\(\int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx\) [234]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=\frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3} \]

[Out]

1/7*a*c*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^5+2/35*a*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^4+2/105*a*cos(f*x+e)^3/c/f/(c
-c*sin(f*x+e))^3

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2815, 2751, 2750} \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5} \]

[In]

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

[Out]

(a*c*Cos[e + f*x]^3)/(7*f*(c - c*Sin[e + f*x])^5) + (2*a*Cos[e + f*x]^3)/(35*f*(c - c*Sin[e + f*x])^4) + (2*a*
Cos[e + f*x]^3)/(105*c*f*(c - c*Sin[e + f*x])^3)

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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]

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[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[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

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}& = (a c) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^5} \, dx \\ & = \frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {1}{7} (2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^4} \, dx \\ & = \frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {(2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{35 c} \\ & = \frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.51 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.18 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=\frac {a \left (70 \cos \left (e+\frac {f x}{2}\right )-21 \cos \left (e+\frac {3 f x}{2}\right )+\cos \left (3 e+\frac {7 f x}{2}\right )+35 \sin \left (\frac {f x}{2}\right )+7 \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{210 c^4 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]

[In]

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

[Out]

(a*(70*Cos[e + (f*x)/2] - 21*Cos[e + (3*f*x)/2] + Cos[3*e + (7*f*x)/2] + 35*Sin[(f*x)/2] + 7*Sin[2*e + (5*f*x)
/2]))/(210*c^4*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82

method	result	size
risch	\(\frac {\frac {4 a}{105}+\frac {4 i a \,{\mathrm e}^{3 i \left (f x +e \right )}}{3}+\frac {8 a \,{\mathrm e}^{4 i \left (f x +e \right )}}{3}+\frac {4 i a \,{\mathrm e}^{i \left (f x +e \right )}}{15}-\frac {4 a \,{\mathrm e}^{2 i \left (f x +e \right )}}{5}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\)	\(75\)
parallelrisch	\(-\frac {2 a \left (105 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-210 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+455 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-350 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+273 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-56 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+23\right )}{105 f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\)	\(101\)
derivativedivides	\(\frac {2 a \left (-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {28}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {16}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {68}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\right )}{f \,c^{4}}\)	\(116\)
default	\(\frac {2 a \left (-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {28}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {16}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {68}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\right )}{f \,c^{4}}\)	\(116\)
norman	\(\frac {-\frac {46 a}{105 c f}-\frac {2 a \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {4 a \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {32 a \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {32 a \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {16 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 c f}-\frac {208 a \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}+\frac {116 a \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {592 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\)	\(201\)

[In]

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

[Out]

4/105*(a+35*I*a*exp(3*I*(f*x+e))+70*a*exp(4*I*(f*x+e))+7*I*a*exp(I*(f*x+e))-21*a*exp(2*I*(f*x+e)))/(exp(I*(f*x
+e))-I)^7/f/c^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (89) = 178\).

Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.24 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=\frac {2 \, a \cos \left (f x + e\right )^{4} + 8 \, a \cos \left (f x + e\right )^{3} - 9 \, a \cos \left (f x + e\right )^{2} + 15 \, a \cos \left (f x + e\right ) - {\left (2 \, a \cos \left (f x + e\right )^{3} - 6 \, a \cos \left (f x + e\right )^{2} - 15 \, a \cos \left (f x + e\right ) - 30 \, a\right )} \sin \left (f x + e\right ) + 30 \, a}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

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

[Out]

1/105*(2*a*cos(f*x + e)^4 + 8*a*cos(f*x + e)^3 - 9*a*cos(f*x + e)^2 + 15*a*cos(f*x + e) - (2*a*cos(f*x + e)^3
- 6*a*cos(f*x + e)^2 - 15*a*cos(f*x + e) - 30*a)*sin(f*x + e) + 30*a)/(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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1061 vs. \(2 (82) = 164\).

Time = 4.99 (sec) , antiderivative size = 1061, normalized size of antiderivative = 11.53 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((-210*a*tan(e/2 + f*x/2)**6/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*
c**4*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*t
an(e/2 + f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) + 420*a*tan(e/2 + f*x/2)**5/(105*c**4*f*tan(e/2
+ f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)
**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 + f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c*
*4*f) - 910*a*tan(e/2 + f*x/2)**4/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4
*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e
/2 + f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) + 700*a*tan(e/2 + f*x/2)**3/(105*c**4*f*tan(e/2 + f
*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)**4
+ 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 + f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f
) - 546*a*tan(e/2 + f*x/2)**2/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*t
an(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 +
f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) + 112*a*tan(e/2 + f*x/2)/(105*c**4*f*tan(e/2 + f*x/2)**
7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)**4 + 3675*
c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 + f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) - 46*
a/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675*c*
*4*f*tan(e/2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 + f*x/2)**2 + 735*c**4*f*tan(
e/2 + f*x/2) - 105*c**4*f), Ne(f, 0)), (x*(a*sin(e) + a)/(-c*sin(e) + c)**4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (89) = 178\).

Time = 0.22 (sec) , antiderivative size = 561, normalized size of antiderivative = 6.10 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=\frac {2 \, {\left (\frac {a {\left (\frac {91 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {168 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {280 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {175 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - 13\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} - \frac {3 \, a {\left (\frac {49 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {147 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {210 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 12\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}}\right )}}{105 \, f} \]

[In]

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

[Out]

2/105*(a*(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/(c
os(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*(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))
/f

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (105 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 210 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 455 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 350 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 273 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 56 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 23 \, a\right )}}{105 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \]

[In]

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

[Out]

-2/105*(105*a*tan(1/2*f*x + 1/2*e)^6 - 210*a*tan(1/2*f*x + 1/2*e)^5 + 455*a*tan(1/2*f*x + 1/2*e)^4 - 350*a*tan
(1/2*f*x + 1/2*e)^3 + 273*a*tan(1/2*f*x + 1/2*e)^2 - 56*a*tan(1/2*f*x + 1/2*e) + 23*a)/(c^4*f*(tan(1/2*f*x + 1
/2*e) - 1)^7)

Mupad [B] (verification not implemented)

Time = 6.96 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx=\frac {\sqrt {2}\,a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {25\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {595\,\sin \left (e+f\,x\right )}{8}-\frac {43\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {353\,\cos \left (e+f\,x\right )}{8}+\frac {77\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {21\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {171}{2}\right )}{840\,c^4\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^7} \]

[In]

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

[Out]

(2^(1/2)*a*cos(e/2 + (f*x)/2)*((25*cos(3*e + 3*f*x))/8 - (595*sin(e + f*x))/8 - (43*cos(2*e + 2*f*x))/2 - (353
*cos(e + f*x))/8 + (77*sin(2*e + 2*f*x))/4 + (21*sin(3*e + 3*f*x))/8 + 171/2))/(840*c^4*f*cos(e/2 + pi/4 + (f*
x)/2)^7)