Integrand size = 26, antiderivative size = 67 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx=\frac {a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac {a^2 c \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5} \]
Time = 1.93 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.75 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-35 \cos \left (\frac {1}{2} (e+f x)\right )+14 \cos \left (\frac {3}{2} (e+f x)\right )+\cos \left (\frac {7}{2} (e+f x)\right )-70 \sin \left (\frac {1}{2} (e+f x)\right )-35 \sin \left (\frac {3}{2} (e+f x)\right )+7 \sin \left (\frac {5}{2} (e+f x)\right )\right )}{140 c^4 f (-1+\sin (e+f x))^4} \]
-1/140*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-35*Cos[(e + f*x)/2] + 14*Cos[(3*(e + f*x))/2] + Cos[(7*(e + f*x))/2] - 70*Sin[(e + f*x)/2] - 35* Sin[(3*(e + f*x))/2] + 7*Sin[(5*(e + f*x))/2]))/(c^4*f*(-1 + Sin[e + f*x]) ^4)
Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 3215, 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.
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^4}dx\) |
\(\Big \downarrow \) 3215 |
\(\displaystyle a^2 c^2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^2 c^2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^6}dx\) |
\(\Big \downarrow \) 3151 |
\(\displaystyle a^2 c^2 \left (\frac {\int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^2 c^2 \left (\frac {\int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle a^2 c^2 \left (\frac {\cos ^5(e+f x)}{35 c f (c-c \sin (e+f x))^5}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )\) |
a^2*c^2*(Cos[e + f*x]^5/(7*f*(c - c*Sin[e + f*x])^6) + Cos[e + f*x]^5/(35* c*f*(c - c*Sin[e + f*x])^5))
3.3.44.3.1 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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.30
method | result | size |
risch | \(-\frac {2 i a^{2} \left (35 i {\mathrm e}^{4 i \left (f x +e \right )}+35 \,{\mathrm e}^{5 i \left (f x +e \right )}-14 i {\mathrm e}^{2 i \left (f x +e \right )}-70 \,{\mathrm e}^{3 i \left (f x +e \right )}-i+7 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{35 f \,c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7}}\) | \(87\) |
parallelrisch | \(-\frac {2 a^{2} \left (35 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-35 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+140 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-70 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+91 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+6\right )}{35 f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(103\) |
derivativedivides | \(\frac {2 a^{2} \left (-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {32}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {24}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{4}}\) | \(118\) |
default | \(\frac {2 a^{2} \left (-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {32}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {24}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{4}}\) | \(118\) |
norman | \(\frac {\frac {2 a^{2} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {12 a^{2}}{35 c f}-\frac {2 a^{2} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5 c f}+\frac {8 a^{2} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {12 a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {24 a^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}+\frac {52 a^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {116 a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {206 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}-\frac {656 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(263\) |
-2/35*I*a^2*(35*I*exp(4*I*(f*x+e))+35*exp(5*I*(f*x+e))-14*I*exp(2*I*(f*x+e ))-70*exp(3*I*(f*x+e))-I+7*exp(I*(f*x+e)))/f/c^4/(exp(I*(f*x+e))-I)^7
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (65) = 130\).
Time = 0.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.31 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx=-\frac {a^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{2} \cos \left (f x + e\right )^{3} + 13 \, a^{2} \cos \left (f x + e\right )^{2} - 10 \, a^{2} \cos \left (f x + e\right ) - 20 \, a^{2} - {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 10 \, a^{2} \cos \left (f x + e\right ) + 20 \, a^{2}\right )} \sin \left (f x + e\right )}{35 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]
-1/35*(a^2*cos(f*x + e)^4 + 4*a^2*cos(f*x + e)^3 + 13*a^2*cos(f*x + e)^2 - 10*a^2*cos(f*x + e) - 20*a^2 - (a^2*cos(f*x + e)^3 - 3*a^2*cos(f*x + e)^2 + 10*a^2*cos(f*x + e) + 20*a^2)*sin(f*x + e))/(c^4*f*cos(f*x + e)^4 - 3*c ^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^ 4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e ) - 8*c^4*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (58) = 116\).
Time = 8.63 (sec) , antiderivative size = 1074, normalized size of antiderivative = 16.03 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
Piecewise((-70*a**2*tan(e/2 + f*x/2)**6/(35*c**4*f*tan(e/2 + f*x/2)**7 - 2 45*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4 *f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan( e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f) + 70*a**2*tan(e /2 + f*x/2)**5/(35*c**4*f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2 )**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2)**2 + 245*c** 4*f*tan(e/2 + f*x/2) - 35*c**4*f) - 280*a**2*tan(e/2 + f*x/2)**4/(35*c**4* f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/ 2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f* x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 3 5*c**4*f) + 140*a**2*tan(e/2 + f*x/2)**3/(35*c**4*f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c** 4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan (e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f) - 182*a**2*tan (e/2 + f*x/2)**2/(35*c**4*f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x /2)**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2)**2 + 245*c **4*f*tan(e/2 + f*x/2) - 35*c**4*f) + 14*a**2*tan(e/2 + f*x/2)/(35*c**4*f* tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e...
Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (65) = 130\).
Time = 0.22 (sec) , antiderivative size = 816, normalized size of antiderivative = 12.18 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
2/105*(2*a^2*(91*sin(f*x + e)/(cos(f*x + e) + 1) - 168*sin(f*x + e)^2/(cos (f*x + e) + 1)^2 + 280*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 175*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 13) /(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos (f*x + e) + 1)^2 - 35*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin (f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1 )^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - c^4*sin(f*x + e)^7/(cos( f*x + e) + 1)^7) - 3*a^2*(49*sin(f*x + e)/(cos(f*x + e) + 1) - 147*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 210*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 2 10*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 35*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 12)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35 *c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) - 4* a^2*(14*sin(f*x + e)/(cos(f*x + e) + 1) - 42*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 35*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 35*sin(f*x + e)^4/(cos(f *x + e) + 1)^4 - 2)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 21*c^4* sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + ...
Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.81 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (35 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 35 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 140 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 70 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 91 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2}\right )}}{35 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \]
-2/35*(35*a^2*tan(1/2*f*x + 1/2*e)^6 - 35*a^2*tan(1/2*f*x + 1/2*e)^5 + 140 *a^2*tan(1/2*f*x + 1/2*e)^4 - 70*a^2*tan(1/2*f*x + 1/2*e)^3 + 91*a^2*tan(1 /2*f*x + 1/2*e)^2 - 7*a^2*tan(1/2*f*x + 1/2*e) + 6*a^2)/(c^4*f*(tan(1/2*f* x + 1/2*e) - 1)^7)
Time = 7.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.48 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx=\frac {\sqrt {2}\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {5\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {105\,\sin \left (e+f\,x\right )}{8}-\frac {27\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {121\,\cos \left (e+f\,x\right )}{8}+\frac {7\,\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {7\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {109}{4}\right )}{280\,c^4\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^7} \]