\(\int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx\) [280]

Optimal result

Integrand size = 26, antiderivative size = 124 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {7 c^4 x}{a^3}-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2} \]

[Out]

-7*c^4*x/a^3-7*c^4*cos(f*x+e)/a^3/f-2/5*a^3*c^4*cos(f*x+e)^7/f/(a+a*sin(f*x+e))^6+14/15*a*c^4*cos(f*x+e)^5/f/(
a+a*sin(f*x+e))^4-14/3*c^4*cos(f*x+e)^3/a/f/(a+a*sin(f*x+e))^2

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2759, 2761, 8} \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}-\frac {7 c^4 x}{a^3}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a \sin (e+f x)+a)^2} \]

[In]

Int[(c - c*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^3,x]

[Out]

(-7*c^4*x)/a^3 - (7*c^4*Cos[e + f*x])/(a^3*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(5*f*(a + a*Sin[e + f*x])^6) + (14*
a*c^4*Cos[e + f*x]^5)/(15*f*(a + a*Sin[e + f*x])^4) - (14*c^4*Cos[e + f*x]^3)/(3*a*f*(a + a*Sin[e + f*x])^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 2815

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[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] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = \left (a^4 c^4\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^7} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}-\frac {1}{5} \left (7 a^2 c^4\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^5} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}+\frac {1}{3} \left (7 c^4\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^3} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2}-\frac {\left (7 c^4\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{a^2} \\ & = -\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2}-\frac {\left (7 c^4\right ) \int 1 \, dx}{a^3} \\ & = -\frac {7 c^4 x}{a^3}-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(124)=248\).

Time = 12.48 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.18 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (96 \sin \left (\frac {1}{2} (e+f x)\right )-48 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-256 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+464 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-105 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-15 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5\right ) (c-c \sin (e+f x))^4}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (a+a \sin (e+f x))^3} \]

[In]

Integrate[(c - c*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^3,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(96*Sin[(e + f*x)/2] - 48*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 256*S
in[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 128*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 464*Si
n[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 105*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5
- 15*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)*(c - c*Sin[e + f*x])^4)/(15*f*(Cos[(e + f*x)/2] -
Sin[(e + f*x)/2])^8*(a + a*Sin[e + f*x])^3)

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {2 c^{4} \left (-\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-7 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {64}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\)	\(102\)
default	\(\frac {2 c^{4} \left (-\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-7 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {64}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\)	\(102\)
risch	\(-\frac {7 c^{4} x}{a^{3}}-\frac {c^{4} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{3} f}-\frac {c^{4} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{3} f}-\frac {16 \left (120 i c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+45 \,{\mathrm e}^{4 i \left (f x +e \right )} c^{4}-100 i c^{4} {\mathrm e}^{i \left (f x +e \right )}-170 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+29 c^{4}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\)	\(137\)
parallelrisch	\(-\frac {c^{4} \left (2100 f x \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+2100 f x \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-210 f x \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+1050 f x \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-210 f x \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-1050 f x \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+2465 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+4215 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-15 \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+205 \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+2685 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-873 \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-15 \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )-655 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )\right )}{30 f \,a^{3} \left (10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\)	\(248\)
norman	\(\frac {-\frac {98 c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {35 c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}-\frac {286 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}-\frac {334 c^{4}}{15 f a}-\frac {497 c^{4} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {588 c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {588 c^{4} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {1366 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {860 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {16036 c^{4} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {2404 c^{4} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {1102 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {638 c^{4} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {66 c^{4} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {16 c^{4} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {210 c^{4} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {357 c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {497 c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {98 c^{4} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {10814 c^{4} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {357 c^{4} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {210 c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {35 c^{4} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {7 c^{4} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {3686 c^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {11524 c^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {7 c^{4} x}{a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\)	\(574\)

[In]

int((c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f*c^4/a^3*(-1/(1+tan(1/2*f*x+1/2*e)^2)-7*arctan(tan(1/2*f*x+1/2*e))-64/5/(tan(1/2*f*x+1/2*e)+1)^5+32/(tan(1/
2*f*x+1/2*e)+1)^4-64/3/(tan(1/2*f*x+1/2*e)+1)^3-8/(tan(1/2*f*x+1/2*e)+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (118) = 236\).

Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.06 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {15 \, c^{4} \cos \left (f x + e\right )^{4} - 420 \, c^{4} f x - 48 \, c^{4} + {\left (105 \, c^{4} f x + 277 \, c^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (315 \, c^{4} f x - 134 \, c^{4}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (35 \, c^{4} f x + 74 \, c^{4}\right )} \cos \left (f x + e\right ) + {\left (15 \, c^{4} \cos \left (f x + e\right )^{3} - 420 \, c^{4} f x + 48 \, c^{4} + {\left (105 \, c^{4} f x - 262 \, c^{4}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (35 \, c^{4} f x + 66 \, c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/15*(15*c^4*cos(f*x + e)^4 - 420*c^4*f*x - 48*c^4 + (105*c^4*f*x + 277*c^4)*cos(f*x + e)^3 + (315*c^4*f*x -
134*c^4)*cos(f*x + e)^2 - 6*(35*c^4*f*x + 74*c^4)*cos(f*x + e) + (15*c^4*cos(f*x + e)^3 - 420*c^4*f*x + 48*c^4
+ (105*c^4*f*x - 262*c^4)*cos(f*x + e)^2 - 6*(35*c^4*f*x + 66*c^4)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x
+ e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x +
e) - 4*a^3*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2314 vs. \(2 (119) = 238\).

Time = 13.82 (sec) , antiderivative size = 2314, normalized size of antiderivative = 18.66 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((c-c*sin(f*x+e))**4/(a+a*sin(f*x+e))**3,x)

[Out]

Piecewise((-105*c**4*f*x*tan(e/2 + f*x/2)**7/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 +
165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*
tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 525*c**4*f*x*tan(e/2 + f*x/2)**6/(15*a**3*f*ta
n(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/
2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3
*f) - 1155*c**4*f*x*tan(e/2 + f*x/2)**5/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a
**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e
/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 1575*c**4*f*x*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(e/
2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**
4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f)
- 1575*c**4*f*x*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*
f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 +
f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 1155*c**4*f*x*tan(e/2 + f*x/2)**2/(15*a**3*f*tan(e/2 +
f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 +
225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 52
5*c**4*f*x*tan(e/2 + f*x/2)/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/
2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)*
*2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 105*c**4*f*x/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2
+ f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**
3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 240*c**4*tan(e/2 + f*x/2)**6/(1
5*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan
(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2
) + 15*a**3*f) - 990*c**4*tan(e/2 + f*x/2)**5/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 +
165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f
*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 2470*c**4*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(
e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)
**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f
) - 2540*c**4*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*
tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f
*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 2684*c**4*tan(e/2 + f*x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)
**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a*
*3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 1430*c**
4*tan(e/2 + f*x/2)/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2
)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a
**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 334*c**4/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6
+ 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*
f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f), Ne(f, 0)), (x*(-c*sin(e) + c)**4/(a*sin(e) +
a)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (118) = 236\).

Time = 0.33 (sec) , antiderivative size = 1096, normalized size of antiderivative = 8.84 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

-2/15*(3*c^4*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 200*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 + 160*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 75*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 +
15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 11*a^3*sin(f*x + e
)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*a^3*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 + 11*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^3*sin(f*x +
e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + 4*c^4*((95*sin(f*x + e)/(cos(f*
x + e) + 1) + 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 + 22)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin
(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + c^4*(20*sin(f*x + e)/(co
s(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 30*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4 + 7)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*
x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*s
in(f*x + e)^5/(cos(f*x + e) + 1)^5) + 12*c^4*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10
*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos
(f*x + e) + 1)^5) - 12*c^4*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5*sin(
f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos
(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^
3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5))/f

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {\frac {105 \, {\left (f x + e\right )} c^{4}}{a^{3}} + \frac {30 \, c^{4}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{3}} + \frac {16 \, {\left (15 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 60 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 130 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 80 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19 \, c^{4}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \]

[In]

integrate((c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

-1/15*(105*(f*x + e)*c^4/a^3 + 30*c^4/((tan(1/2*f*x + 1/2*e)^2 + 1)*a^3) + 16*(15*c^4*tan(1/2*f*x + 1/2*e)^4 +
60*c^4*tan(1/2*f*x + 1/2*e)^3 + 130*c^4*tan(1/2*f*x + 1/2*e)^2 + 80*c^4*tan(1/2*f*x + 1/2*e) + 19*c^4)/(a^3*(
tan(1/2*f*x + 1/2*e) + 1)^5))/f

Mupad [B] (verification not implemented)

Time = 10.57 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.34 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\frac {7\,c^4\,\left (e+f\,x\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (35\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (525\,e+525\,f\,x+1430\right )}{15}\right )-\frac {c^4\,\left (105\,e+105\,f\,x+334\right )}{15}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (35\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (525\,e+525\,f\,x+240\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (77\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1155\,e+1155\,f\,x+990\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (77\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1155\,e+1155\,f\,x+2684\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (105\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1575\,e+1575\,f\,x+2470\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (105\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1575\,e+1575\,f\,x+2540\right )}{15}\right )}{a^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {7\,c^4\,x}{a^3} \]

[In]

int((c - c*sin(e + f*x))^4/(a + a*sin(e + f*x))^3,x)

[Out]

(7*c^4*(e + f*x) + tan(e/2 + (f*x)/2)*(35*c^4*(e + f*x) - (c^4*(525*e + 525*f*x + 1430))/15) - (c^4*(105*e + 1
05*f*x + 334))/15 + tan(e/2 + (f*x)/2)^6*(35*c^4*(e + f*x) - (c^4*(525*e + 525*f*x + 240))/15) + tan(e/2 + (f*
x)/2)^5*(77*c^4*(e + f*x) - (c^4*(1155*e + 1155*f*x + 990))/15) + tan(e/2 + (f*x)/2)^2*(77*c^4*(e + f*x) - (c^
4*(1155*e + 1155*f*x + 2684))/15) + tan(e/2 + (f*x)/2)^4*(105*c^4*(e + f*x) - (c^4*(1575*e + 1575*f*x + 2470))
/15) + tan(e/2 + (f*x)/2)^3*(105*c^4*(e + f*x) - (c^4*(1575*e + 1575*f*x + 2540))/15))/(a^3*f*(tan(e/2 + (f*x)
/2) + 1)^5*(tan(e/2 + (f*x)/2)^2 + 1)) - (7*c^4*x)/a^3