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

Optimal. Leaf size=34 \[ \frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]

[Out]

(a^3*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7)

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Rubi [A]  time = 0.0902245, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2736, 2671} \[ \frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7)

Rule 2736

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]))

Rule 2671

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]

Rubi steps

\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}\\ \end{align*}

Mathematica [B]  time = 0.776164, size = 93, normalized size = 2.74 \[ \frac{a^3 \left (35 \cos \left (\frac{1}{2} (e+f x)\right )-21 \cos \left (\frac{3}{2} (e+f x)\right )-7 \cos \left (\frac{5}{2} (e+f x)\right )+\cos \left (\frac{7}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{28 c^4 f (\sin (e+f x)-1)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)

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Maple [B]  time = 0.099, size = 118, normalized size = 3.5 \begin{align*} 2\,{\frac{{a}^{3}}{f{c}^{4}} \left ( -{\frac{64}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-48\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-5}-6\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}-20\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-3}-40\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}- \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-1}-32\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-6} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*a^3/c^4*(-64/7/(tan(1/2*f*x+1/2*e)-1)^7-48/(tan(1/2*f*x+1/2*e)-1)^5-6/(tan(1/2*f*x+1/2*e)-1)^2-20/(tan(1/2
*f*x+1/2*e)-1)^3-40/(tan(1/2*f*x+1/2*e)-1)^4-1/(tan(1/2*f*x+1/2*e)-1)-32/(tan(1/2*f*x+1/2*e)-1)^6)

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Maxima [B]  time = 1.54743, size = 1411, normalized size = 41.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*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^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/(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) + 2*a^3*(7*sin(f*x + e)/(cos(f*x + e) + 1) - 21*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + 35*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 1)/(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

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Fricas [B]  time = 1.36482, size = 528, normalized size = 15.53 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*
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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 2.04922, size = 104, normalized size = 3.06 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)