3.289 \(\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx\)

Optimal. Leaf size=167 \[ \frac{16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{16 \tan (e+f x)}{33 a^3 c^6 f}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3} \]

[Out]

Sec[e + f*x]^5/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^3 - c^3*Sin[e + f*x])^2
) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (16*Tan[e + f*x])/(33*a^3*c^6*f) + (32*Tan[e + f*
x]^3)/(99*a^3*c^6*f) + (16*Tan[e + f*x]^5)/(165*a^3*c^6*f)

________________________________________________________________________________________

Rubi [A]  time = 0.217262, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2672, 3767} \[ \frac{16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{16 \tan (e+f x)}{33 a^3 c^6 f}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]

[Out]

Sec[e + f*x]^5/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^3 - c^3*Sin[e + f*x])^2
) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (16*Tan[e + f*x])/(33*a^3*c^6*f) + (32*Tan[e + f*
x]^3)/(99*a^3*c^6*f) + (16*Tan[e + f*x]^5)/(165*a^3*c^6*f)

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 2672

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] + 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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx &=\frac{\int \frac{\sec ^6(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{a^3 c^3}\\ &=\frac{\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{8 \int \frac{\sec ^6(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{11 a^3 c^4}\\ &=\frac{\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{56 \int \frac{\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{99 a^3 c^5}\\ &=\frac{\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{16 \int \sec ^6(e+f x) \, dx}{33 a^3 c^6}\\ &=\frac{\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}-\frac{16 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{33 a^3 c^6 f}\\ &=\frac{\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac{16 \tan (e+f x)}{33 a^3 c^6 f}+\frac{32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{16 \tan ^5(e+f x)}{165 a^3 c^6 f}\\ \end{align*}

Mathematica [A]  time = 1.56477, size = 233, normalized size = 1.4 \[ \frac{\left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (1802240 \sin (e+f x)+247170 \sin (2 (e+f x))+557056 \sin (3 (e+f x))+187250 \sin (4 (e+f x))-163840 \sin (5 (e+f x))+37450 \sin (6 (e+f x))-98304 \sin (7 (e+f x))-3745 \sin (8 (e+f x))-411950 \cos (e+f x)+1081344 \cos (2 (e+f x))-127330 \cos (3 (e+f x))+819200 \cos (4 (e+f x))+37450 \cos (5 (e+f x))+163840 \cos (6 (e+f x))+22470 \cos (7 (e+f x))-16384 \cos (8 (e+f x)))}{8110080 f (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]

[Out]

((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-411950*Cos[e + f*x] + 1081344*C
os[2*(e + f*x)] - 127330*Cos[3*(e + f*x)] + 819200*Cos[4*(e + f*x)] + 37450*Cos[5*(e + f*x)] + 163840*Cos[6*(e
+ f*x)] + 22470*Cos[7*(e + f*x)] - 16384*Cos[8*(e + f*x)] + 1802240*Sin[e + f*x] + 247170*Sin[2*(e + f*x)] +
557056*Sin[3*(e + f*x)] + 187250*Sin[4*(e + f*x)] - 163840*Sin[5*(e + f*x)] + 37450*Sin[6*(e + f*x)] - 98304*S
in[7*(e + f*x)] - 3745*Sin[8*(e + f*x)]))/(8110080*f*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6)

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Maple [A]  time = 0.076, size = 253, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{f{c}^{6}{a}^{3}} \left ( -4/11\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-11}-2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-10}-{\frac{53}{9\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-23/2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-8}-{\frac{33}{2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-{\frac{217}{12\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}}-{\frac{623}{40\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}}-{\frac{169}{16\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-{\frac{365}{64\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-{\frac{303}{128\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-{\frac{219}{256\,\tan \left ( 1/2\,fx+e/2 \right ) -256}}-{\frac{1}{80\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{5}}}+1/32\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-4}-{\frac{7}{96\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{3}}}+{\frac{5}{64\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{2}}}-{\frac{37}{256\,\tan \left ( 1/2\,fx+e/2 \right ) +256}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x)

[Out]

2/f/c^6/a^3*(-4/11/(tan(1/2*f*x+1/2*e)-1)^11-2/(tan(1/2*f*x+1/2*e)-1)^10-53/9/(tan(1/2*f*x+1/2*e)-1)^9-23/2/(t
an(1/2*f*x+1/2*e)-1)^8-33/2/(tan(1/2*f*x+1/2*e)-1)^7-217/12/(tan(1/2*f*x+1/2*e)-1)^6-623/40/(tan(1/2*f*x+1/2*e
)-1)^5-169/16/(tan(1/2*f*x+1/2*e)-1)^4-365/64/(tan(1/2*f*x+1/2*e)-1)^3-303/128/(tan(1/2*f*x+1/2*e)-1)^2-219/25
6/(tan(1/2*f*x+1/2*e)-1)-1/80/(tan(1/2*f*x+1/2*e)+1)^5+1/32/(tan(1/2*f*x+1/2*e)+1)^4-7/96/(tan(1/2*f*x+1/2*e)+
1)^3+5/64/(tan(1/2*f*x+1/2*e)+1)^2-37/256/(tan(1/2*f*x+1/2*e)+1))

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Maxima [B]  time = 1.35087, size = 949, normalized size = 5.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-2/495*(255*sin(f*x + e)/(cos(f*x + e) + 1) + 235*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 3065*sin(f*x + e)^3/(c
os(f*x + e) + 1)^3 + 3775*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 667*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 8217
*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2035*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 8745*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 - 11715*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 33*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 4917*sin
(f*x + e)^11/(cos(f*x + e) + 1)^11 - 2475*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 1815*sin(f*x + e)^13/(cos(f*
x + e) + 1)^13 + 1485*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 495*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - 125)
/((a^3*c^6 - 6*a^3*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a
^3*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 50*a^3*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 34*a^3*c^6*sin(f
*x + e)^5/(cos(f*x + e) + 1)^5 + 66*a^3*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 110*a^3*c^6*sin(f*x + e)^7/(
cos(f*x + e) + 1)^7 + 110*a^3*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 66*a^3*c^6*sin(f*x + e)^10/(cos(f*x +
e) + 1)^10 - 34*a^3*c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 50*a^3*c^6*sin(f*x + e)^12/(cos(f*x + e) + 1)^
12 - 10*a^3*c^6*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 10*a^3*c^6*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 + 6*a
^3*c^6*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - a^3*c^6*sin(f*x + e)^16/(cos(f*x + e) + 1)^16)*f)

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Fricas [A]  time = 1.64909, size = 412, normalized size = 2.47 \begin{align*} \frac{128 \, \cos \left (f x + e\right )^{8} - 576 \, \cos \left (f x + e\right )^{6} + 240 \, \cos \left (f x + e\right )^{4} + 56 \, \cos \left (f x + e\right )^{2} + 8 \,{\left (48 \, \cos \left (f x + e\right )^{6} - 40 \, \cos \left (f x + e\right )^{4} - 14 \, \cos \left (f x + e\right )^{2} - 9\right )} \sin \left (f x + e\right ) + 27}{495 \,{\left (3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5} -{\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5}\right )} \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

1/495*(128*cos(f*x + e)^8 - 576*cos(f*x + e)^6 + 240*cos(f*x + e)^4 + 56*cos(f*x + e)^2 + 8*(48*cos(f*x + e)^6
- 40*cos(f*x + e)^4 - 14*cos(f*x + e)^2 - 9)*sin(f*x + e) + 27)/(3*a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos
(f*x + e)^5 - (a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos(f*x + e)^5)*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(1/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**6,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.10262, size = 331, normalized size = 1.98 \begin{align*} -\frac{\frac{33 \,{\left (555 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 1920 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 2710 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1760 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 463\right )}}{a^{3} c^{6}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{5}} + \frac{108405 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 784080 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 2901195 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 6652800 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 10407474 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 11435424 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 8949270 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 4899840 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 1816265 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 411664 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 47279}{a^{3} c^{6}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{11}}}{63360 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="giac")

[Out]

-1/63360*(33*(555*tan(1/2*f*x + 1/2*e)^4 + 1920*tan(1/2*f*x + 1/2*e)^3 + 2710*tan(1/2*f*x + 1/2*e)^2 + 1760*ta
n(1/2*f*x + 1/2*e) + 463)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) + 1)^5) + (108405*tan(1/2*f*x + 1/2*e)^10 - 784080*ta
n(1/2*f*x + 1/2*e)^9 + 2901195*tan(1/2*f*x + 1/2*e)^8 - 6652800*tan(1/2*f*x + 1/2*e)^7 + 10407474*tan(1/2*f*x
+ 1/2*e)^6 - 11435424*tan(1/2*f*x + 1/2*e)^5 + 8949270*tan(1/2*f*x + 1/2*e)^4 - 4899840*tan(1/2*f*x + 1/2*e)^3
+ 1816265*tan(1/2*f*x + 1/2*e)^2 - 411664*tan(1/2*f*x + 1/2*e) + 47279)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) - 1)^1
1))/f