3.5.72 \(\int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx\) [472]

Optimal. Leaf size=195 \[ \frac {(4 c-3 d) d^3 x}{a^3}+\frac {d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{15 a^3 f}-\frac {(c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3} \]

[Out]

(4*c-3*d)*d^3*x/a^3+1/15*d^2*(2*c^2+10*c*d-27*d^2)*cos(f*x+e)/a^3/f-1/15*(c-d)^2*(2*c^2+12*c*d+45*d^2)*cos(f*x
+e)/f/(a^3+a^3*sin(f*x+e))-1/15*(c-d)*(2*c+9*d)*cos(f*x+e)*(c+d*sin(f*x+e))^2/a/f/(a+a*sin(f*x+e))^2-1/5*(c-d)
*cos(f*x+e)*(c+d*sin(f*x+e))^3/f/(a+a*sin(f*x+e))^3

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Rubi [A]
time = 0.42, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2844, 3056, 3047, 3102, 2814, 2727} \begin {gather*} \frac {d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{15 a^3 f}-\frac {(c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {d^3 x (4 c-3 d)}{a^3}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a \sin (e+f x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*c - 3*d)*d^3*x)/a^3 + (d^2*(2*c^2 + 10*c*d - 27*d^2)*Cos[e + f*x])/(15*a^3*f) - ((c - d)^2*(2*c^2 + 12*c*d
+ 45*d^2)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*(2*c + 9*d)*Cos[e + f*x]*(c + d*Sin[e + f*
x])^2)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(5*f*(a + a*Sin[e + f*x
])^3)

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2814

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

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 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {(c+d \sin (e+f x))^2 \left (-a \left (2 c^2+6 c d-3 d^2\right )+a (c-6 d) d \sin (e+f x)\right )}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {(c+d \sin (e+f x)) \left (-a^2 \left (2 c^3+8 c^2 d+23 c d^2-18 d^3\right )+a^2 d \left (2 c^2+10 c d-27 d^2\right ) \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{15 a^4}\\ &=-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {-a^2 c \left (2 c^3+8 c^2 d+23 c d^2-18 d^3\right )+\left (a^2 c d \left (2 c^2+10 c d-27 d^2\right )-a^2 d \left (2 c^3+8 c^2 d+23 c d^2-18 d^3\right )\right ) \sin (e+f x)+a^2 d^2 \left (2 c^2+10 c d-27 d^2\right ) \sin ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{15 a^4}\\ &=\frac {d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{15 a^3 f}-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {-a^3 c \left (2 c^3+8 c^2 d+23 c d^2-18 d^3\right )-15 a^3 (4 c-3 d) d^3 \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{15 a^5}\\ &=\frac {(4 c-3 d) d^3 x}{a^3}+\frac {d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{15 a^3 f}-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3}+\frac {\left ((c-d)^2 \left (2 c^2+12 c d+45 d^2\right )\right ) \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2}\\ &=\frac {(4 c-3 d) d^3 x}{a^3}+\frac {d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{15 a^3 f}-\frac {(c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(683\) vs. \(2(195)=390\).
time = 0.94, size = 683, normalized size = 3.50 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (15 d \left (16 c^3+48 c^2 d-15 d^3 (-5+4 e+4 f x)+16 c d^2 (-9+5 e+5 f x)\right ) \cos \left (\frac {1}{2} (e+f x)\right )-5 \left (8 c^4+48 c^3 d+96 c^2 d^2-9 d^4 (-27+10 e+10 f x)+8 c d^3 (-46+15 e+15 f x)\right ) \cos \left (\frac {3}{2} (e+f x)\right )+75 d^4 \cos \left (\frac {5}{2} (e+f x)\right )-120 c d^3 e \cos \left (\frac {5}{2} (e+f x)\right )+90 d^4 e \cos \left (\frac {5}{2} (e+f x)\right )-120 c d^3 f x \cos \left (\frac {5}{2} (e+f x)\right )+90 d^4 f x \cos \left (\frac {5}{2} (e+f x)\right )+15 d^4 \cos \left (\frac {7}{2} (e+f x)\right )+80 c^4 \sin \left (\frac {1}{2} (e+f x)\right )+240 c^3 d \sin \left (\frac {1}{2} (e+f x)\right )+960 c^2 d^2 \sin \left (\frac {1}{2} (e+f x)\right )-2960 c d^3 \sin \left (\frac {1}{2} (e+f x)\right )+1755 d^4 \sin \left (\frac {1}{2} (e+f x)\right )+1200 c d^3 e \sin \left (\frac {1}{2} (e+f x)\right )-900 d^4 e \sin \left (\frac {1}{2} (e+f x)\right )+1200 c d^3 f x \sin \left (\frac {1}{2} (e+f x)\right )-900 d^4 f x \sin \left (\frac {1}{2} (e+f x)\right )+360 c^2 d^2 \sin \left (\frac {3}{2} (e+f x)\right )-720 c d^3 \sin \left (\frac {3}{2} (e+f x)\right )+225 d^4 \sin \left (\frac {3}{2} (e+f x)\right )+600 c d^3 e \sin \left (\frac {3}{2} (e+f x)\right )-450 d^4 e \sin \left (\frac {3}{2} (e+f x)\right )+600 c d^3 f x \sin \left (\frac {3}{2} (e+f x)\right )-450 d^4 f x \sin \left (\frac {3}{2} (e+f x)\right )-8 c^4 \sin \left (\frac {5}{2} (e+f x)\right )-48 c^3 d \sin \left (\frac {5}{2} (e+f x)\right )-168 c^2 d^2 \sin \left (\frac {5}{2} (e+f x)\right )+512 c d^3 \sin \left (\frac {5}{2} (e+f x)\right )-363 d^4 \sin \left (\frac {5}{2} (e+f x)\right )-120 c d^3 e \sin \left (\frac {5}{2} (e+f x)\right )+90 d^4 e \sin \left (\frac {5}{2} (e+f x)\right )-120 c d^3 f x \sin \left (\frac {5}{2} (e+f x)\right )+90 d^4 f x \sin \left (\frac {5}{2} (e+f x)\right )+15 d^4 \sin \left (\frac {7}{2} (e+f x)\right )\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])^4/(a + a*Sin[e + f*x])^3,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(15*d*(16*c^3 + 48*c^2*d - 15*d^3*(-5 + 4*e + 4*f*x) + 16*c*d^2*(-9 + 5
*e + 5*f*x))*Cos[(e + f*x)/2] - 5*(8*c^4 + 48*c^3*d + 96*c^2*d^2 - 9*d^4*(-27 + 10*e + 10*f*x) + 8*c*d^3*(-46
+ 15*e + 15*f*x))*Cos[(3*(e + f*x))/2] + 75*d^4*Cos[(5*(e + f*x))/2] - 120*c*d^3*e*Cos[(5*(e + f*x))/2] + 90*d
^4*e*Cos[(5*(e + f*x))/2] - 120*c*d^3*f*x*Cos[(5*(e + f*x))/2] + 90*d^4*f*x*Cos[(5*(e + f*x))/2] + 15*d^4*Cos[
(7*(e + f*x))/2] + 80*c^4*Sin[(e + f*x)/2] + 240*c^3*d*Sin[(e + f*x)/2] + 960*c^2*d^2*Sin[(e + f*x)/2] - 2960*
c*d^3*Sin[(e + f*x)/2] + 1755*d^4*Sin[(e + f*x)/2] + 1200*c*d^3*e*Sin[(e + f*x)/2] - 900*d^4*e*Sin[(e + f*x)/2
] + 1200*c*d^3*f*x*Sin[(e + f*x)/2] - 900*d^4*f*x*Sin[(e + f*x)/2] + 360*c^2*d^2*Sin[(3*(e + f*x))/2] - 720*c*
d^3*Sin[(3*(e + f*x))/2] + 225*d^4*Sin[(3*(e + f*x))/2] + 600*c*d^3*e*Sin[(3*(e + f*x))/2] - 450*d^4*e*Sin[(3*
(e + f*x))/2] + 600*c*d^3*f*x*Sin[(3*(e + f*x))/2] - 450*d^4*f*x*Sin[(3*(e + f*x))/2] - 8*c^4*Sin[(5*(e + f*x)
)/2] - 48*c^3*d*Sin[(5*(e + f*x))/2] - 168*c^2*d^2*Sin[(5*(e + f*x))/2] + 512*c*d^3*Sin[(5*(e + f*x))/2] - 363
*d^4*Sin[(5*(e + f*x))/2] - 120*c*d^3*e*Sin[(5*(e + f*x))/2] + 90*d^4*e*Sin[(5*(e + f*x))/2] - 120*c*d^3*f*x*S
in[(5*(e + f*x))/2] + 90*d^4*f*x*Sin[(5*(e + f*x))/2] + 15*d^4*Sin[(7*(e + f*x))/2]))/(120*a^3*f*(1 + Sin[e +
f*x])^3)

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Maple [A]
time = 0.56, size = 248, normalized size = 1.27 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/a^3*(-(c^4-4*c*d^3+3*d^4)/(tan(1/2*f*x+1/2*e)+1)-1/2*(-4*c^4+8*c^3*d-8*c*d^3+4*d^4)/(tan(1/2*f*x+1/2*e)+1)
^2-1/4*(-8*c^4+32*c^3*d-48*c^2*d^2+32*c*d^3-8*d^4)/(tan(1/2*f*x+1/2*e)+1)^4-1/5*(4*c^4-16*c^3*d+24*c^2*d^2-16*
c*d^3+4*d^4)/(tan(1/2*f*x+1/2*e)+1)^5-8/3*c*(c^3-3*c^2*d+3*c*d^2-d^3)/(tan(1/2*f*x+1/2*e)+1)^3+d^3*(-d/(1+tan(
1/2*f*x+1/2*e)^2)+(4*c-3*d)*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. 1197 vs. \(2 (196) = 392\).
time = 0.53, size = 1197, normalized size = 6.14 \begin {gather*} -\frac {2 \, {\left (3 \, d^{4} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {189 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {160 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {75 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 24}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {11 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {11 \, a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} + \frac {15 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} - 4 \, c d^{3} {\left (\frac {\frac {95 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {145 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 22}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {15 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + \frac {c^{4} {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {12 \, c^{2} d^{2} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {12 \, c^{3} d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(3*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) - 4*c*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*s
in(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)/(
cos(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
*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 12*c^2*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) + 12*c^3*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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (196) = 392\).
time = 0.40, size = 508, normalized size = 2.61 \begin {gather*} -\frac {15 \, d^{4} \cos \left (f x + e\right )^{4} - 3 \, c^{4} + 12 \, c^{3} d - 18 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4} + {\left (2 \, c^{4} + 12 \, c^{3} d + 42 \, c^{2} d^{2} - 128 \, c d^{3} + 117 \, d^{4} - 15 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )^{3} + 60 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x - {\left (4 \, c^{4} + 24 \, c^{3} d - 6 \, c^{2} d^{2} - 76 \, c d^{3} + 84 \, d^{4} + 45 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (3 \, c^{4} + 8 \, c^{3} d + 18 \, c^{2} d^{2} - 72 \, c d^{3} + 63 \, d^{4} - 10 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right ) + {\left (15 \, d^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} - 12 \, c^{3} d + 18 \, c^{2} d^{2} - 12 \, c d^{3} + 3 \, d^{4} + 60 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x - {\left (2 \, c^{4} + 12 \, c^{3} d + 42 \, c^{2} d^{2} - 128 \, c d^{3} + 102 \, d^{4} + 15 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (c^{4} + 6 \, c^{3} d + 6 \, c^{2} d^{2} - 34 \, c d^{3} + 31 \, d^{4} - 5 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(15*d^4*cos(f*x + e)^4 - 3*c^4 + 12*c^3*d - 18*c^2*d^2 + 12*c*d^3 - 3*d^4 + (2*c^4 + 12*c^3*d + 42*c^2*d
^2 - 128*c*d^3 + 117*d^4 - 15*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e)^3 + 60*(4*c*d^3 - 3*d^4)*f*x - (4*c^4 + 24*c
^3*d - 6*c^2*d^2 - 76*c*d^3 + 84*d^4 + 45*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e)^2 - 3*(3*c^4 + 8*c^3*d + 18*c^2*
d^2 - 72*c*d^3 + 63*d^4 - 10*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e) + (15*d^4*cos(f*x + e)^3 + 3*c^4 - 12*c^3*d +
18*c^2*d^2 - 12*c*d^3 + 3*d^4 + 60*(4*c*d^3 - 3*d^4)*f*x - (2*c^4 + 12*c^3*d + 42*c^2*d^2 - 128*c*d^3 + 102*d
^4 + 15*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e)^2 - 6*(c^4 + 6*c^3*d + 6*c^2*d^2 - 34*c*d^3 + 31*d^4 - 5*(4*c*d^3
- 3*d^4)*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)*sin(f*x + e))

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 7373 vs. \(2 (177) = 354\).
time = 19.03, size = 7373, normalized size = 37.81 \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))**4/(a+a*sin(f*x+e))**3,x)

[Out]

Piecewise((-30*c**4*tan(e/2 + f*x/2)**6/(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) - 60*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 + 22
5*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) - 110*
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) - 100*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) - 94*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 + 22
5*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) - 40*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 + 16
5*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 14*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) - 120*c**3*d
*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) - 120*c**3*d*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) - 240*c**3*d*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 + 2
25*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*ta
n(e/2 + f*x/2) + 15*a**3*f) - 144*c**3*d*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) - 120*c**3*d*tan(e/2 + f*x/2)/(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) - 24*c**3*d/(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**2*d**2*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) - 120*c**2*d*
*2*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) - 264*c**2*d**2*tan(e/2 + f*x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**7 + 7
5*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*ta
n(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) - 120*c**2*d**2*t
an(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) - 24*c**2*d**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) + 60*c*d**3*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 + 16...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (196) = 392\).
time = 0.46, size = 395, normalized size = 2.03 \begin {gather*} -\frac {\frac {30 \, d^{4}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{3}} - \frac {15 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} {\left (f x + e\right )}}{a^{3}} + \frac {2 \, {\left (15 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 60 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 \, c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 300 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 210 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 60 \, c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 120 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 580 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 360 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 380 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 240 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{4} + 12 \, c^{3} d + 12 \, c^{2} d^{2} - 88 \, c d^{3} + 57 \, d^{4}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(30*d^4/((tan(1/2*f*x + 1/2*e)^2 + 1)*a^3) - 15*(4*c*d^3 - 3*d^4)*(f*x + e)/a^3 + 2*(15*c^4*tan(1/2*f*x
+ 1/2*e)^4 - 60*c*d^3*tan(1/2*f*x + 1/2*e)^4 + 45*d^4*tan(1/2*f*x + 1/2*e)^4 + 30*c^4*tan(1/2*f*x + 1/2*e)^3 +
60*c^3*d*tan(1/2*f*x + 1/2*e)^3 - 300*c*d^3*tan(1/2*f*x + 1/2*e)^3 + 210*d^4*tan(1/2*f*x + 1/2*e)^3 + 40*c^4*
tan(1/2*f*x + 1/2*e)^2 + 60*c^3*d*tan(1/2*f*x + 1/2*e)^2 + 120*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 - 580*c*d^3*tan(
1/2*f*x + 1/2*e)^2 + 360*d^4*tan(1/2*f*x + 1/2*e)^2 + 20*c^4*tan(1/2*f*x + 1/2*e) + 60*c^3*d*tan(1/2*f*x + 1/2
*e) + 60*c^2*d^2*tan(1/2*f*x + 1/2*e) - 380*c*d^3*tan(1/2*f*x + 1/2*e) + 240*d^4*tan(1/2*f*x + 1/2*e) + 7*c^4
+ 12*c^3*d + 12*c^2*d^2 - 88*c*d^3 + 57*d^4)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f

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Mupad [B]
time = 9.03, size = 440, normalized size = 2.26 \begin {gather*} \frac {2\,d^3\,\mathrm {atan}\left (\frac {2\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,c-3\,d\right )}{8\,c\,d^3-6\,d^4}\right )\,\left (4\,c-3\,d\right )}{a^3\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {22\,c^4}{3}+8\,c^3\,d+16\,c^2\,d^2-\frac {256\,c\,d^3}{3}+64\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {20\,c^4}{3}+16\,c^3\,d+8\,c^2\,d^2-\frac {272\,c\,d^3}{3}+80\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {94\,c^4}{15}+\frac {48\,c^3\,d}{5}+\frac {88\,c^2\,d^2}{5}-\frac {1336\,c\,d^3}{15}+\frac {378\,d^4}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (4\,c^4+8\,c^3\,d-40\,c\,d^3+30\,d^4\right )-\frac {176\,c\,d^3}{15}+\frac {8\,c^3\,d}{5}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,c^4-8\,c\,d^3+6\,d^4\right )+\frac {14\,c^4}{15}+\frac {48\,d^4}{5}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {8\,c^4}{3}+8\,c^3\,d+8\,c^2\,d^2-\frac {152\,c\,d^3}{3}+42\,d^4\right )+\frac {8\,c^2\,d^2}{5}}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+5\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+11\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+15\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+15\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+11\,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))^4/(a + a*sin(e + f*x))^3,x)

[Out]

(2*d^3*atan((2*d^3*tan(e/2 + (f*x)/2)*(4*c - 3*d))/(8*c*d^3 - 6*d^4))*(4*c - 3*d))/(a^3*f) - (tan(e/2 + (f*x)/
2)^4*(8*c^3*d - (256*c*d^3)/3 + (22*c^4)/3 + 64*d^4 + 16*c^2*d^2) + tan(e/2 + (f*x)/2)^3*(16*c^3*d - (272*c*d^
3)/3 + (20*c^4)/3 + 80*d^4 + 8*c^2*d^2) + tan(e/2 + (f*x)/2)^2*((48*c^3*d)/5 - (1336*c*d^3)/15 + (94*c^4)/15 +
(378*d^4)/5 + (88*c^2*d^2)/5) + tan(e/2 + (f*x)/2)^5*(8*c^3*d - 40*c*d^3 + 4*c^4 + 30*d^4) - (176*c*d^3)/15 +
(8*c^3*d)/5 + tan(e/2 + (f*x)/2)^6*(2*c^4 - 8*c*d^3 + 6*d^4) + (14*c^4)/15 + (48*d^4)/5 + tan(e/2 + (f*x)/2)*
(8*c^3*d - (152*c*d^3)/3 + (8*c^4)/3 + 42*d^4 + 8*c^2*d^2) + (8*c^2*d^2)/5)/(f*(11*a^3*tan(e/2 + (f*x)/2)^2 +
15*a^3*tan(e/2 + (f*x)/2)^3 + 15*a^3*tan(e/2 + (f*x)/2)^4 + 11*a^3*tan(e/2 + (f*x)/2)^5 + 5*a^3*tan(e/2 + (f*x
)/2)^6 + a^3*tan(e/2 + (f*x)/2)^7 + a^3 + 5*a^3*tan(e/2 + (f*x)/2)))

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