Integrand size = 25, antiderivative size = 73 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\cos ^2(e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 \sin (e+f x)}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \]
1/3*cos(f*x+e)^2*sin(f*x+e)/a/f/(a+b*sin(f*x+e)^2)^(3/2)+2/3*sin(f*x+e)/a^ 2/f/(a+b*sin(f*x+e)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {3 a \sin (e+f x)-(a-2 b) \sin ^3(e+f x)}{3 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3669, 292, 208}
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.
\(\displaystyle \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (e+f x)^3}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {1-\sin ^2(e+f x)}{\left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 292 |
\(\displaystyle \frac {\frac {2 \int \frac {1}{\left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{f}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\frac {2 \sin (e+f x)}{3 a^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (1-\sin ^2(e+f x)\right ) \sin (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{f}\) |
((Sin[e + f*x]*(1 - Sin[e + f*x]^2))/(3*a*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*Sin[e + f*x])/(3*a^2*Sqrt[a + b*Sin[e + f*x]^2]))/f
3.4.64.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(65)=130\).
Time = 0.35 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.89
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (f x +e \right )}{3 a {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}+\frac {2 \sin \left (f x +e \right )}{3 a^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}+\frac {\sin \left (f x +e \right )}{2 b {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}-\frac {a \left (\frac {\sin \left (f x +e \right )}{3 a {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}+\frac {2 \sin \left (f x +e \right )}{3 a^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}\right )}{2 b}}{f}\) | \(138\) |
default | \(\frac {\frac {\sin \left (f x +e \right )}{3 a {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}+\frac {2 \sin \left (f x +e \right )}{3 a^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}+\frac {\sin \left (f x +e \right )}{2 b {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}-\frac {a \left (\frac {\sin \left (f x +e \right )}{3 a {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}+\frac {2 \sin \left (f x +e \right )}{3 a^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}\right )}{2 b}}{f}\) | \(138\) |
1/f*(1/3*sin(f*x+e)/a/(a+b*sin(f*x+e)^2)^(3/2)+2/3/a^2*sin(f*x+e)/(a+b*sin (f*x+e)^2)^(1/2)+1/2*sin(f*x+e)/b/(a+b*sin(f*x+e)^2)^(3/2)-1/2*a/b*(1/3*si n(f*x+e)/a/(a+b*sin(f*x+e)^2)^(3/2)+2/3/a^2*sin(f*x+e)/(a+b*sin(f*x+e)^2)^ (1/2)))
Time = 0.46 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.47 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {{\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{2} + 2 \, a + 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{3 \, {\left (a^{2} b^{2} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{3} b + a^{2} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} f\right )}} \]
1/3*((a - 2*b)*cos(f*x + e)^2 + 2*a + 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b) *sin(f*x + e)/(a^2*b^2*f*cos(f*x + e)^4 - 2*(a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^4 + 2*a^3*b + a^2*b^2)*f)
Timed out. \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.47 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\frac {2 \, \sin \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2}} + \frac {\sin \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a} + \frac {\sin \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b} - \frac {\sin \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a b}}{3 \, f} \]
1/3*(2*sin(f*x + e)/(sqrt(b*sin(f*x + e)^2 + a)*a^2) + sin(f*x + e)/((b*si n(f*x + e)^2 + a)^(3/2)*a) + sin(f*x + e)/((b*sin(f*x + e)^2 + a)^(3/2)*b) - sin(f*x + e)/(sqrt(b*sin(f*x + e)^2 + a)*a*b))/f
Leaf count of result is larger than twice the leaf count of optimal. 35378 vs. \(2 (65) = 130\).
Time = 104.20 (sec) , antiderivative size = 35378, normalized size of antiderivative = 484.63 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
-2/3*(((((((3*a^8*b^4*tan(1/2*e)^57 + 12*a^7*b^5*tan(1/2*e)^57 + 18*a^6*b^ 6*tan(1/2*e)^57 + 12*a^5*b^7*tan(1/2*e)^57 + 3*a^4*b^8*tan(1/2*e)^57 + 80* a^8*b^4*tan(1/2*e)^55 + 340*a^7*b^5*tan(1/2*e)^55 + 560*a^6*b^6*tan(1/2*e) ^55 + 440*a^5*b^7*tan(1/2*e)^55 + 160*a^4*b^8*tan(1/2*e)^55 + 20*a^3*b^9*t an(1/2*e)^55 + 1030*a^8*b^4*tan(1/2*e)^53 + 4624*a^7*b^5*tan(1/2*e)^53 + 8 228*a^6*b^6*tan(1/2*e)^53 + 7272*a^5*b^7*tan(1/2*e)^53 + 3238*a^4*b^8*tan( 1/2*e)^53 + 632*a^3*b^9*tan(1/2*e)^53 + 32*a^2*b^10*tan(1/2*e)^53 + 8528*a ^8*b^4*tan(1/2*e)^51 + 40228*a^7*b^5*tan(1/2*e)^51 + 76400*a^6*b^6*tan(1/2 *e)^51 + 73880*a^5*b^7*tan(1/2*e)^51 + 37600*a^4*b^8*tan(1/2*e)^51 + 9188* a^3*b^9*tan(1/2*e)^51 + 768*a^2*b^10*tan(1/2*e)^51 + 51025*a^8*b^4*tan(1/2 *e)^49 + 251684*a^7*b^5*tan(1/2*e)^49 + 505318*a^6*b^6*tan(1/2*e)^49 + 524 932*a^5*b^7*tan(1/2*e)^49 + 294353*a^4*b^8*tan(1/2*e)^49 + 82912*a^3*b^9*t an(1/2*e)^49 + 8832*a^2*b^10*tan(1/2*e)^49 + 235040*a^8*b^4*tan(1/2*e)^47 + 1206776*a^7*b^5*tan(1/2*e)^47 + 2541472*a^6*b^6*tan(1/2*e)^47 + 2798928* a^5*b^7*tan(1/2*e)^47 + 1690112*a^4*b^8*tan(1/2*e)^47 + 525688*a^3*b^9*tan (1/2*e)^47 + 64768*a^2*b^10*tan(1/2*e)^47 + 867100*a^8*b^4*tan(1/2*e)^45 + 4613984*a^7*b^5*tan(1/2*e)^45 + 10124968*a^6*b^6*tan(1/2*e)^45 + 11702032 *a^5*b^7*tan(1/2*e)^45 + 7489628*a^4*b^8*tan(1/2*e)^45 + 2505712*a^3*b^9*t an(1/2*e)^45 + 340032*a^2*b^10*tan(1/2*e)^45 + 2631200*a^8*b^4*tan(1/2*e)^ 43 + 14449336*a^7*b^5*tan(1/2*e)^43 + 32845472*a^6*b^6*tan(1/2*e)^43 + ...
Time = 21.07 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.51 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )\,\sqrt {a+b\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,\left (a\,1{}\mathrm {i}-b\,2{}\mathrm {i}+a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,10{}\mathrm {i}+a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}+b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}-b\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,2{}\mathrm {i}\right )}{3\,a^2\,f\,{\left (b-4\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-2\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+b\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\right )}^2} \]
-(2*exp(e*1i + f*x*1i)*(exp(e*2i + f*x*2i) - 1)*(a + b*((exp(- e*1i - f*x* 1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*(a*1i - b*2i + a*exp(e*2i + f*x*2i)*10i + a*exp(e*4i + f*x*4i)*1i + b*exp(e*2i + f*x*2i)*4i - b*exp( e*4i + f*x*4i)*2i))/(3*a^2*f*(b - 4*a*exp(e*2i + f*x*2i) - 2*b*exp(e*2i + f*x*2i) + b*exp(e*4i + f*x*4i))^2)