Integrand size = 25, antiderivative size = 195 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\frac {(4 c-3 d) d^3 x}{a^3}+\frac {d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{15 a^3 f}-\frac {(c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a+a \sin (e+f x))^3} \] Output:
(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)*c os(f*x+e)*(c+d*sin(f*x+e))^3/f/(a+a*sin(f*x+e))^3
Leaf count is larger than twice the leaf count of optimal. \(683\) vs. \(2(195)=390\).
Time = 1.45 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.50 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\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} \] Input:
Integrate[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^3,x]
Output:
((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] + 12 00*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*Sin[(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)
Time = 1.41 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3244, 25, 3042, 3456, 3042, 3447, 3042, 3502, 3042, 3214, 3042, 3127}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^4}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^4}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int -\frac {(c+d \sin (e+f x))^2 \left (a \left (2 c^2+6 d c-3 d^2\right )-a (c-6 d) d \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^2 \left (a \left (2 c^2+6 d c-3 d^2\right )-a (c-6 d) d \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^2 \left (a \left (2 c^2+6 d c-3 d^2\right )-a (c-6 d) d \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x)) \left (a^2 \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right )-a^2 d \left (2 c^2+10 d c-27 d^2\right ) \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x)) \left (a^2 \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right )-a^2 d \left (2 c^2+10 d c-27 d^2\right ) \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\int \frac {-d^2 \left (2 c^2+10 d c-27 d^2\right ) \sin ^2(e+f x) a^2+c \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right ) a^2+\left (a^2 d \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right )-a^2 c d \left (2 c^2+10 d c-27 d^2\right )\right ) \sin (e+f x)}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-d^2 \left (2 c^2+10 d c-27 d^2\right ) \sin (e+f x)^2 a^2+c \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right ) a^2+\left (a^2 d \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right )-a^2 c d \left (2 c^2+10 d c-27 d^2\right )\right ) \sin (e+f x)}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {c \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right ) a^3+15 (4 c-3 d) d^3 \sin (e+f x) a^3}{\sin (e+f x) a+a}dx}{a}+\frac {a d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{f}}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {c \left (2 c^3+8 d c^2+23 d^2 c-18 d^3\right ) a^3+15 (4 c-3 d) d^3 \sin (e+f x) a^3}{\sin (e+f x) a+a}dx}{a}+\frac {a d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{f}}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {a^3 (c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \int \frac {1}{\sin (e+f x) a+a}dx+15 a^2 d^3 x (4 c-3 d)}{a}+\frac {a d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{f}}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^3 (c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \int \frac {1}{\sin (e+f x) a+a}dx+15 a^2 d^3 x (4 c-3 d)}{a}+\frac {a d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{f}}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {\frac {\frac {15 a^2 d^3 x (4 c-3 d)-\frac {a^3 (c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{f (a \sin (e+f x)+a)}}{a}+\frac {a d^2 \left (2 c^2+10 c d-27 d^2\right ) \cos (e+f x)}{f}}{3 a^2}-\frac {a (c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}\) |
Input:
Int[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^3,x]
Output:
-1/5*((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]) ^3) + (-1/3*(a*(c - d)*(2*c + 9*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(f *(a + a*Sin[e + f*x])^2) + ((a*d^2*(2*c^2 + 10*c*d - 27*d^2)*Cos[e + f*x]) /f + (15*a^2*(4*c - 3*d)*d^3*x - (a^3*(c - d)^2*(2*c^2 + 12*c*d + 45*d^2)* Cos[e + f*x])/(f*(a + a*Sin[e + f*x])))/a)/(3*a^2))/(5*a^2)
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] + Simp[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]))
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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(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] - Simp[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] /; Fre eQ[{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] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[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]
Time = 10.75 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (c^{4}-4 c \,d^{3}+3 d^{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{4}+8 c^{3} d -8 c \,d^{3}+4 d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-8 c^{4}+32 c^{3} d -48 c^{2} d^{2}+32 c \,d^{3}-8 d^{4}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{4}-16 c^{3} d +24 c^{2} d^{2}-16 c \,d^{3}+4 d^{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {16 c \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{3} \left (-\frac {d}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}+\left (4 c -3 d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{a^{3} f}\) | \(248\) |
| default | \(\frac {-\frac {2 \left (c^{4}-4 c \,d^{3}+3 d^{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{4}+8 c^{3} d -8 c \,d^{3}+4 d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-8 c^{4}+32 c^{3} d -48 c^{2} d^{2}+32 c \,d^{3}-8 d^{4}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{4}-16 c^{3} d +24 c^{2} d^{2}-16 c \,d^{3}+4 d^{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {16 c \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{3} \left (-\frac {d}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}+\left (4 c -3 d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{a^{3} f}\) | \(248\) |
| risch | \(\frac {4 d^{3} x c}{a^{3}}-\frac {3 d^{4} x}{a^{3}}-\frac {d^{4} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{3} f}-\frac {d^{4} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{3} f}+\frac {36 i d^{4} {\mathrm e}^{i \left (f x +e \right )}-\frac {48 d^{4}}{5}-24 i c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+16 i c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+8 i c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}-\frac {184 i c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}}{3}-\frac {8 c^{3} d}{5}-\frac {28 c^{2} d^{2}}{5}+\frac {256 c \,d^{3}}{15}+72 i c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-12 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+56 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-12 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+32 d^{2} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+\frac {4 i c^{4} {\mathrm e}^{i \left (f x +e \right )}}{3}+\frac {8 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}}{3}+24 c \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+8 d \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {296 d^{3} c \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}-40 i d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-8 i c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}-\frac {4 c^{4}}{15}}{f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(380\) |
| parallelrisch | \(\frac {\left (\left (-900 f x -1755\right ) d^{4}+\left (1200 f x +2160\right ) c \,d^{3}-360 c^{2} d^{2}-240 c^{3} d -180 c^{4}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (90 f x +363\right ) d^{4}+\left (-120 f x -432\right ) c \,d^{3}+108 c^{2} d^{2}+48 c^{3} d +18 c^{4}\right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\left (-900 f x -1125\right ) d^{4}+\left (1200 f x +1360\right ) c \,d^{3}-120 c^{2} d^{2}-240 c^{3} d -100 c^{4}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-450 f x -1215\right ) d^{4}+\left (600 f x +1440\right ) c \,d^{3}-180 c^{2} d^{2}-240 c^{3} d -90 c^{4}\right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (450 f x +225\right ) d^{4}+\left (-600 f x -320\right ) c \,d^{3}+60 c^{2} d^{2}+50 c^{4}\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (90 f x -75\right ) d^{4}+\left (-120 f x +80\right ) c \,d^{3}-60 c^{2} d^{2}+10 c^{4}\right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+15 d^{4} \left (\cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+\sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )\right )}{30 f \,a^{3} \left (10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\) | \(397\) |
| norman | \(\text {Expression too large to display}\) | \(1010\) |
Input:
int((c+d*sin(f*x+e))^4/(a+sin(f*x+e)*a)^3,x,method=_RETURNVERBOSE)
Output:
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))))
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (187) = 374\).
Time = 0.09 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.53 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {15 \, d^{4} \cos \left (f x + e\right )^{4} - 3 \, c^{4} + 12 \, c^{3} d - 18 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4} + {\left (2 \, c^{4} + 12 \, c^{3} d + 42 \, c^{2} d^{2} - 128 \, c d^{3} + 117 \, d^{4} - 15 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )^{3} + 60 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x - {\left (4 \, c^{4} + 24 \, c^{3} d - 6 \, c^{2} d^{2} - 76 \, c d^{3} + 84 \, d^{4} + 45 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (3 \, c^{4} + 8 \, c^{3} d + 18 \, c^{2} d^{2} - 72 \, c d^{3} + 63 \, d^{4} - 10 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right ) + {\left (15 \, d^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} - 12 \, c^{3} d + 18 \, c^{2} d^{2} - 12 \, c d^{3} + 3 \, d^{4} + 60 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x - {\left (2 \, c^{4} + 12 \, c^{3} d + 42 \, c^{2} d^{2} - 128 \, c d^{3} + 102 \, d^{4} + 15 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (c^{4} + 6 \, c^{3} d + 6 \, c^{2} d^{2} - 34 \, c d^{3} + 31 \, d^{4} - 5 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-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))
Leaf count of result is larger than twice the leaf count of optimal. 7373 vs. \(2 (177) = 354\).
Time = 16.25 (sec) , antiderivative size = 7373, normalized size of antiderivative = 37.81 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((c+d*sin(f*x+e))**4/(a+a*sin(f*x+e))**3,x)
Output:
Piecewise((-30*c**4*tan(e/2 + f*x/2)**6/(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*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 + 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) - 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 + 1 65*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*ta n(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 + 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) - 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...
Leaf count of result is larger than twice the leaf count of optimal. 1101 vs. \(2 (187) = 374\).
Time = 0.14 (sec) , antiderivative size = 1101, normalized size of antiderivative = 5.65 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
Output:
-2/15*(3*d^4*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x + e)^2/(c os(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*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arcta n(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/(c os(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 12*c^2*...
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (187) = 374\).
Time = 0.16 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.92 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {\frac {30 \, d^{4}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{3}} - \frac {15 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} {\left (f x + e\right )}}{a^{3}} + \frac {2 \, {\left (15 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 60 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 \, c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 300 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 210 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 60 \, c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 120 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 580 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 360 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 380 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 240 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{4} + 12 \, c^{3} d + 12 \, c^{2} d^{2} - 88 \, c d^{3} + 57 \, d^{4}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \] Input:
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="giac")
Output:
-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 + 6 0*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) - 38 0*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
Time = 17.83 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.26 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\frac {2\,d^3\,\mathrm {atan}\left (\frac {2\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,c-3\,d\right )}{8\,c\,d^3-6\,d^4}\right )\,\left (4\,c-3\,d\right )}{a^3\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {22\,c^4}{3}+8\,c^3\,d+16\,c^2\,d^2-\frac {256\,c\,d^3}{3}+64\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {20\,c^4}{3}+16\,c^3\,d+8\,c^2\,d^2-\frac {272\,c\,d^3}{3}+80\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {94\,c^4}{15}+\frac {48\,c^3\,d}{5}+\frac {88\,c^2\,d^2}{5}-\frac {1336\,c\,d^3}{15}+\frac {378\,d^4}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (4\,c^4+8\,c^3\,d-40\,c\,d^3+30\,d^4\right )-\frac {176\,c\,d^3}{15}+\frac {8\,c^3\,d}{5}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,c^4-8\,c\,d^3+6\,d^4\right )+\frac {14\,c^4}{15}+\frac {48\,d^4}{5}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {8\,c^4}{3}+8\,c^3\,d+8\,c^2\,d^2-\frac {152\,c\,d^3}{3}+42\,d^4\right )+\frac {8\,c^2\,d^2}{5}}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+5\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+11\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+15\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+15\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+11\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^3\right )} \] Input:
int((c + d*sin(e + f*x))^4/(a + a*sin(e + f*x))^3,x)
Output:
(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)))
Time = 0.22 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.79 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:
int((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x)
Output:
(15*cos(e + f*x)*sin(e + f*x)**3*d**4 + cos(e + f*x)*sin(e + f*x)**2*c**4 + 30*cos(e + f*x)*sin(e + f*x)**2*c**2*d**2 + 60*cos(e + f*x)*sin(e + f*x) **2*c*d**3*f*x - 64*cos(e + f*x)*sin(e + f*x)**2*c*d**3 - 45*cos(e + f*x)* sin(e + f*x)**2*d**4*f*x + 63*cos(e + f*x)*sin(e + f*x)**2*d**4 + 4*cos(e + f*x)*sin(e + f*x)*c**4 + 12*cos(e + f*x)*sin(e + f*x)*c**3*d + 12*cos(e + f*x)*sin(e + f*x)*c**2*d**2 + 120*cos(e + f*x)*sin(e + f*x)*c*d**3*f*x - 76*cos(e + f*x)*sin(e + f*x)*c*d**3 - 90*cos(e + f*x)*sin(e + f*x)*d**4*f *x + 63*cos(e + f*x)*sin(e + f*x)*d**4 + 6*cos(e + f*x)*c**4 + 60*cos(e + f*x)*c*d**3*f*x - 24*cos(e + f*x)*c*d**3 - 45*cos(e + f*x)*d**4*f*x + 18*c os(e + f*x)*d**4 + 15*sin(e + f*x)**4*d**4 + 3*sin(e + f*x)**3*c**4 + 24*s in(e + f*x)**3*c**3*d + 54*sin(e + f*x)**3*c**2*d**2 - 60*sin(e + f*x)**3* c*d**3*f*x - 192*sin(e + f*x)**3*c*d**3 + 45*sin(e + f*x)**3*d**4*f*x + 15 6*sin(e + f*x)**3*d**4 + 7*sin(e + f*x)**2*c**4 + 60*sin(e + f*x)**2*c**3* d + 30*sin(e + f*x)**2*c**2*d**2 - 180*sin(e + f*x)**2*c*d**3*f*x - 268*si n(e + f*x)**2*c*d**3 + 135*sin(e + f*x)**2*d**4*f*x + 216*sin(e + f*x)**2* d**4 + 4*sin(e + f*x)*c**4 + 12*sin(e + f*x)*c**3*d + 12*sin(e + f*x)*c**2 *d**2 - 180*sin(e + f*x)*c*d**3*f*x - 76*sin(e + f*x)*c*d**3 + 135*sin(e + f*x)*d**4*f*x + 63*sin(e + f*x)*d**4 - 6*c**4 - 60*c*d**3*f*x + 24*c*d**3 + 45*d**4*f*x - 18*d**4)/(15*a**3*f*(cos(e + f*x)*sin(e + f*x)**2 + 2*cos (e + f*x)*sin(e + f*x) + cos(e + f*x) - sin(e + f*x)**3 - 3*sin(e + f*x...