Integrand size = 27, antiderivative size = 115 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}-\frac {16 \sqrt {e \cos (c+d x)}}{45 a^2 d e \sqrt {a+a \sin (c+d x)}} \] Output:
-2/9*(e*cos(d*x+c))^(1/2)/d/e/(a+a*sin(d*x+c))^(5/2)-8/45*(e*cos(d*x+c))^( 1/2)/a/d/e/(a+a*sin(d*x+c))^(3/2)-16/45*(e*cos(d*x+c))^(1/2)/a^2/d/e/(a+a* sin(d*x+c))^(1/2)
Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \sqrt {e \cos (c+d x)} \left (17+20 \sin (c+d x)+8 \sin ^2(c+d x)\right )}{45 d e (a (1+\sin (c+d x)))^{5/2}} \] Input:
Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)),x]
Output:
(-2*Sqrt[e*Cos[c + d*x]]*(17 + 20*Sin[c + d*x] + 8*Sin[c + d*x]^2))/(45*d* e*(a*(1 + Sin[c + d*x]))^(5/2))
Time = 0.55 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {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.
\(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}}dx\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {4 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^{3/2}}dx}{9 a}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^{3/2}}dx}{9 a}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {4 \left (\frac {2 \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\sin (c+d x) a+a}}dx}{5 a}-\frac {2 \sqrt {e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}}\right )}{9 a}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \left (\frac {2 \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\sin (c+d x) a+a}}dx}{5 a}-\frac {2 \sqrt {e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}}\right )}{9 a}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle \frac {4 \left (-\frac {4 \sqrt {e \cos (c+d x)}}{5 a d e \sqrt {a \sin (c+d x)+a}}-\frac {2 \sqrt {e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}}\right )}{9 a}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}\) |
Input:
Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)),x]
Output:
(-2*Sqrt[e*Cos[c + d*x]])/(9*d*e*(a + a*Sin[c + d*x])^(5/2)) + (4*((-2*Sqr t[e*Cos[c + d*x]])/(5*d*e*(a + a*Sin[c + d*x])^(3/2)) - (4*Sqrt[e*Cos[c + d*x]])/(5*a*d*e*Sqrt[a + a*Sin[c + d*x]])))/(9*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]
Leaf count of result is larger than twice the leaf count of optimal. \(206\) vs. \(2(97)=194\).
Time = 11.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.80
method | result | size |
default | \(-\frac {2 \left (32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-32 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-72 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+23 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+17 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, a^{2} \sqrt {\left (1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}}\) | \(207\) |
Input:
int(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/45/d*(32*cos(1/2*d*x+1/2*c)^5-32*sin(1/2*d*x+1/2*c)*cos(1/2*d*x+1/2*c)^ 4-72*cos(1/2*d*x+1/2*c)^3-8*sin(1/2*d*x+1/2*c)*cos(1/2*d*x+1/2*c)^2+23*cos (1/2*d*x+1/2*c)+17*sin(1/2*d*x+1/2*c))/(2*cos(1/2*d*x+1/2*c)^3-2*sin(1/2*d *x+1/2*c)*cos(1/2*d*x+1/2*c)^2-3*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c))/(e *(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/a^2/((1+2*cos(1/2*d*x+1/2*c)*sin(1/2*d* x+1/2*c))*a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} - 20 \, \sin \left (d x + c\right ) - 25\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45 \, {\left (3 \, a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e + {\left (a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fric as")
Output:
-2/45*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 - 20*sin(d*x + c) - 25)*sqrt( a*sin(d*x + c) + a)/(3*a^3*d*e*cos(d*x + c)^2 - 4*a^3*d*e + (a^3*d*e*cos(d *x + c)^2 - 4*a^3*d*e)*sin(d*x + c))
Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))**(1/2)/(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. 287 vs. \(2 (97) = 194\).
Time = 0.16 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.50 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (17 \, \sqrt {a} \sqrt {e} + \frac {40 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {49 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {49 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {40 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {17 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{45 \, {\left (a^{3} e + \frac {3 \, a^{3} e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} e \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} e \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \] Input:
integrate(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxi ma")
Output:
-2/45*(17*sqrt(a)*sqrt(e) + 40*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 49*sqrt(a)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 49*sqrt(a) *sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 40*sqrt(a)*sqrt(e)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 17*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^3/((a^3*e + 3*a^3*e* sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^3*e*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a^3*e*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos( d*x + c) + 1) + 1)^(11/2)*sqrt(-sin(d*x + c)/(cos(d*x + c) + 1) + 1))
Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac ")
Output:
Timed out
Time = 17.73 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (137\,\cos \left (c+d\,x\right )-71\,\cos \left (3\,c+3\,d\,x\right )+2\,\cos \left (5\,c+5\,d\,x\right )+144\,\sin \left (2\,c+2\,d\,x\right )-18\,\sin \left (4\,c+4\,d\,x\right )\right )}{45\,a^3\,d\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (210\,\sin \left (c+d\,x\right )-120\,\cos \left (2\,c+2\,d\,x\right )+10\,\cos \left (4\,c+4\,d\,x\right )-45\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+126\right )} \] Input:
int(1/((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^(5/2)),x)
Output:
-(8*(a*(sin(c + d*x) + 1))^(1/2)*(137*cos(c + d*x) - 71*cos(3*c + 3*d*x) + 2*cos(5*c + 5*d*x) + 144*sin(2*c + 2*d*x) - 18*sin(4*c + 4*d*x)))/(45*a^3 *d*(e*cos(c + d*x))^(1/2)*(210*sin(c + d*x) - 120*cos(2*c + 2*d*x) + 10*co s(4*c + 4*d*x) - 45*sin(3*c + 3*d*x) + sin(5*c + 5*d*x) + 126))
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos \left (d x +c \right )}d x \right )}{a^{3} e} \] Input:
int(1/(e*cos(d*x+c))^(1/2)/(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)*sin(c + d*x)**3 + 3*cos(c + d*x)*sin(c + d*x)**2 + 3*cos(c + d*x)*sin (c + d*x) + cos(c + d*x)),x))/(a**3*e)