Integrand size = 27, antiderivative size = 58 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {\cos ^5(e+f x)}{7 f (a+a \sin (e+f x))^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a+a \sin (e+f x))^5} \]
Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(58)=116\).
Time = 1.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.47 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {4585 \cos \left (e+\frac {f x}{2}\right )-2982 \cos \left (e+\frac {3 f x}{2}\right )-1148 \cos \left (3 e+\frac {5 f x}{2}\right )+197 \cos \left (3 e+\frac {7 f x}{2}\right )+2275 \sin \left (\frac {f x}{2}\right )+1134 \sin \left (2 e+\frac {3 f x}{2}\right )-224 \sin \left (2 e+\frac {5 f x}{2}\right )+\sin \left (4 e+\frac {7 f x}{2}\right )}{4620 a^6 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
(4585*Cos[e + (f*x)/2] - 2982*Cos[e + (3*f*x)/2] - 1148*Cos[3*e + (5*f*x)/ 2] + 197*Cos[3*e + (7*f*x)/2] + 2275*Sin[(f*x)/2] + 1134*Sin[2*e + (3*f*x) /2] - 224*Sin[2*e + (5*f*x)/2] + Sin[4*e + (7*f*x)/2])/(4620*a^6*f*(Cos[e/ 2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)
Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3042, 3338, 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.
\(\displaystyle \int \frac {\sin (e+f x) \cos ^4(e+f x)}{(a \sin (e+f x)+a)^6} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (e+f x) \cos (e+f x)^4}{(a \sin (e+f x)+a)^6}dx\) |
\(\Big \downarrow \) 3338 |
\(\displaystyle \frac {6 \int \frac {\cos ^4(e+f x)}{(\sin (e+f x) a+a)^5}dx}{7 a}+\frac {\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {6 \int \frac {\cos (e+f x)^4}{(\sin (e+f x) a+a)^5}dx}{7 a}+\frac {\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle \frac {\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a \sin (e+f x)+a)^5}\) |
Cos[e + f*x]^5/(7*f*(a + a*Sin[e + f*x])^6) - (6*Cos[e + f*x]^5)/(35*a*f*( a + a*Sin[e + f*x])^5)
3.5.40.3.1 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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 1, 0]
Time = 0.53 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.50
method | result | size |
parallelrisch | \(\frac {-\frac {2}{35}-2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {4 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}}{f \,a^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}\) | \(87\) |
risch | \(-\frac {2 \left (35 i {\mathrm e}^{5 i \left (f x +e \right )}+35 \,{\mathrm e}^{6 i \left (f x +e \right )}-70 i {\mathrm e}^{3 i \left (f x +e \right )}-140 \,{\mathrm e}^{4 i \left (f x +e \right )}+7 i {\mathrm e}^{i \left (f x +e \right )}+91 \,{\mathrm e}^{2 i \left (f x +e \right )}-6\right )}{35 f \,a^{6} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{7}}\) | \(94\) |
derivativedivides | \(\frac {\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {224}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{6}}\) | \(100\) |
default | \(\frac {\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {224}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{6}}\) | \(100\) |
2/35*(-1-35*tan(1/2*f*x+1/2*e)^5+35*tan(1/2*f*x+1/2*e)^4-70*tan(1/2*f*x+1/ 2*e)^3+14*tan(1/2*f*x+1/2*e)^2-7*tan(1/2*f*x+1/2*e))/f/a^6/(tan(1/2*f*x+1/ 2*e)+1)^7
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (54) = 108\).
Time = 0.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 3.36 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {6 \, \cos \left (f x + e\right )^{4} - 11 \, \cos \left (f x + e\right )^{3} - 27 \, \cos \left (f x + e\right )^{2} + {\left (6 \, \cos \left (f x + e\right )^{3} + 17 \, \cos \left (f x + e\right )^{2} - 10 \, \cos \left (f x + e\right ) - 20\right )} \sin \left (f x + e\right ) + 10 \, \cos \left (f x + e\right ) + 20}{35 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} - 3 \, a^{6} f \cos \left (f x + e\right )^{3} - 8 \, a^{6} f \cos \left (f x + e\right )^{2} + 4 \, a^{6} f \cos \left (f x + e\right ) + 8 \, a^{6} f - {\left (a^{6} f \cos \left (f x + e\right )^{3} + 4 \, a^{6} f \cos \left (f x + e\right )^{2} - 4 \, a^{6} f \cos \left (f x + e\right ) - 8 \, a^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
1/35*(6*cos(f*x + e)^4 - 11*cos(f*x + e)^3 - 27*cos(f*x + e)^2 + (6*cos(f* x + e)^3 + 17*cos(f*x + e)^2 - 10*cos(f*x + e) - 20)*sin(f*x + e) + 10*cos (f*x + e) + 20)/(a^6*f*cos(f*x + e)^4 - 3*a^6*f*cos(f*x + e)^3 - 8*a^6*f*c os(f*x + e)^2 + 4*a^6*f*cos(f*x + e) + 8*a^6*f - (a^6*f*cos(f*x + e)^3 + 4 *a^6*f*cos(f*x + e)^2 - 4*a^6*f*cos(f*x + e) - 8*a^6*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (49) = 98\).
Time = 93.37 (sec) , antiderivative size = 901, normalized size of antiderivative = 15.53 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\text {Too large to display} \]
Piecewise((-70*tan(e/2 + f*x/2)**5/(35*a**6*f*tan(e/2 + f*x/2)**7 + 245*a* *6*f*tan(e/2 + f*x/2)**6 + 735*a**6*f*tan(e/2 + f*x/2)**5 + 1225*a**6*f*ta n(e/2 + f*x/2)**4 + 1225*a**6*f*tan(e/2 + f*x/2)**3 + 735*a**6*f*tan(e/2 + f*x/2)**2 + 245*a**6*f*tan(e/2 + f*x/2) + 35*a**6*f) + 70*tan(e/2 + f*x/2 )**4/(35*a**6*f*tan(e/2 + f*x/2)**7 + 245*a**6*f*tan(e/2 + f*x/2)**6 + 735 *a**6*f*tan(e/2 + f*x/2)**5 + 1225*a**6*f*tan(e/2 + f*x/2)**4 + 1225*a**6* f*tan(e/2 + f*x/2)**3 + 735*a**6*f*tan(e/2 + f*x/2)**2 + 245*a**6*f*tan(e/ 2 + f*x/2) + 35*a**6*f) - 140*tan(e/2 + f*x/2)**3/(35*a**6*f*tan(e/2 + f*x /2)**7 + 245*a**6*f*tan(e/2 + f*x/2)**6 + 735*a**6*f*tan(e/2 + f*x/2)**5 + 1225*a**6*f*tan(e/2 + f*x/2)**4 + 1225*a**6*f*tan(e/2 + f*x/2)**3 + 735*a **6*f*tan(e/2 + f*x/2)**2 + 245*a**6*f*tan(e/2 + f*x/2) + 35*a**6*f) + 28* tan(e/2 + f*x/2)**2/(35*a**6*f*tan(e/2 + f*x/2)**7 + 245*a**6*f*tan(e/2 + f*x/2)**6 + 735*a**6*f*tan(e/2 + f*x/2)**5 + 1225*a**6*f*tan(e/2 + f*x/2)* *4 + 1225*a**6*f*tan(e/2 + f*x/2)**3 + 735*a**6*f*tan(e/2 + f*x/2)**2 + 24 5*a**6*f*tan(e/2 + f*x/2) + 35*a**6*f) - 14*tan(e/2 + f*x/2)/(35*a**6*f*ta n(e/2 + f*x/2)**7 + 245*a**6*f*tan(e/2 + f*x/2)**6 + 735*a**6*f*tan(e/2 + f*x/2)**5 + 1225*a**6*f*tan(e/2 + f*x/2)**4 + 1225*a**6*f*tan(e/2 + f*x/2) **3 + 735*a**6*f*tan(e/2 + f*x/2)**2 + 245*a**6*f*tan(e/2 + f*x/2) + 35*a* *6*f) - 2/(35*a**6*f*tan(e/2 + f*x/2)**7 + 245*a**6*f*tan(e/2 + f*x/2)**6 + 735*a**6*f*tan(e/2 + f*x/2)**5 + 1225*a**6*f*tan(e/2 + f*x/2)**4 + 12...
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (54) = 108\).
Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.64 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (\frac {7 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {14 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {70 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + 1\right )}}{35 \, {\left (a^{6} + \frac {7 \, a^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, a^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, a^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, a^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {21 \, a^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, a^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {a^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} f} \]
-2/35*(7*sin(f*x + e)/(cos(f*x + e) + 1) - 14*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 70*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 35*sin(f*x + e)^4/(cos( f*x + e) + 1)^4 + 35*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1)/((a^6 + 7*a^ 6*sin(f*x + e)/(cos(f*x + e) + 1) + 21*a^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 35*a^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*a^6*sin(f*x + e)^4/ (cos(f*x + e) + 1)^4 + 21*a^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 7*a^6* sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^6*sin(f*x + e)^7/(cos(f*x + e) + 1 )^7)*f)
Time = 0.49 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 70 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 14 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{35 \, a^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{7}} \]
-2/35*(35*tan(1/2*f*x + 1/2*e)^5 - 35*tan(1/2*f*x + 1/2*e)^4 + 70*tan(1/2* f*x + 1/2*e)^3 - 14*tan(1/2*f*x + 1/2*e)^2 + 7*tan(1/2*f*x + 1/2*e) + 1)/( a^6*f*(tan(1/2*f*x + 1/2*e) + 1)^7)
Time = 9.86 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.71 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-14\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+70\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-35\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+35\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}{35\,a^6\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^7} \]
-(2*cos(e/2 + (f*x)/2)^2*(cos(e/2 + (f*x)/2)^5 + 35*sin(e/2 + (f*x)/2)^5 - 35*cos(e/2 + (f*x)/2)*sin(e/2 + (f*x)/2)^4 + 7*cos(e/2 + (f*x)/2)^4*sin(e /2 + (f*x)/2) + 70*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^3 - 14*cos(e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2)^2))/(35*a^6*f*(cos(e/2 + (f*x)/2) + sin(e/ 2 + (f*x)/2))^7)