Integrand size = 27, antiderivative size = 115 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {2 \sqrt {a+a \sin (c+d x)}}{3 d e (e \cos (c+d x))^{5/2}}+\frac {8 (a+a \sin (c+d x))^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac {16 (a+a \sin (c+d x))^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}} \] Output:
-2/3*(a+a*sin(d*x+c))^(1/2)/d/e/(e*cos(d*x+c))^(5/2)+8/3*(a+a*sin(d*x+c))^ (3/2)/a/d/e/(e*cos(d*x+c))^(5/2)-16/15*(a+a*sin(d*x+c))^(5/2)/a^2/d/e/(e*c os(d*x+c))^(5/2)
Time = 0.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 \sqrt {a (1+\sin (c+d x))} (3+4 \cos (2 (c+d x))+4 \sin (c+d x))}{15 d e (e \cos (c+d x))^{5/2}} \] Input:
Integrate[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(7/2),x]
Output:
(2*Sqrt[a*(1 + Sin[c + d*x])]*(3 + 4*Cos[2*(c + d*x)] + 4*Sin[c + d*x]))/( 15*d*e*(e*Cos[c + d*x])^(5/2))
Time = 0.55 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{(e \cos (c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{(e \cos (c+d x))^{7/2}}dx\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {4 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{7/2}}dx}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{7/2}}dx}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{7/2}}dx}{a}\right )}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{7/2}}dx}{a}\right )}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle \frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}}\right )}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\) |
Input:
Int[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(7/2),x]
Output:
(-2*Sqrt[a + a*Sin[c + d*x]])/(3*d*e*(e*Cos[c + d*x])^(5/2)) + (4*((2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(5/2)) - (4*(a + a*Sin[c + d *x])^(5/2))/(5*a*d*e*(e*Cos[c + d*x])^(5/2))))/(3*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 = 7.63 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {2 \left (32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+7\right ) \sqrt {\left (1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}}{15 e^{3} d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2} \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}}\) | \(119\) |
Input:
int((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
2/15/e^3/d*(32*cos(1/2*d*x+1/2*c)^4-32*cos(1/2*d*x+1/2*c)^2+8*cos(1/2*d*x+ 1/2*c)*sin(1/2*d*x+1/2*c)+7)*((1+2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c))* a)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)^2/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{15 \, d e^{4} \cos \left (d x + c\right )^{3}} \] Input:
integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x, algorithm="fricas ")
Output:
2/15*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 + 4*sin(d*x + c) - 1)*sqrt(a*s in(d*x + c) + a)/(d*e^4*cos(d*x + c)^3)
Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(d*x+c))**(1/2)/(e*cos(d*x+c))**(7/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (97) = 194\).
Time = 0.15 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.45 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 \, {\left (7 \, \sqrt {a} \sqrt {e} + \frac {8 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {25 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {8 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {7 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, {\left (e^{4} + \frac {3 \, e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {e^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}}} \] Input:
integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x, algorithm="maxima ")
Output:
2/15*(7*sqrt(a)*sqrt(e) + 8*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1 ) - 25*sqrt(a)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 25*sqrt(a)*sq rt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 8*sqrt(a)*sqrt(e)*sin(d*x + c) ^5/(cos(d*x + c) + 1)^5 - 7*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/((e^4 + 3*e^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*e^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + e ^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(5/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))
Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x, algorithm="giac")
Output:
Timed out
Time = 27.66 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (2\,\sin \left (c+d\,x\right )+7\,\cos \left (2\,c+2\,d\,x\right )+2\,\cos \left (4\,c+4\,d\,x\right )+2\,\sin \left (3\,c+3\,d\,x\right )+5\right )}{15\,d\,e^3\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+3\right )} \] Input:
int((a + a*sin(c + d*x))^(1/2)/(e*cos(c + d*x))^(7/2),x)
Output:
(8*(a*(sin(c + d*x) + 1))^(1/2)*(2*sin(c + d*x) + 7*cos(2*c + 2*d*x) + 2*c os(4*c + 4*d*x) + 2*sin(3*c + 3*d*x) + 5))/(15*d*e^3*(e*cos(c + d*x))^(1/2 )*(4*cos(2*c + 2*d*x) + cos(4*c + 4*d*x) + 3))
\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right )}{e^{4}} \] Input:
int((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x)
Output:
(sqrt(e)*sqrt(a)*int((sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d *x)**4,x))/e**4