Integrand size = 27, antiderivative size = 154 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a^2 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^3 d e \sqrt {e \cos (c+d x)}} \] Output:
-2/11/d/e/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2)-12/77/a/d/e/(e*cos(d *x+c))^(1/2)/(a+a*sin(d*x+c))^(3/2)-16/77/a^2/d/e/(e*cos(d*x+c))^(1/2)/(a+ a*sin(d*x+c))^(1/2)+32/77*(a+a*sin(d*x+c))^(1/2)/a^3/d/e/(e*cos(d*x+c))^(1 /2)
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {-10+52 \sin (c+d x)+80 \sin ^2(c+d x)+32 \sin ^3(c+d x)}{77 d e \sqrt {e \cos (c+d x)} (a (1+\sin (c+d x)))^{5/2}} \] Input:
Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]
Output:
(-10 + 52*Sin[c + d*x] + 80*Sin[c + d*x]^2 + 32*Sin[c + d*x]^3)/(77*d*e*Sq rt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d*x]))^(5/2))
Time = 0.73 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06, 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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {6 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^{3/2}}dx}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \int \frac {1}{(e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^{3/2}}dx}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {6 \left (\frac {4 \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {\sin (c+d x) a+a}}dx}{7 a}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \left (\frac {4 \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {\sin (c+d x) a+a}}dx}{7 a}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (c+d x) a+a}}{(e \cos (c+d x))^{3/2}}dx}{3 a}-\frac {2}{3 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}\right )}{7 a}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (c+d x) a+a}}{(e \cos (c+d x))^{3/2}}dx}{3 a}-\frac {2}{3 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}\right )}{7 a}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {4 \sqrt {a \sin (c+d x)+a}}{3 a d e \sqrt {e \cos (c+d x)}}-\frac {2}{3 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}\right )}{7 a}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}\) |
Input:
Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]
Output:
-2/(11*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)) + (6*(-2/(7*d* e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)) + (4*(-2/(3*d*e*Sqrt[e* Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (4*Sqrt[a + a*Sin[c + d*x]])/(3* a*d*e*Sqrt[e*Cos[c + d*x]])))/(7*a)))/(11*a)
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 = 12.82 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.19
| method | result | size |
| default | \(-\frac {2 \left (5+4 \left (32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-13 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{77 d e \,a^{2} \left (1+4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, \sqrt {\left (1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}}\) | \(184\) |
Input:
int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/77/d/e/a^2*(5+4*(32*cos(1/2*d*x+1/2*c)^5-32*cos(1/2*d*x+1/2*c)^3-13*cos (1/2*d*x+1/2*c))*sin(1/2*d*x+1/2*c)+160*cos(1/2*d*x+1/2*c)^4-160*cos(1/2*d *x+1/2*c)^2)/(1+4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)-4*cos(1/2*d*x+1/2* c)^4+4*cos(1/2*d*x+1/2*c)^2)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/((1+2*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c))*a)^(1/2)
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (40 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 21\right )} \sin \left (d x + c\right ) - 35\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{77 \, {\left (3 \, a^{3} d e^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{3} d e^{2} \cos \left (d x + c\right ) + {\left (a^{3} d e^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{3} d e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fric as")
Output:
2/77*sqrt(e*cos(d*x + c))*(40*cos(d*x + c)^2 + 2*(8*cos(d*x + c)^2 - 21)*s in(d*x + c) - 35)*sqrt(a*sin(d*x + c) + a)/(3*a^3*d*e^2*cos(d*x + c)^3 - 4 *a^3*d*e^2*cos(d*x + c) + (a^3*d*e^2*cos(d*x + c)^3 - 4*a^3*d*e^2*cos(d*x + c))*sin(d*x + c))
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**(5/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (130) = 260\).
Time = 0.17 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (5 \, \sqrt {a} \sqrt {e} - \frac {52 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {150 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {180 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {180 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {150 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {52 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {5 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{77 \, {\left (a^{3} e^{2} + \frac {4 \, a^{3} e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{3} e^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{3} e^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} e^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxi ma")
Output:
-2/77*(5*sqrt(a)*sqrt(e) - 52*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 150*sqrt(a)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 180*sqrt(a )*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 180*sqrt(a)*sqrt(e)*sin(d* x + c)^5/(cos(d*x + c) + 1)^5 + 150*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d* x + c) + 1)^6 + 52*sqrt(a)*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 5 *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^3*e^2 + 4*a^3*e^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a^3*e^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a^3*e^2*sin(d* x + c)^6/(cos(d*x + c) + 1)^6 + a^3*e^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^ 8)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2))
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac ")
Output:
Timed out
Time = 20.56 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {76\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+30\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-40\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-8\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {385\,a^3\,d\,e\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}+\frac {1155\,a^3\,d\,e\,\sin \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {231\,a^3\,d\,e\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}-\frac {77\,a^3\,d\,e\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \] Input:
int(1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(5/2)),x)
Output:
(76*sin(c + d*x)*(a + a*sin(c + d*x))^(1/2) + 30*(a + a*sin(c + d*x))^(1/2 ) - 40*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))/((385*a^3*d*e*((e*exp(- c*1i - d*x*1i))/2 + (e*e xp(c*1i + d*x*1i))/2)^(1/2))/2 + (1155*a^3*d*e*sin(c + d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/4 - (231*a^3*d*e*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 - (77*a^3*d*e*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)
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+\cos \left (d x +c \right )^{2}}d x \right )}{a^{3} e^{2}} \] Input:
int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x)
Output:
(sqrt(e)*sqrt(a)*int((sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x)))/(cos(c + d*x)**2*sin(c + d*x)**3 + 3*cos(c + d*x)**2*sin(c + d*x)**2 + 3*cos(c + d* x)**2*sin(c + d*x) + cos(c + d*x)**2),x))/(a**3*e**2)