Integrand size = 26, antiderivative size = 34 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \] Output:
1/7*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^7
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).
Time = 4.58 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.74 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 \left (35 \cos \left (\frac {1}{2} (e+f x)\right )-21 \cos \left (\frac {3}{2} (e+f x)\right )-7 \cos \left (\frac {5}{2} (e+f x)\right )+\cos \left (\frac {7}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{28 c^4 f (-1+\sin (e+f x))^4} \] Input:
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4,x]
Output:
(a^3*(35*Cos[(e + f*x)/2] - 21*Cos[(3*(e + f*x))/2] - 7*Cos[(5*(e + f*x))/ 2] + Cos[(7*(e + f*x))/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/(28*c^4* f*(-1 + Sin[e + f*x])^4)
Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3042, 3215, 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.
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^4}dx\) |
\(\Big \downarrow \) 3215 |
\(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^7}dx\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle \frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}\) |
Input:
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4,x]
Output:
(a^3*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7)
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[((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]))
Time = 1.75 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.88
method | result | size |
parallelrisch | \(-\frac {2 \left (\frac {1}{7}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) a^{3}}{f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(64\) |
risch | \(-\frac {2 \left (7 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-35 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+21 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-a^{3}\right )}{7 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\) | \(71\) |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{4}}\) | \(118\) |
default | \(\frac {2 a^{3} \left (-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{4}}\) | \(118\) |
norman | \(\frac {-\frac {2 a^{3}}{7 c f}-\frac {48 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{7 c f}-\frac {202 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{7 c f}-\frac {352 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{7 c f}-\frac {42 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{c f}-\frac {16 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{f c}-\frac {2 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{f c}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(177\) |
Input:
int((a+sin(f*x+e)*a)^3/(c-c*sin(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
-2*(1/7+tan(1/2*f*x+1/2*e)^6+5*tan(1/2*f*x+1/2*e)^4+3*tan(1/2*f*x+1/2*e)^2 )*a^3/f/c^4/(tan(1/2*f*x+1/2*e)-1)^7
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (33) = 66\).
Time = 0.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.53 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^{3} \cos \left (f x + e\right )^{4} - 3 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} - {\left (a^{3} \cos \left (f x + e\right )^{3} + 4 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) - 8 \, a^{3}\right )} \sin \left (f x + e\right )}{7 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="fricas")
Output:
1/7*(a^3*cos(f*x + e)^4 - 3*a^3*cos(f*x + e)^3 - 8*a^3*cos(f*x + e)^2 + 4* a^3*cos(f*x + e) + 8*a^3 - (a^3*cos(f*x + e)^3 + 4*a^3*cos(f*x + e)^2 - 4* a^3*cos(f*x + e) - 8*a^3)*sin(f*x + e))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*co s(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. 619 vs. \(2 (29) = 58\).
Time = 15.48 (sec) , antiderivative size = 619, normalized size of antiderivative = 18.21 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**4,x)
Output:
Piecewise((-14*a**3*tan(e/2 + f*x/2)**6/(7*c**4*f*tan(e/2 + f*x/2)**7 - 49 *c**4*f*tan(e/2 + f*x/2)**6 + 147*c**4*f*tan(e/2 + f*x/2)**5 - 245*c**4*f* tan(e/2 + f*x/2)**4 + 245*c**4*f*tan(e/2 + f*x/2)**3 - 147*c**4*f*tan(e/2 + f*x/2)**2 + 49*c**4*f*tan(e/2 + f*x/2) - 7*c**4*f) - 70*a**3*tan(e/2 + f *x/2)**4/(7*c**4*f*tan(e/2 + f*x/2)**7 - 49*c**4*f*tan(e/2 + f*x/2)**6 + 1 47*c**4*f*tan(e/2 + f*x/2)**5 - 245*c**4*f*tan(e/2 + f*x/2)**4 + 245*c**4* f*tan(e/2 + f*x/2)**3 - 147*c**4*f*tan(e/2 + f*x/2)**2 + 49*c**4*f*tan(e/2 + f*x/2) - 7*c**4*f) - 42*a**3*tan(e/2 + f*x/2)**2/(7*c**4*f*tan(e/2 + f* x/2)**7 - 49*c**4*f*tan(e/2 + f*x/2)**6 + 147*c**4*f*tan(e/2 + f*x/2)**5 - 245*c**4*f*tan(e/2 + f*x/2)**4 + 245*c**4*f*tan(e/2 + f*x/2)**3 - 147*c** 4*f*tan(e/2 + f*x/2)**2 + 49*c**4*f*tan(e/2 + f*x/2) - 7*c**4*f) - 2*a**3/ (7*c**4*f*tan(e/2 + f*x/2)**7 - 49*c**4*f*tan(e/2 + f*x/2)**6 + 147*c**4*f *tan(e/2 + f*x/2)**5 - 245*c**4*f*tan(e/2 + f*x/2)**4 + 245*c**4*f*tan(e/2 + f*x/2)**3 - 147*c**4*f*tan(e/2 + f*x/2)**2 + 49*c**4*f*tan(e/2 + f*x/2) - 7*c**4*f), Ne(f, 0)), (x*(a*sin(e) + a)**3/(-c*sin(e) + c)**4, True))
Leaf count of result is larger than twice the leaf count of optimal. 1045 vs. \(2 (33) = 66\).
Time = 0.06 (sec) , antiderivative size = 1045, normalized size of antiderivative = 30.74 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="maxima")
Output:
2/35*(a^3*(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) - a^3*(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 - 210*si n(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^3*( 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 + e)^5/...
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (33) = 66\).
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.15 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (7 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 35 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 21 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3}\right )}}{7 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="giac")
Output:
-2/7*(7*a^3*tan(1/2*f*x + 1/2*e)^6 + 35*a^3*tan(1/2*f*x + 1/2*e)^4 + 21*a^ 3*tan(1/2*f*x + 1/2*e)^2 + a^3)/(c^4*f*(tan(1/2*f*x + 1/2*e) - 1)^7)
Time = 17.95 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.41 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {2\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+21\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{7\,c^4\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^7} \] Input:
int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^4,x)
Output:
(2*a^3*cos(e/2 + (f*x)/2)*(cos(e/2 + (f*x)/2)^6 + 7*sin(e/2 + (f*x)/2)^6 + 35*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^4 + 21*cos(e/2 + (f*x)/2)^4*si n(e/2 + (f*x)/2)^2))/(7*c^4*f*(cos(e/2 + (f*x)/2) - sin(e/2 + (f*x)/2))^7)
Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 4.35 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {2 a^{3} \left (3 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+\cos \left (f x +e \right )-\sin \left (f x +e \right )^{4}-6 \sin \left (f x +e \right )^{2}-1\right )}{7 c^{4} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-3 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+3 \cos \left (f x +e \right ) \sin \left (f x +e \right )-\cos \left (f x +e \right )+\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1\right )} \] Input:
int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x)
Output:
(2*a**3*(3*cos(e + f*x)*sin(e + f*x)**2 + cos(e + f*x) - sin(e + f*x)**4 - 6*sin(e + f*x)**2 - 1))/(7*c**4*f*(cos(e + f*x)*sin(e + f*x)**3 - 3*cos(e + f*x)*sin(e + f*x)**2 + 3*cos(e + f*x)*sin(e + f*x) - cos(e + f*x) + sin (e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1) )