Integrand size = 27, antiderivative size = 193 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {256 (a+a \sin (c+d x))^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}} \]
-2/11/d/e/(e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^(3/2)+128/77*(a+a*sin(d*x+ c))^(3/2)/a^3/d/e/(e*cos(d*x+c))^(5/2)-256/385*(a+a*sin(d*x+c))^(5/2)/a^4/ d/e/(e*cos(d*x+c))^(5/2)-16/77/a/d/e/(e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c)) ^(1/2)-32/77*(a+a*sin(d*x+c))^(1/2)/a^2/d/e/(e*cos(d*x+c))^(5/2)
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 (45+8 \cos (2 (c+d x))-16 \cos (4 (c+d x))+104 \sin (c+d x)+48 \sin (3 (c+d x)))}{385 d e (e \cos (c+d x))^{5/2} (a (1+\sin (c+d x)))^{3/2}} \]
(2*(45 + 8*Cos[2*(c + d*x)] - 16*Cos[4*(c + d*x)] + 104*Sin[c + d*x] + 48* Sin[3*(c + d*x)]))/(385*d*e*(e*Cos[c + d*x])^(5/2)*(a*(1 + Sin[c + d*x]))^ (3/2))
Time = 0.88 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3151, 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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {8 \int \frac {1}{(e \cos (c+d x))^{7/2} \sqrt {\sin (c+d x) a+a}}dx}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8 \int \frac {1}{(e \cos (c+d x))^{7/2} \sqrt {\sin (c+d x) a+a}}dx}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {8 \left (\frac {6 \int \frac {\sqrt {\sin (c+d x) a+a}}{(e \cos (c+d x))^{7/2}}dx}{7 a}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8 \left (\frac {6 \int \frac {\sqrt {\sin (c+d x) a+a}}{(e \cos (c+d x))^{7/2}}dx}{7 a}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{7/2}}dx}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\right )}{7 a}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \int \frac {(\sin (c+d x) a+a)^{3/2}}{(e \cos (c+d x))^{7/2}}dx}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\right )}{7 a}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{7/2}}dx}{a}\right )}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\right )}{7 a}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{5/2}}{(e \cos (c+d x))^{7/2}}dx}{a}\right )}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\right )}{7 a}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}}\right )}{3 a}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}\right )}{7 a}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}\right )}{11 a}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}\) |
-2/(11*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)) + (8*(-2/(7* d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]) + (6*((-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)))/(7*a)))/(11*a)
3.4.12.3.1 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 = 2.81 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.46
| method | result | size |
| default | \(-\frac {2 \left (128 \left (\cos ^{2}\left (d x +c \right )\right )-192 \sin \left (d x +c \right )-144-56 \sec \left (d x +c \right ) \tan \left (d x +c \right )-21 \left (\sec ^{2}\left (d x +c \right )\right )\right )}{385 d a \,e^{3} \left (1+\sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {e \cos \left (d x +c \right )}}\) | \(88\) |
-2/385/d/a/e^3/(1+sin(d*x+c))/(a*(1+sin(d*x+c)))^(1/2)/(e*cos(d*x+c))^(1/2 )*(128*cos(d*x+c)^2-192*sin(d*x+c)-144-56*sec(d*x+c)*tan(d*x+c)-21*sec(d*x +c)^2)
Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (128 \, \cos \left (d x + c\right )^{4} - 144 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (24 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) - 21\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{385 \, {\left (a^{2} d e^{4} \cos \left (d x + c\right )^{5} - 2 \, a^{2} d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d e^{4} \cos \left (d x + c\right )^{3}\right )}} \]
2/385*(128*cos(d*x + c)^4 - 144*cos(d*x + c)^2 - 8*(24*cos(d*x + c)^2 + 7) *sin(d*x + c) - 21)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(a^2*d*e ^4*cos(d*x + c)^5 - 2*a^2*d*e^4*cos(d*x + c)^3*sin(d*x + c) - 2*a^2*d*e^4* cos(d*x + c)^3)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (163) = 326\).
Time = 0.34 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (37 \, \sqrt {a} \sqrt {e} + \frac {496 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {559 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {544 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1526 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1526 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {544 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {559 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {496 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {37 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5}}{385 \, {\left (a^{2} e^{4} + \frac {5 \, a^{2} e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{2} e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{2} e^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{2} e^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{2} e^{4} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\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 {7}{2}}} \]
2/385*(37*sqrt(a)*sqrt(e) + 496*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 559*sqrt(a)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 544*sqrt (a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1526*sqrt(a)*sqrt(e)*sin (d*x + c)^4/(cos(d*x + c) + 1)^4 + 1526*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(co s(d*x + c) + 1)^6 + 544*sqrt(a)*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^ 7 - 559*sqrt(a)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 496*sqrt(a)* sqrt(e)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 37*sqrt(a)*sqrt(e)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/ ((a^2*e^4 + 5*a^2*e^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^2*e^4*sin (d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a^2*e^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*a^2*e^4*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^2*e^4*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13 /2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Time = 11.77 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.14 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}\,\left (\frac {288\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}}{77\,a^2\,d\,e^3}+\frac {256\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )}{385\,a^2\,d\,e^3}-\frac {512\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )}{385\,a^2\,d\,e^3}+\frac {1536\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )}{385\,a^2\,d\,e^3}+\frac {3328\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )}{385\,a^2\,d\,e^3}\right )}{10\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}-2\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}} \]
((a + a*sin(c + d*x))^(1/2)*((288*exp(c*4i + d*x*4i))/(77*a^2*d*e^3) + (25 6*exp(c*4i + d*x*4i)*cos(2*c + 2*d*x))/(385*a^2*d*e^3) - (512*exp(c*4i + d *x*4i)*cos(4*c + 4*d*x))/(385*a^2*d*e^3) + (1536*exp(c*4i + d*x*4i)*sin(3* c + 3*d*x))/(385*a^2*d*e^3) + (3328*exp(c*4i + d*x*4i)*sin(c + d*x))/(385* a^2*d*e^3)))/(10*exp(c*4i + d*x*4i)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2) + 8*exp(c*4i + d*x*4i)*sin(c + d*x)*(e*(exp(- c*1i - d *x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2) + 8*exp(c*4i + d*x*4i)*cos(2*c + 2 *d*x)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2) - 2*exp(c* 4i + d*x*4i)*cos(4*c + 4*d*x)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x* 1i)/2))^(1/2) + 8*exp(c*4i + d*x*4i)*sin(3*c + 3*d*x)*(e*(exp(- c*1i - d*x *1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2))