sqrt(a)*sqrt(e)*sin(d* x + c)^5/(cos(d*x + c) + 1)^5 + 58*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 12*sqrt(a)*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 19 *sqrt(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/((a^2*e^3 + 4*a^2*e^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a^2*e^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a^2*e^3*sin(d* x + c)^6/(cos(d*x + c) + 1)^6 + a^2*e^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^ 8)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(5/2))
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Time = 10.55 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {14\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-12\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+24\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+8\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {225\,a^2\,d\,e^2\,\cos \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {45\,a^2\,d\,e^2\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}+45\,a^2\,d\,e^2\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}} \]
-(14*(a + a*sin(c + d*x))^(1/2) - 12*sin(c + d*x)*(a + a*sin(c + d*x))^(1/ 2) + 24*cos(2*c + 2*d*x)*(a + a*sin(c + d*x))^(1/2) + 8*sin(3*c + 3*d*x)*( a + a*sin(c + d*x))^(1/2))/((225*a^2*d*e^2*cos(c + d*x)*((e*exp(- c*1i - d *x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/4 - (45*a^2*d*e^2*cos(3*c + 3 *d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/4 + 4 5*a^2*d*e^2*sin(2*c + 2*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d *x*1i))/2)^(1/2))