Integrand size = 29, antiderivative size = 235 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a \left (8 a^4-4 a^2 b^2-b^4\right ) x}{8 b^6}-\frac {2 a^4 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 d}+\frac {\left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{15 b^5 d}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac {\left (5 a^2-b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^3 d}-\frac {a \cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b d} \] Output:
1/8*a*(8*a^4-4*a^2*b^2-b^4)*x/b^6-2*a^4*(a^2-b^2)^(1/2)*arctan((b+a*tan(1/ 2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^6/d+1/15*(15*a^4-5*a^2*b^2-2*b^4)*cos(d*x +c)/b^5/d-1/8*a*(4*a^2-b^2)*cos(d*x+c)*sin(d*x+c)/b^4/d+1/15*(5*a^2-b^2)*c os(d*x+c)*sin(d*x+c)^2/b^3/d-1/4*a*cos(d*x+c)*sin(d*x+c)^3/b^2/d+1/5*cos(d *x+c)*sin(d*x+c)^4/b/d
Time = 2.41 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-960 a^4 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-60 b \left (-8 a^4+2 a^2 b^2+b^4\right ) \cos (c+d x)-10 \left (4 a^2 b^3+b^5\right ) \cos (3 (c+d x))+6 b^5 \cos (5 (c+d x))+15 a \left (4 \left (8 a^4-4 a^2 b^2-b^4\right ) (c+d x)-8 a^2 b^2 \sin (2 (c+d x))+b^4 \sin (4 (c+d x))\right )}{480 b^6 d} \] Input:
Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + b*Sin[c + d*x]),x]
Output:
(-960*a^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 60*b*(-8*a^4 + 2*a^2*b^2 + b^4)*Cos[c + d*x] - 10*(4*a^2*b^3 + b^5)*Cos [3*(c + d*x)] + 6*b^5*Cos[5*(c + d*x)] + 15*a*(4*(8*a^4 - 4*a^2*b^2 - b^4) *(c + d*x) - 8*a^2*b^2*Sin[2*(c + d*x)] + b^4*Sin[4*(c + d*x)]))/(480*b^6* d)
Time = 1.90 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.17, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {3042, 3368, 3042, 3529, 25, 3042, 3528, 25, 3042, 3528, 25, 3042, 3528, 25, 3042, 3502, 27, 3042, 3214, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 \cos (c+d x)^2}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \frac {\sin ^4(c+d x) \left (1-\sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 \left (1-\sin (c+d x)^2\right )}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {\int -\frac {\sin ^3(c+d x) \left (-5 a \sin ^2(c+d x)-b \sin (c+d x)+4 a\right )}{a+b \sin (c+d x)}dx}{5 b}+\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\int \frac {\sin ^3(c+d x) \left (-5 a \sin ^2(c+d x)-b \sin (c+d x)+4 a\right )}{a+b \sin (c+d x)}dx}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\int \frac {\sin (c+d x)^3 \left (-5 a \sin (c+d x)^2-b \sin (c+d x)+4 a\right )}{a+b \sin (c+d x)}dx}{5 b}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {\int -\frac {\sin ^2(c+d x) \left (15 a^2-b \sin (c+d x) a-4 \left (5 a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b}+\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}}{5 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\int \frac {\sin ^2(c+d x) \left (15 a^2-b \sin (c+d x) a-4 \left (5 a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\int \frac {\sin (c+d x)^2 \left (15 a^2-b \sin (c+d x) a-4 \left (5 a^2-b^2\right ) \sin (c+d x)^2\right )}{a+b \sin (c+d x)}dx}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {\int -\frac {\sin (c+d x) \left (-15 a \left (4 a^2-b^2\right ) \sin ^2(c+d x)-b \left (5 a^2+8 b^2\right ) \sin (c+d x)+8 a \left (5 a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-15 a \left (4 a^2-b^2\right ) \sin ^2(c+d x)-b \left (5 a^2+8 b^2\right ) \sin (c+d x)+8 a \left (5 a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-15 a \left (4 a^2-b^2\right ) \sin (c+d x)^2-b \left (5 a^2+8 b^2\right ) \sin (c+d x)+8 a \left (5 a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {15 \left (4 a^2-b^2\right ) a^2-b \left (20 a^2-b^2\right ) \sin (c+d x) a-8 \left (15 a^4-5 b^2 a^2-2 b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)}dx}{2 b}+\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {15 \left (4 a^2-b^2\right ) a^2-b \left (20 a^2-b^2\right ) \sin (c+d x) a-8 \left (15 a^4-5 b^2 a^2-2 b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)}dx}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {15 \left (4 a^2-b^2\right ) a^2-b \left (20 a^2-b^2\right ) \sin (c+d x) a-8 \left (15 a^4-5 b^2 a^2-2 b^4\right ) \sin (c+d x)^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {15 \left (b \left (4 a^2-b^2\right ) a^2+\left (8 a^4-4 b^2 a^2-b^4\right ) \sin (c+d x) a\right )}{a+b \sin (c+d x)}dx}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {b \left (4 a^2-b^2\right ) a^2+\left (8 a^4-4 b^2 a^2-b^4\right ) \sin (c+d x) a}{a+b \sin (c+d x)}dx}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {b \left (4 a^2-b^2\right ) a^2+\left (8 a^4-4 b^2 a^2-b^4\right ) \sin (c+d x) a}{a+b \sin (c+d x)}dx}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a x \left (8 a^4-4 a^2 b^2-b^4\right )}{b}-\frac {8 a^4 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a x \left (8 a^4-4 a^2 b^2-b^4\right )}{b}-\frac {8 a^4 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a x \left (8 a^4-4 a^2 b^2-b^4\right )}{b}-\frac {16 a^4 \left (a^2-b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {32 a^4 \left (a^2-b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a x \left (8 a^4-4 a^2 b^2-b^4\right )}{b}\right )}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {5 a \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {\frac {4 \left (5 a^2-b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a \left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a x \left (8 a^4-4 a^2 b^2-b^4\right )}{b}-\frac {16 a^4 \sqrt {a^2-b^2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}\right )}{b}+\frac {8 \left (15 a^4-5 a^2 b^2-2 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{5 b}\) |
Input:
Int[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + b*Sin[c + d*x]),x]
Output:
(Cos[c + d*x]*Sin[c + d*x]^4)/(5*b*d) - ((5*a*Cos[c + d*x]*Sin[c + d*x]^3) /(4*b*d) - ((4*(5*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d) - (-1/2* ((15*((a*(8*a^4 - 4*a^2*b^2 - b^4)*x)/b - (16*a^4*Sqrt[a^2 - b^2]*ArcTan[( 2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*d)))/b + (8*(15*a^4 - 5*a^2*b^2 - 2*b^4)*Cos[c + d*x])/(b*d))/b + (15*a*(4*a^2 - b^2)*Cos[c + d *x]*Sin[c + d*x])/(2*b*d))/(3*b))/(4*b))/(5*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [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[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Result contains complex when optimal does not.
Time = 1.12 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.44
| method | result | size |
| risch | \(\frac {a^{5} x}{b^{6}}-\frac {a^{3} x}{2 b^{4}}-\frac {a x}{8 b^{2}}+\frac {{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{5} d}-\frac {{\mathrm e}^{i \left (d x +c \right )} a^{2}}{8 b^{3} d}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{16 b d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{5} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{8 b^{3} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{16 b d}+\frac {\sqrt {-a^{2}+b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}-\frac {\sqrt {-a^{2}+b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}+\frac {\cos \left (5 d x +5 c \right )}{80 b d}+\frac {a \sin \left (4 d x +4 c \right )}{32 b^{2} d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{12 b^{3} d}-\frac {\cos \left (3 d x +3 c \right )}{48 b d}-\frac {a^{3} \sin \left (2 d x +2 c \right )}{4 b^{4} d}\) | \(338\) |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{3} b^{2}-\frac {1}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (a^{4} b -a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (a^{3} b^{2}+\frac {3}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (4 a^{4} b -2 a^{2} b^{3}-2 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (6 a^{4} b +\frac {2}{3} b^{5}-\frac {4}{3} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-a^{3} b^{2}-\frac {3}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (4 a^{4} b -\frac {2}{3} a^{2} b^{3}-\frac {2}{3} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {1}{2} a^{3} b^{2}+\frac {1}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{4} b -\frac {a^{2} b^{3}}{3}-\frac {2 b^{5}}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {a \left (8 a^{4}-4 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{6}}-\frac {2 a^{4} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{6}}}{d}\) | \(355\) |
| default | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{3} b^{2}-\frac {1}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (a^{4} b -a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (a^{3} b^{2}+\frac {3}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (4 a^{4} b -2 a^{2} b^{3}-2 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (6 a^{4} b +\frac {2}{3} b^{5}-\frac {4}{3} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-a^{3} b^{2}-\frac {3}{4} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (4 a^{4} b -\frac {2}{3} a^{2} b^{3}-\frac {2}{3} b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {1}{2} a^{3} b^{2}+\frac {1}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{4} b -\frac {a^{2} b^{3}}{3}-\frac {2 b^{5}}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {a \left (8 a^{4}-4 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{6}}-\frac {2 a^{4} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{6}}}{d}\) | \(355\) |
Input:
int(cos(d*x+c)^2*sin(d*x+c)^4/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
a^5*x/b^6-1/2*a^3*x/b^4-1/8*a*x/b^2+1/2/b^5/d*exp(I*(d*x+c))*a^4-1/8/b^3/d *exp(I*(d*x+c))*a^2-1/16/b/d*exp(I*(d*x+c))+1/2/b^5/d*exp(-I*(d*x+c))*a^4- 1/8/b^3/d*exp(-I*(d*x+c))*a^2-1/16/b/d*exp(-I*(d*x+c))+(-a^2+b^2)^(1/2)*a^ 4/d/b^6*ln(exp(I*(d*x+c))-(-I*a+(-a^2+b^2)^(1/2))/b)-(-a^2+b^2)^(1/2)*a^4/ d/b^6*ln(exp(I*(d*x+c))+(I*a+(-a^2+b^2)^(1/2))/b)+1/80/b/d*cos(5*d*x+5*c)+ 1/32*a/b^2/d*sin(4*d*x+4*c)-1/12/b^3/d*cos(3*d*x+3*c)*a^2-1/48/b/d*cos(3*d *x+3*c)-1/4*a^3/b^4/d*sin(2*d*x+2*c)
Time = 0.11 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.82 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\left [\frac {24 \, b^{5} \cos \left (d x + c\right )^{5} + 120 \, a^{4} b \cos \left (d x + c\right ) + 60 \, \sqrt {-a^{2} + b^{2}} a^{4} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 40 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} - 4 \, a^{3} b^{2} - a b^{4}\right )} d x + 15 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, b^{6} d}, \frac {24 \, b^{5} \cos \left (d x + c\right )^{5} + 120 \, a^{4} b \cos \left (d x + c\right ) + 120 \, \sqrt {a^{2} - b^{2}} a^{4} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 40 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} - 4 \, a^{3} b^{2} - a b^{4}\right )} d x + 15 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, b^{6} d}\right ] \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
[1/120*(24*b^5*cos(d*x + c)^5 + 120*a^4*b*cos(d*x + c) + 60*sqrt(-a^2 + b^ 2)*a^4*log(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2* cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 40*(a^2*b^3 + b^5)*cos (d*x + c)^3 + 15*(8*a^5 - 4*a^3*b^2 - a*b^4)*d*x + 15*(2*a*b^4*cos(d*x + c )^3 - (4*a^3*b^2 + a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^6*d), 1/120*(24*b ^5*cos(d*x + c)^5 + 120*a^4*b*cos(d*x + c) + 120*sqrt(a^2 - b^2)*a^4*arcta n(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 40*(a^2*b^3 + b^ 5)*cos(d*x + c)^3 + 15*(8*a^5 - 4*a^3*b^2 - a*b^4)*d*x + 15*(2*a*b^4*cos(d *x + c)^3 - (4*a^3*b^2 + a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^6*d)]
Timed out. \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*sin(d*x+c)**4/(a+b*sin(d*x+c)),x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (218) = 436\).
Time = 0.22 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.99 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/120*(15*(8*a^5 - 4*a^3*b^2 - a*b^4)*(d*x + c)/b^6 - 240*(a^6 - a^4*b^2)* (pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^6) + 2*(60*a^3*b*tan(1/2*d*x + 1 /2*c)^9 - 15*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*a^4*tan(1/2*d*x + 1/2*c)^8 - 120*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 120*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 90*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 480*a^4*tan(1/2*d*x + 1/2*c)^6 - 240*a^ 2*b^2*tan(1/2*d*x + 1/2*c)^6 - 240*b^4*tan(1/2*d*x + 1/2*c)^6 + 720*a^4*ta n(1/2*d*x + 1/2*c)^4 - 160*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 80*b^4*tan(1/2 *d*x + 1/2*c)^4 - 120*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 90*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 480*a^4*tan(1/2*d*x + 1/2*c)^2 - 80*a^2*b^2*tan(1/2*d*x + 1/2 *c)^2 - 80*b^4*tan(1/2*d*x + 1/2*c)^2 - 60*a^3*b*tan(1/2*d*x + 1/2*c) + 15 *a*b^3*tan(1/2*d*x + 1/2*c) + 120*a^4 - 40*a^2*b^2 - 16*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*b^5))/d
Time = 20.87 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a^4\,\cos \left (c+d\,x\right )}{b^5\,d}-\frac {\frac {\cos \left (c+d\,x\right )}{8}+\frac {\cos \left (3\,c+3\,d\,x\right )}{48}-\frac {\cos \left (5\,c+5\,d\,x\right )}{80}}{b\,d}-\frac {\frac {a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}-\frac {a\,\sin \left (4\,c+4\,d\,x\right )}{32}}{b^2\,d}-\frac {\frac {a^2\,\cos \left (c+d\,x\right )}{4}+\frac {a^2\,\cos \left (3\,c+3\,d\,x\right )}{12}}{b^3\,d}-\frac {a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+\frac {a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}}{b^4\,d}+\frac {2\,a^5\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^6\,d}-\frac {2\,a^4\,\mathrm {atanh}\left (\frac {2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}-a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )\,\sqrt {b^2-a^2}}{b^6\,d} \] Input:
int((cos(c + d*x)^2*sin(c + d*x)^4)/(a + b*sin(c + d*x)),x)
Output:
(a^4*cos(c + d*x))/(b^5*d) - (cos(c + d*x)/8 + cos(3*c + 3*d*x)/48 - cos(5 *c + 5*d*x)/80)/(b*d) - ((a*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/4 - (a*sin(4*c + 4*d*x))/32)/(b^2*d) - ((a^2*cos(c + d*x))/4 + (a^2*cos(3*c + 3*d*x))/12)/(b^3*d) - (a^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + (a^3*sin(2*c + 2*d*x))/4)/(b^4*d) + (2*a^5*atan(sin(c/2 + (d*x)/2)/cos(c /2 + (d*x)/2)))/(b^6*d) - (2*a^4*atanh((2*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^ 2)^(1/2) - a^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2) + a*b*cos(c/2 + (d*x)/ 2)*(b^2 - a^2)^(1/2))/(a^3*cos(c/2 + (d*x)/2) - 2*b^3*sin(c/2 + (d*x)/2) - a*b^2*cos(c/2 + (d*x)/2) + 2*a^2*b*sin(c/2 + (d*x)/2)))*(b^2 - a^2)^(1/2) )/(b^6*d)
Time = 0.20 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-240 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{4}+24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{5}-30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{4}+40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2} b^{3}-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{5}-60 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b^{2}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{4}+120 \cos \left (d x +c \right ) a^{4} b -40 \cos \left (d x +c \right ) a^{2} b^{3}-16 \cos \left (d x +c \right ) b^{5}+120 a^{5} c +120 a^{5} d x +72 a^{4} b -60 a^{3} b^{2} c -60 a^{3} b^{2} d x +8 a^{2} b^{3}-15 a \,b^{4} c -15 a \,b^{4} d x -16 b^{5}}{120 b^{6} d} \] Input:
int(cos(d*x+c)^2*sin(d*x+c)^4/(a+b*sin(d*x+c)),x)
Output:
( - 240*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2)) *a**4 + 24*cos(c + d*x)*sin(c + d*x)**4*b**5 - 30*cos(c + d*x)*sin(c + d*x )**3*a*b**4 + 40*cos(c + d*x)*sin(c + d*x)**2*a**2*b**3 - 8*cos(c + d*x)*s in(c + d*x)**2*b**5 - 60*cos(c + d*x)*sin(c + d*x)*a**3*b**2 + 15*cos(c + d*x)*sin(c + d*x)*a*b**4 + 120*cos(c + d*x)*a**4*b - 40*cos(c + d*x)*a**2* b**3 - 16*cos(c + d*x)*b**5 + 120*a**5*c + 120*a**5*d*x + 72*a**4*b - 60*a **3*b**2*c - 60*a**3*b**2*d*x + 8*a**2*b**3 - 15*a*b**4*c - 15*a*b**4*d*x - 16*b**5)/(120*b**6*d)