3.5.70 \(\int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx\) [470]

Optimal. Leaf size=354 \[ \frac {d^3 \left (40 c^3-90 c^2 d+78 c d^2-23 d^3\right ) x}{2 a^3}+\frac {2 d \left (2 c^5+18 c^4 d+107 c^3 d^2-472 c^2 d^3+456 c d^4-136 d^5\right ) \cos (e+f x)}{15 a^3 f}+\frac {d^2 \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \cos (e+f x) \sin (e+f x)}{30 a^3 f}+\frac {d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}-\frac {(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3} \]

[Out]

1/2*d^3*(40*c^3-90*c^2*d+78*c*d^2-23*d^3)*x/a^3+2/15*d*(2*c^5+18*c^4*d+107*c^3*d^2-472*c^2*d^3+456*c*d^4-136*d
^5)*cos(f*x+e)/a^3/f+1/30*d^2*(4*c^4+36*c^3*d+216*c^2*d^2-626*c*d^3+345*d^4)*cos(f*x+e)*sin(f*x+e)/a^3/f+1/15*
d*(2*c^3+18*c^2*d+111*c*d^2-136*d^3)*cos(f*x+e)*(c+d*sin(f*x+e))^2/a^3/f-1/15*(c-d)*(2*c^2+18*c*d+115*d^2)*cos
(f*x+e)*(c+d*sin(f*x+e))^3/f/(a^3+a^3*sin(f*x+e))-1/15*(c-d)*(2*c+13*d)*cos(f*x+e)*(c+d*sin(f*x+e))^4/a/f/(a+a
*sin(f*x+e))^2-1/5*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^5/f/(a+a*sin(f*x+e))^3

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Rubi [A]
time = 0.53, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2844, 3056, 2832, 2813} \begin {gather*} -\frac {(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}+\frac {d^3 x \left (40 c^3-90 c^2 d+78 c d^2-23 d^3\right )}{2 a^3}+\frac {d^2 \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac {2 d \left (2 c^5+18 c^4 d+107 c^3 d^2-472 c^2 d^3+456 c d^4-136 d^5\right ) \cos (e+f x)}{15 a^3 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}-\frac {(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a \sin (e+f x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*x)/(2*a^3) + (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 +
456*c*d^4 - 136*d^5)*Cos[e + f*x])/(15*a^3*f) + (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*C
os[e + f*x]*Sin[e + f*x])/(30*a^3*f) + (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*Cos[e + f*x]*(c + d*Sin[e +
f*x])^2)/(15*a^3*f) - ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*f*(a^3 + a
^3*Sin[e + f*x])) - ((c - d)*(2*c + 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(15*a*f*(a + a*Sin[e + f*x])^2)
- ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(5*f*(a + a*Sin[e + f*x])^3)

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2832

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2844

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rubi steps

\begin {align*} \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {(c+d \sin (e+f x))^4 \left (-a \left (2 c^2+8 c d-5 d^2\right )+a (3 c-8 d) d \sin (e+f x)\right )}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac {(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {(c+d \sin (e+f x))^3 \left (-a^2 \left (2 c^3+10 c^2 d+55 c d^2-52 d^3\right )+3 a^2 d \left (2 c^2+14 c d-21 d^2\right ) \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{15 a^4}\\ &=-\frac {(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac {\int (c+d \sin (e+f x))^2 \left (-3 a^3 d^2 \left (2 c^2+118 c d-115 d^2\right )+3 a^3 d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \sin (e+f x)\right ) \, dx}{15 a^6}\\ &=\frac {d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}-\frac {(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}-\frac {\int (c+d \sin (e+f x)) \left (-3 a^3 d^2 \left (2 c^3+318 c^2 d-567 c d^2+272 d^3\right )+3 a^3 d \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \sin (e+f x)\right ) \, dx}{45 a^6}\\ &=\frac {d^3 \left (40 c^3-90 c^2 d+78 c d^2-23 d^3\right ) x}{2 a^3}+\frac {2 d \left (2 c^5+18 c^4 d+107 c^3 d^2-472 c^2 d^3+456 c d^4-136 d^5\right ) \cos (e+f x)}{15 a^3 f}+\frac {d^2 \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \cos (e+f x) \sin (e+f x)}{30 a^3 f}+\frac {d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}-\frac {(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.74, size = 560, normalized size = 1.58 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (48 (c-d)^6 \sin \left (\frac {1}{2} (e+f x)\right )-24 (c-d)^6 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+32 (c-d)^5 (c+14 d) \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 )^2-16 (c-d)^5 (c+14 d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+16 (c-d)^4 \left (2 c^2+26 c d+197 d^2\right ) \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 )^4-60 d^3 \left (-40 c^3+90 c^2 d-78 c d^2+23 d^3\right ) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+10 d^6 \cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-45 d^4 \left (20 c^2-24 c d+9 d^2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (e+f x)-i \sin (e+f x))-45 d^4 \left (20 c^2-24 c d+9 d^2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (e+f x)+i \sin (e+f x))-45 i (2 c-d) d^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (2 (e+f x))-i \sin (2 (e+f x)))+45 i (2 c-d) d^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (2 (e+f x))+i \sin (2 (e+f x)))\right )}{120 a^3 f (1+\sin (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(48*(c - d)^6*Sin[(e + f*x)/2] - 24*(c - d)^6*(Cos[(e + f*x)/2] + Sin[(
e + f*x)/2]) + 32*(c - d)^5*(c + 14*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 16*(c - d)^5
*(c + 14*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 16*(c - d)^4*(2*c^2 + 26*c*d + 197*d^2)*Sin[(e + f*x)/2]
*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 60*d^3*(-40*c^3 + 90*c^2*d - 78*c*d^2 + 23*d^3)*(e + f*x)*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2])^5 + 10*d^6*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 45*d^4*(20*
c^2 - 24*c*d + 9*d^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[e + f*x] - I*Sin[e + f*x]) - 45*d^4*(20*c^2
- 24*c*d + 9*d^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[e + f*x] + I*Sin[e + f*x]) - (45*I)*(2*c - d)*
d^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]) + (45*I)*(2*c - d)*d^5*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])))/(120*a^3*f*(1 + Sin[e + f*x])^3
)

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Maple [A]
time = 0.63, size = 468, normalized size = 1.32 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^6/(a+a*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f/a^3*(-(c^6-20*c^3*d^3+45*c^2*d^4-36*c*d^5+10*d^6)/(tan(1/2*f*x+1/2*e)+1)-1/2*(-4*c^6+12*c^5*d-40*c^3*d^3+6
0*c^2*d^4-36*c*d^5+8*d^6)/(tan(1/2*f*x+1/2*e)+1)^2-1/3*(8*c^6-36*c^5*d+60*c^4*d^2-40*c^3*d^3+12*c*d^5-4*d^6)/(
tan(1/2*f*x+1/2*e)+1)^3-1/4*(-8*c^6+48*c^5*d-120*c^4*d^2+160*c^3*d^3-120*c^2*d^4+48*c*d^5-8*d^6)/(tan(1/2*f*x+
1/2*e)+1)^4-1/5*(4*c^6-24*c^5*d+60*c^4*d^2-80*c^3*d^3+60*c^2*d^4-24*c*d^5+4*d^6)/(tan(1/2*f*x+1/2*e)+1)^5+d^3*
(((3*c*d^2-3/2*d^3)*tan(1/2*f*x+1/2*e)^5+(-15*c^2*d+18*c*d^2-6*d^3)*tan(1/2*f*x+1/2*e)^4+(-30*c^2*d+36*c*d^2-1
4*d^3)*tan(1/2*f*x+1/2*e)^2+(-3*c*d^2+3/2*d^3)*tan(1/2*f*x+1/2*e)-15*c^2*d+18*c*d^2-20/3*d^3)/(1+tan(1/2*f*x+1
/2*e)^2)^3+1/2*(40*c^3-90*c^2*d+78*c*d^2-23*d^3)*arctan(tan(1/2*f*x+1/2*e))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2169 vs. \(2 (354) = 708\).
time = 0.55, size = 2169, normalized size = 6.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(d^6*((2375*sin(f*x + e)/(cos(f*x + e) + 1) + 5347*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9230*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 12622*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 13340*sin(f*x + e)^5/(cos(f*x + e) + 1
)^5 + 11684*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 8050*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 4370*sin(f*x + e)
^8/(cos(f*x + e) + 1)^8 + 1725*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 345*sin(f*x + e)^10/(cos(f*x + e) + 1)^10
+ 544)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 13*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 25*a^3*sin
(f*x + e)^3/(cos(f*x + e) + 1)^3 + 38*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 46*a^3*sin(f*x + e)^5/(cos(f*x
+ e) + 1)^5 + 46*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 38*a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 25*a^
3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 13*a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 5*a^3*sin(f*x + e)^10/(co
s(f*x + e) + 1)^10 + a^3*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 345*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/
a^3) - 6*c*d^5*((1325*sin(f*x + e)/(cos(f*x + e) + 1) + 2673*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3805*sin(f*
x + e)^3/(cos(f*x + e) + 1)^3 + 4329*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3575*sin(f*x + e)^5/(cos(f*x + e) +
1)^5 + 2275*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 975*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 195*sin(f*x + e)^
8/(cos(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 26*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26*a^3*s
in(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12*a^3*sin(f*x + e)^7/(cos(f
*x + e) + 1)^7 + 5*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 195*ar
ctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + 90*c^2*d^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) - 40*c^3*d^3*((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)/(co
s(f*x + e) + 1))/a^3) + 2*c^6*(20*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3
0*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*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 60*c^4*d^2*(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) + 36*c^5*d*(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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (354) = 708\).
time = 0.44, size = 841, normalized size = 2.38 \begin {gather*} \frac {10 \, d^{6} \cos \left (f x + e\right )^{6} + 6 \, c^{6} - 36 \, c^{5} d + 90 \, c^{4} d^{2} - 120 \, c^{3} d^{3} + 90 \, c^{2} d^{4} - 36 \, c d^{5} + 6 \, d^{6} + 15 \, {\left (6 \, c d^{5} - d^{6}\right )} \cos \left (f x + e\right )^{5} - 10 \, {\left (45 \, c^{2} d^{4} - 36 \, c d^{5} + 14 \, d^{6}\right )} \cos \left (f x + e\right )^{4} - {\left (4 \, c^{6} + 36 \, c^{5} d + 210 \, c^{4} d^{2} - 1280 \, c^{3} d^{3} + 3510 \, c^{2} d^{4} - 2694 \, c d^{5} + 839 \, d^{6} - 15 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} f x\right )} \cos \left (f x + e\right )^{3} - 60 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} f x + {\left (8 \, c^{6} + 72 \, c^{5} d - 30 \, c^{4} d^{2} - 760 \, c^{3} d^{3} + 2520 \, c^{2} d^{4} - 2148 \, c d^{5} + 668 \, d^{6} + 45 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} f x\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (3 \, c^{6} + 12 \, c^{5} d + 45 \, c^{4} d^{2} - 360 \, c^{3} d^{3} + 945 \, c^{2} d^{4} - 768 \, c d^{5} + 233 \, d^{6} - 5 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} f x\right )} \cos \left (f x + e\right ) + {\left (10 \, d^{6} \cos \left (f x + e\right )^{5} - 6 \, c^{6} + 36 \, c^{5} d - 90 \, c^{4} d^{2} + 120 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 36 \, c d^{5} - 6 \, d^{6} - 5 \, {\left (18 \, c d^{5} - 5 \, d^{6}\right )} \cos \left (f x + e\right )^{4} - 5 \, {\left (90 \, c^{2} d^{4} - 54 \, c d^{5} + 23 \, d^{6}\right )} \cos \left (f x + e\right )^{3} - 60 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} f x + {\left (4 \, c^{6} + 36 \, c^{5} d + 210 \, c^{4} d^{2} - 1280 \, c^{3} d^{3} + 3060 \, c^{2} d^{4} - 2424 \, c d^{5} + 724 \, d^{6} + 15 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} f x\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (2 \, c^{6} + 18 \, c^{5} d + 30 \, c^{4} d^{2} - 340 \, c^{3} d^{3} + 930 \, c^{2} d^{4} - 762 \, c d^{5} + 232 \, d^{6} - 5 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \, {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(10*d^6*cos(f*x + e)^6 + 6*c^6 - 36*c^5*d + 90*c^4*d^2 - 120*c^3*d^3 + 90*c^2*d^4 - 36*c*d^5 + 6*d^6 + 15
*(6*c*d^5 - d^6)*cos(f*x + e)^5 - 10*(45*c^2*d^4 - 36*c*d^5 + 14*d^6)*cos(f*x + e)^4 - (4*c^6 + 36*c^5*d + 210
*c^4*d^2 - 1280*c^3*d^3 + 3510*c^2*d^4 - 2694*c*d^5 + 839*d^6 - 15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^
6)*f*x)*cos(f*x + e)^3 - 60*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x + (8*c^6 + 72*c^5*d - 30*c^4*d^2
- 760*c^3*d^3 + 2520*c^2*d^4 - 2148*c*d^5 + 668*d^6 + 45*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*c
os(f*x + e)^2 + 6*(3*c^6 + 12*c^5*d + 45*c^4*d^2 - 360*c^3*d^3 + 945*c^2*d^4 - 768*c*d^5 + 233*d^6 - 5*(40*c^3
*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e) + (10*d^6*cos(f*x + e)^5 - 6*c^6 + 36*c^5*d - 90*c^4*
d^2 + 120*c^3*d^3 - 90*c^2*d^4 + 36*c*d^5 - 6*d^6 - 5*(18*c*d^5 - 5*d^6)*cos(f*x + e)^4 - 5*(90*c^2*d^4 - 54*c
*d^5 + 23*d^6)*cos(f*x + e)^3 - 60*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x + (4*c^6 + 36*c^5*d + 210
*c^4*d^2 - 1280*c^3*d^3 + 3060*c^2*d^4 - 2424*c*d^5 + 724*d^6 + 15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^
6)*f*x)*cos(f*x + e)^2 + 6*(2*c^6 + 18*c^5*d + 30*c^4*d^2 - 340*c^3*d^3 + 930*c^2*d^4 - 762*c*d^5 + 232*d^6 -
5*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*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)*s
in(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 28065 vs. \(2 (340) = 680\).
time = 51.50, size = 28065, normalized size = 79.28 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))**6/(a+a*sin(f*x+e))**3,x)

[Out]

Piecewise((-60*c**6*tan(e/2 + f*x/2)**10/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 3
90*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f
*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2
+ f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 120*c**6*tan(e/2 + f
*x/2)**9/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 +
750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3
*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2
+ f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 340*c**6*tan(e/2 + f*x/2)**8/(30*a**3*f*tan(e/2 + f*
x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 +
1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a*
*3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/
2 + f*x/2) + 30*a**3*f) - 440*c**6*tan(e/2 + f*x/2)**7/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 +
f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7
+ 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a
**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 688*c*
*6*tan(e/2 + f*x/2)**6/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2
+ f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)*
*6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*
a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 600*c**6*tan(e/2 + f*x/2)**5/(30*a**3*
f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2
+ f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2
)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150
*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 624*c**6*tan(e/2 + f*x/2)**4/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**
3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e
/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x
/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a*
*3*f) - 360*c**6*tan(e/2 + f*x/2)**3/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a
**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan
(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*
x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 244*c**6*tan(e/2 + f*x/2
)**2/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*
a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*t
an(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f
*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 80*c**6*tan(e/2 + f*x/2)/(30*a**3*f*tan(e/2 + f*x/2)**11
+ 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a*
*3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan
(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/
2) + 30*a**3*f) - 28*c**6/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e
/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/
2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 3
90*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 360*c**5*d*tan(e/2 + f*x/2)**9/(30*
a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*ta
n(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 +
f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2
+ 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 360*c**5*d*tan(e/2 + f*x/2)**8/(30*a**3*f*tan(e/2 + f*x/2)**11 +
150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*ta...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (354) = 708\).
time = 0.52, size = 776, normalized size = 2.19 \begin {gather*} \frac {\frac {15 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} {\left (f x + e\right )}}{a^{3}} + \frac {10 \, {\left (18 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 9 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 90 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 108 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 36 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 180 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 216 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 84 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 18 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 90 \, c^{2} d^{4} + 108 \, c d^{5} - 40 \, d^{6}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} a^{3}} - \frac {4 \, {\left (15 \, c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 300 \, c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 675 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 540 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 150 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 90 \, c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1500 \, c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3150 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2430 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 660 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 90 \, c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 300 \, c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2900 \, c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5400 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3990 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1060 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 90 \, c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 150 \, c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1900 \, c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3600 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2670 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 710 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{6} + 18 \, c^{5} d + 30 \, c^{4} d^{2} - 440 \, c^{3} d^{3} + 855 \, c^{2} d^{4} - 642 \, c d^{5} + 172 \, d^{6}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{30 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*(f*x + e)/a^3 + 10*(18*c*d^5*tan(1/2*f*x + 1/2*e)^5 - 9
*d^6*tan(1/2*f*x + 1/2*e)^5 - 90*c^2*d^4*tan(1/2*f*x + 1/2*e)^4 + 108*c*d^5*tan(1/2*f*x + 1/2*e)^4 - 36*d^6*ta
n(1/2*f*x + 1/2*e)^4 - 180*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 216*c*d^5*tan(1/2*f*x + 1/2*e)^2 - 84*d^6*tan(1/2*
f*x + 1/2*e)^2 - 18*c*d^5*tan(1/2*f*x + 1/2*e) + 9*d^6*tan(1/2*f*x + 1/2*e) - 90*c^2*d^4 + 108*c*d^5 - 40*d^6)
/((tan(1/2*f*x + 1/2*e)^2 + 1)^3*a^3) - 4*(15*c^6*tan(1/2*f*x + 1/2*e)^4 - 300*c^3*d^3*tan(1/2*f*x + 1/2*e)^4
+ 675*c^2*d^4*tan(1/2*f*x + 1/2*e)^4 - 540*c*d^5*tan(1/2*f*x + 1/2*e)^4 + 150*d^6*tan(1/2*f*x + 1/2*e)^4 + 30*
c^6*tan(1/2*f*x + 1/2*e)^3 + 90*c^5*d*tan(1/2*f*x + 1/2*e)^3 - 1500*c^3*d^3*tan(1/2*f*x + 1/2*e)^3 + 3150*c^2*
d^4*tan(1/2*f*x + 1/2*e)^3 - 2430*c*d^5*tan(1/2*f*x + 1/2*e)^3 + 660*d^6*tan(1/2*f*x + 1/2*e)^3 + 40*c^6*tan(1
/2*f*x + 1/2*e)^2 + 90*c^5*d*tan(1/2*f*x + 1/2*e)^2 + 300*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 - 2900*c^3*d^3*tan(1/
2*f*x + 1/2*e)^2 + 5400*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 - 3990*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 1060*d^6*tan(1/2*
f*x + 1/2*e)^2 + 20*c^6*tan(1/2*f*x + 1/2*e) + 90*c^5*d*tan(1/2*f*x + 1/2*e) + 150*c^4*d^2*tan(1/2*f*x + 1/2*e
) - 1900*c^3*d^3*tan(1/2*f*x + 1/2*e) + 3600*c^2*d^4*tan(1/2*f*x + 1/2*e) - 2670*c*d^5*tan(1/2*f*x + 1/2*e) +
710*d^6*tan(1/2*f*x + 1/2*e) + 7*c^6 + 18*c^5*d + 30*c^4*d^2 - 440*c^3*d^3 + 855*c^2*d^4 - 642*c*d^5 + 172*d^6
)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f

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Mupad [B]
time = 9.71, size = 898, normalized size = 2.54 \begin {gather*} \frac {d^3\,\mathrm {atan}\left (\frac {d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (40\,c^3-90\,c^2\,d+78\,c\,d^2-23\,d^3\right )}{40\,c^3\,d^3-90\,c^2\,d^4+78\,c\,d^5-23\,d^6}\right )\,\left (40\,c^3-90\,c^2\,d+78\,c\,d^2-23\,d^3\right )}{a^3\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (4\,c^6+12\,c^5\,d-200\,c^3\,d^3+450\,c^2\,d^4-390\,c\,d^5+115\,d^6\right )-\frac {608\,c\,d^5}{5}+\frac {12\,c^5\,d}{5}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (2\,c^6-40\,c^3\,d^3+90\,c^2\,d^4-78\,c\,d^5+23\,d^6\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {8\,c^6}{3}+12\,c^5\,d+20\,c^4\,d^2-\frac {760\,c^3\,d^3}{3}+630\,c^2\,d^4-530\,c\,d^5+\frac {475\,d^6}{3}\right )+\frac {14\,c^6}{15}+\frac {544\,d^6}{15}+144\,c^2\,d^4-\frac {176\,c^3\,d^3}{3}+4\,c^4\,d^2+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {34\,c^6}{3}+12\,c^5\,d+40\,c^4\,d^2-\frac {1520\,c^3\,d^3}{3}+1140\,c^2\,d^4-988\,c\,d^5+\frac {874\,d^6}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (12\,c^6+48\,c^5\,d+60\,c^4\,d^2-960\,c^3\,d^3+2460\,c^2\,d^4-2052\,c\,d^5+\frac {1846\,d^6}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {44\,c^6}{3}+48\,c^5\,d+20\,c^4\,d^2-\frac {2560\,c^3\,d^3}{3}+2100\,c^2\,d^4-1820\,c\,d^5+\frac {1610\,d^6}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (20\,c^6+72\,c^5\,d+60\,c^4\,d^2-1360\,c^3\,d^3+3480\,c^2\,d^4-2952\,c\,d^5+\frac {2668\,d^6}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {122\,c^6}{15}+\frac {96\,c^5\,d}{5}+52\,c^4\,d^2-\frac {1688\,c^3\,d^3}{3}+1422\,c^2\,d^4-\frac {5954\,c\,d^5}{5}+\frac {5347\,d^6}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {104\,c^6}{5}+\frac {216\,c^5\,d}{5}+132\,c^4\,d^2-1376\,c^3\,d^3+3372\,c^2\,d^4-\frac {14004\,c\,d^5}{5}+\frac {12622\,d^6}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {344\,c^6}{15}+\frac {192\,c^5\,d}{5}+124\,c^4\,d^2-\frac {4016\,c^3\,d^3}{3}+3144\,c^2\,d^4-\frac {13208\,c\,d^5}{5}+\frac {11684\,d^6}{15}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+5\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+13\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+25\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+38\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+46\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+46\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+38\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+25\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+13\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^3*atan((d^3*tan(e/2 + (f*x)/2)*(78*c*d^2 - 90*c^2*d + 40*c^3 - 23*d^3))/(78*c*d^5 - 23*d^6 - 90*c^2*d^4 + 4
0*c^3*d^3))*(78*c*d^2 - 90*c^2*d + 40*c^3 - 23*d^3))/(a^3*f) - (tan(e/2 + (f*x)/2)^9*(12*c^5*d - 390*c*d^5 + 4
*c^6 + 115*d^6 + 450*c^2*d^4 - 200*c^3*d^3) - (608*c*d^5)/5 + (12*c^5*d)/5 + tan(e/2 + (f*x)/2)^10*(2*c^6 - 78
*c*d^5 + 23*d^6 + 90*c^2*d^4 - 40*c^3*d^3) + tan(e/2 + (f*x)/2)*(12*c^5*d - 530*c*d^5 + (8*c^6)/3 + (475*d^6)/
3 + 630*c^2*d^4 - (760*c^3*d^3)/3 + 20*c^4*d^2) + (14*c^6)/15 + (544*d^6)/15 + 144*c^2*d^4 - (176*c^3*d^3)/3 +
4*c^4*d^2 + tan(e/2 + (f*x)/2)^8*(12*c^5*d - 988*c*d^5 + (34*c^6)/3 + (874*d^6)/3 + 1140*c^2*d^4 - (1520*c^3*
d^3)/3 + 40*c^4*d^2) + tan(e/2 + (f*x)/2)^3*(48*c^5*d - 2052*c*d^5 + 12*c^6 + (1846*d^6)/3 + 2460*c^2*d^4 - 96
0*c^3*d^3 + 60*c^4*d^2) + tan(e/2 + (f*x)/2)^7*(48*c^5*d - 1820*c*d^5 + (44*c^6)/3 + (1610*d^6)/3 + 2100*c^2*d
^4 - (2560*c^3*d^3)/3 + 20*c^4*d^2) + tan(e/2 + (f*x)/2)^5*(72*c^5*d - 2952*c*d^5 + 20*c^6 + (2668*d^6)/3 + 34
80*c^2*d^4 - 1360*c^3*d^3 + 60*c^4*d^2) + tan(e/2 + (f*x)/2)^2*((96*c^5*d)/5 - (5954*c*d^5)/5 + (122*c^6)/15 +
(5347*d^6)/15 + 1422*c^2*d^4 - (1688*c^3*d^3)/3 + 52*c^4*d^2) + tan(e/2 + (f*x)/2)^4*((216*c^5*d)/5 - (14004*
c*d^5)/5 + (104*c^6)/5 + (12622*d^6)/15 + 3372*c^2*d^4 - 1376*c^3*d^3 + 132*c^4*d^2) + tan(e/2 + (f*x)/2)^6*((
192*c^5*d)/5 - (13208*c*d^5)/5 + (344*c^6)/15 + (11684*d^6)/15 + 3144*c^2*d^4 - (4016*c^3*d^3)/3 + 124*c^4*d^2
))/(f*(13*a^3*tan(e/2 + (f*x)/2)^2 + 25*a^3*tan(e/2 + (f*x)/2)^3 + 38*a^3*tan(e/2 + (f*x)/2)^4 + 46*a^3*tan(e/
2 + (f*x)/2)^5 + 46*a^3*tan(e/2 + (f*x)/2)^6 + 38*a^3*tan(e/2 + (f*x)/2)^7 + 25*a^3*tan(e/2 + (f*x)/2)^8 + 13*
a^3*tan(e/2 + (f*x)/2)^9 + 5*a^3*tan(e/2 + (f*x)/2)^10 + a^3*tan(e/2 + (f*x)/2)^11 + a^3 + 5*a^3*tan(e/2 + (f*
x)/2)))

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