Integrand size = 27, antiderivative size = 113 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{7/2}}+\frac {8 (a+a \sin (c+d x))^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac {16 (a+a \sin (c+d x))^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}} \]
-2*(a+a*sin(d*x+c))^(3/2)/d/e/(e*cos(d*x+c))^(7/2)+8/3*(a+a*sin(d*x+c))^(5 /2)/a/d/e/(e*cos(d*x+c))^(7/2)-16/21*(a+a*sin(d*x+c))^(7/2)/a^2/d/e/(e*cos (d*x+c))^(7/2)
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx=\frac {2 \sqrt {e \cos (c+d x)} \sec ^4(c+d x) (a (1+\sin (c+d x)))^{3/2} (-5+4 \cos (2 (c+d x))+12 \sin (c+d x))}{21 d e^5} \]
(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*(a*(1 + Sin[c + d*x]))^(3/2)*(-5 + 4*Cos[2*(c + d*x)] + 12*Sin[c + d*x]))/(21*d*e^5)
Time = 0.56 (sec) , antiderivative size = 116, 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.
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^{3/2}}{(e \cos (c+d x))^{9/2}}dx\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {4 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{9/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{9/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle \frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{9/2}}dx}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{9/2}}dx}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle \frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}\) |
(-2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(7/2)) + (4*((2*(a + a*Sin[c + d*x])^(5/2))/(3*d*e*(e*Cos[c + d*x])^(7/2)) - (4*(a + a*Sin[c + d*x])^(7/2))/(21*a*d*e*(e*Cos[c + d*x])^(7/2))))/a
3.3.87.3.1 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 = 2.52 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {2 \left (\sec ^{3}\left (d x +c \right )\right ) \left (8 \left (\cos ^{2}\left (d x +c \right )\right )+12 \sin \left (d x +c \right )-9\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a \left (1+\sin \left (d x +c \right )\right )}{21 d \sqrt {e \cos \left (d x +c \right )}\, e^{4}}\) | \(68\) |
2/21/d*sec(d*x+c)^3*(8*cos(d*x+c)^2+12*sin(d*x+c)-9)*(a*(1+sin(d*x+c)))^(1 /2)*a*(1+sin(d*x+c))/(e*cos(d*x+c))^(1/2)/e^4
Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \, {\left (8 \, a \cos \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) - 9 \, a\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (d e^{5} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d e^{5} \cos \left (d x + c\right )^{2}\right )}} \]
-2/21*(8*a*cos(d*x + c)^2 + 12*a*sin(d*x + c) - 9*a)*sqrt(e*cos(d*x + c))* sqrt(a*sin(d*x + c) + a)/(d*e^5*cos(d*x + c)^2*sin(d*x + c) - d*e^5*cos(d* x + c)^2)
Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (97) = 194\).
Time = 0.32 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.49 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \, {\left (a^{\frac {3}{2}} \sqrt {e} - \frac {24 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {33 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {33 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {24 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{21 \, {\left (e^{5} + \frac {3 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {e^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]
-2/21*(a^(3/2)*sqrt(e) - 24*a^(3/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1 ) + 33*a^(3/2)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 33*a^(3/2)*sq rt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 24*a^(3/2)*sqrt(e)*sin(d*x + c )^5/(cos(d*x + c) + 1)^5 - a^(3/2)*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^5 + 3*e^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*e^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + e^ 5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2))
Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \]
Time = 6.96 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx=\frac {8\,a\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (12\,\cos \left (c+d\,x\right )-10\,\cos \left (3\,c+3\,d\,x\right )-17\,\sin \left (2\,c+2\,d\,x\right )+2\,\sin \left (4\,c+4\,d\,x\right )\right )}{21\,d\,e^4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\sin \left (c+d\,x\right )-4\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+4\,\sin \left (3\,c+3\,d\,x\right )-5\right )} \]