Integrand size = 12, antiderivative size = 182 \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx=\frac {a \left (2 a^2+3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}+\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}+\frac {5 a b \cos (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \] Output:
a*(2*a^2+3*b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2) ^(7/2)/d+1/3*b*cos(d*x+c)/(a^2-b^2)/d/(a+b*sin(d*x+c))^3+5/6*a*b*cos(d*x+c )/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^2+1/6*b*(11*a^2+4*b^2)*cos(d*x+c)/(a^2-b^ 2)^3/d/(a+b*sin(d*x+c))
Time = 1.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx=\frac {\frac {6 a \left (2 a^2+3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {b \cos (c+d x) \left (18 a^4-5 a^2 b^2+2 b^4+3 a b \left (9 a^2+b^2\right ) \sin (c+d x)+b^2 \left (11 a^2+4 b^2\right ) \sin ^2(c+d x)\right )}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^3}}{6 d} \] Input:
Integrate[(a + b*Sin[c + d*x])^(-4),x]
Output:
((6*a*(2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a ^2 - b^2)^(7/2) + (b*Cos[c + d*x]*(18*a^4 - 5*a^2*b^2 + 2*b^4 + 3*a*b*(9*a ^2 + b^2)*Sin[c + d*x] + b^2*(11*a^2 + 4*b^2)*Sin[c + d*x]^2))/((a - b)^3* (a + b)^3*(a + b*Sin[c + d*x])^3))/(6*d)
Time = 0.77 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3143, 25, 3042, 3233, 25, 3042, 3233, 27, 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.
| \(\displaystyle \int \frac {1}{(a+b \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle \frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}-\frac {\int -\frac {3 a-2 b \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 a-2 b \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a-2 b \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {\int -\frac {2 \left (3 a^2+2 b^2\right )-5 a b \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (3 a^2+2 b^2\right )-5 a b \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (3 a^2+2 b^2\right )-5 a b \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\int -\frac {3 a \left (2 a^2+3 b^2\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 a \left (2 a^2+3 b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a \left (2 a^2+3 b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {6 a \left (2 a^2+3 b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {12 a \left (2 a^2+3 b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {6 a \left (2 a^2+3 b^2\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{3/2}}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {5 a b \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\) |
Input:
Int[(a + b*Sin[c + d*x])^(-4),x]
Output:
(b*Cos[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^3) + ((5*a*b*Cos[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((6*a*(2*a^2 + 3*b^2)*A rcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2 )*d) + (b*(11*a^2 + 4*b^2)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x ])))/(2*(a^2 - b^2)))/(3*(a^2 - b^2))
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp [1/((n + 1)*(a^2 - b^2)) Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(171)=342\).
Time = 1.06 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.80
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {b^{2} \left (9 a^{4}-6 b^{2} a^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {b \left (6 a^{6}+27 a^{4} b^{2}-12 a^{2} b^{4}+4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b^{2} \left (54 a^{6}+21 a^{4} b^{2}-4 a^{2} b^{4}+4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b \left (6 a^{6}+20 a^{4} b^{2}-3 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {b^{2} \left (27 a^{4}-4 b^{2} a^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b \left (18 a^{4}-5 b^{2} a^{2}+2 b^{4}\right )}{6 a^{6}-18 a^{4} b^{2}+18 a^{2} b^{4}-6 b^{6}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{3}}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {a^{2}-b^{2}}}}{d}\) | \(510\) |
| default | \(\frac {\frac {\frac {b^{2} \left (9 a^{4}-6 b^{2} a^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {b \left (6 a^{6}+27 a^{4} b^{2}-12 a^{2} b^{4}+4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b^{2} \left (54 a^{6}+21 a^{4} b^{2}-4 a^{2} b^{4}+4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b \left (6 a^{6}+20 a^{4} b^{2}-3 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {b^{2} \left (27 a^{4}-4 b^{2} a^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b \left (18 a^{4}-5 b^{2} a^{2}+2 b^{4}\right )}{6 a^{6}-18 a^{4} b^{2}+18 a^{2} b^{4}-6 b^{6}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{3}}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {a^{2}-b^{2}}}}{d}\) | \(510\) |
| risch | \(-\frac {i \left (-30 i b \,a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-45 i a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-6 a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-9 a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+102 i b \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+36 i b^{3} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12 i b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+44 a^{5} {\mathrm e}^{3 i \left (d x +c \right )}+82 a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+24 a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-11 i a^{2} b^{3}-4 i b^{5}-60 a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}-15 a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left (-i {\mathrm e}^{2 i \left (d x +c \right )} b +i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{3} \left (a^{2}-b^{2}\right )^{3} d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}\) | \(585\) |
Input:
int(1/(a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(2*(1/2*b^2*(9*a^4-6*a^2*b^2+2*b^4)/a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*ta n(1/2*d*x+1/2*c)^5+1/2*b*(6*a^6+27*a^4*b^2-12*a^2*b^4+4*b^6)/a^2/(a^6-3*a^ 4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^4+1/3/a^3*b^2*(54*a^6+21*a^4*b^2-4 *a^2*b^4+4*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^3+1/a^2*b *(6*a^6+20*a^4*b^2-3*a^2*b^4+2*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2* d*x+1/2*c)^2+1/2*b^2*(27*a^4-4*a^2*b^2+2*b^4)/a/(a^6-3*a^4*b^2+3*a^2*b^4-b ^6)*tan(1/2*d*x+1/2*c)+1/6*b*(18*a^4-5*a^2*b^2+2*b^4)/(a^6-3*a^4*b^2+3*a^2 *b^4-b^6))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^3+a*(2*a^2+3* b^2)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2 *d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (171) = 342\).
Time = 0.15 (sec) , antiderivative size = 965, normalized size of antiderivative = 5.30 \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*sin(d*x+c))^4,x, algorithm="fricas")
Output:
[1/12*(2*(11*a^4*b^3 - 7*a^2*b^5 - 4*b^7)*cos(d*x + c)^3 - 6*(9*a^5*b^2 - 8*a^3*b^4 - a*b^6)*cos(d*x + c)*sin(d*x + c) - 3*(2*a^6 + 9*a^4*b^2 + 9*a^ 2*b^4 - 3*(2*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^2 + (6*a^5*b + 11*a^3*b^3 + 3*a*b^5 - (2*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*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)) - 12*(3*a^6*b - 2*a^4*b^ 3 - b^7)*cos(d*x + c))/(3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a *b^10)*d*cos(d*x + c)^2 - (a^11 - a^9*b^2 - 6*a^7*b^4 + 14*a^5*b^6 - 11*a^ 3*b^8 + 3*a*b^10)*d + ((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11 )*d*cos(d*x + c)^2 - (3*a^10*b - 11*a^8*b^3 + 14*a^6*b^5 - 6*a^4*b^7 - a^2 *b^9 + b^11)*d)*sin(d*x + c)), 1/6*((11*a^4*b^3 - 7*a^2*b^5 - 4*b^7)*cos(d *x + c)^3 - 3*(9*a^5*b^2 - 8*a^3*b^4 - a*b^6)*cos(d*x + c)*sin(d*x + c) + 3*(2*a^6 + 9*a^4*b^2 + 9*a^2*b^4 - 3*(2*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^ 2 + (6*a^5*b + 11*a^3*b^3 + 3*a*b^5 - (2*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^2 )*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b ^2)*cos(d*x + c))) - 6*(3*a^6*b - 2*a^4*b^3 - b^7)*cos(d*x + c))/(3*(a^9*b ^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cos(d*x + c)^2 - (a^11 - a^9*b^2 - 6*a^7*b^4 + 14*a^5*b^6 - 11*a^3*b^8 + 3*a*b^10)*d + ((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cos(d*x + c)^2 - (3*a^10*...
Timed out. \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(d*x+c))**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(a+b*sin(d*x+c))^4,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. 510 vs. \(2 (171) = 342\).
Time = 0.14 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.80 \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/3*(3*(2*a^3 + 3*a*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)))/((a^6 - 3*a^4*b^2 + 3*a^2* b^4 - b^6)*sqrt(a^2 - b^2)) + (27*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 - 18*a^4* b^4*tan(1/2*d*x + 1/2*c)^5 + 6*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 18*a^7*b*t an(1/2*d*x + 1/2*c)^4 + 81*a^5*b^3*tan(1/2*d*x + 1/2*c)^4 - 36*a^3*b^5*tan (1/2*d*x + 1/2*c)^4 + 12*a*b^7*tan(1/2*d*x + 1/2*c)^4 + 108*a^6*b^2*tan(1/ 2*d*x + 1/2*c)^3 + 42*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 8*a^2*b^6*tan(1/2*d *x + 1/2*c)^3 + 8*b^8*tan(1/2*d*x + 1/2*c)^3 + 36*a^7*b*tan(1/2*d*x + 1/2* c)^2 + 120*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 - 18*a^3*b^5*tan(1/2*d*x + 1/2*c )^2 + 12*a*b^7*tan(1/2*d*x + 1/2*c)^2 + 81*a^6*b^2*tan(1/2*d*x + 1/2*c) - 12*a^4*b^4*tan(1/2*d*x + 1/2*c) + 6*a^2*b^6*tan(1/2*d*x + 1/2*c) + 18*a^7* b - 5*a^5*b^3 + 2*a^3*b^5)/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(a*tan (1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^3))/d
Time = 19.86 (sec) , antiderivative size = 708, normalized size of antiderivative = 3.89 \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx=\frac {\frac {18\,a^4\,b-5\,a^2\,b^3+2\,b^5}{3\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (6\,a^6\,b+20\,a^4\,b^3-3\,a^2\,b^5+2\,b^7\right )}{a^2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (6\,a^6\,b+27\,a^4\,b^3-12\,a^2\,b^5+4\,b^7\right )}{a^2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (27\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )}{a\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (9\,a^4\,b-6\,a^2\,b^3+2\,b^5\right )}{a\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^2+2\,b^2\right )\,\left (18\,a^4\,b-5\,a^2\,b^3+2\,b^5\right )}{3\,a^3\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (3\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (3\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^2\,b+8\,b^3\right )+a^3+6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {a\,\mathrm {atan}\left (\frac {\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+3\,b^2\right )}{{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}+\frac {a\,\left (2\,a^2+3\,b^2\right )\,\left (2\,a^6\,b-6\,a^4\,b^3+6\,a^2\,b^5-2\,b^7\right )}{2\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}{2\,a^3+3\,a\,b^2}\right )\,\left (2\,a^2+3\,b^2\right )}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}} \] Input:
int(1/(a + b*sin(c + d*x))^4,x)
Output:
((18*a^4*b + 2*b^5 - 5*a^2*b^3)/(3*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (2*tan(c/2 + (d*x)/2)^2*(6*a^6*b + 2*b^7 - 3*a^2*b^5 + 20*a^4*b^3))/(a^2*( a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (tan(c/2 + (d*x)/2)^4*(6*a^6*b + 4*b ^7 - 12*a^2*b^5 + 27*a^4*b^3))/(a^2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (b*tan(c/2 + (d*x)/2)*(27*a^4*b + 2*b^5 - 4*a^2*b^3))/(a*(a^6 - b^6 + 3*a ^2*b^4 - 3*a^4*b^2)) + (b*tan(c/2 + (d*x)/2)^5*(9*a^4*b + 2*b^5 - 6*a^2*b^ 3))/(a*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (2*b*tan(c/2 + (d*x)/2)^3*(3 *a^2 + 2*b^2)*(18*a^4*b + 2*b^5 - 5*a^2*b^3))/(3*a^3*(a^6 - b^6 + 3*a^2*b^ 4 - 3*a^4*b^2)))/(d*(a^3*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^2*(12*a *b^2 + 3*a^3) + tan(c/2 + (d*x)/2)^4*(12*a*b^2 + 3*a^3) + tan(c/2 + (d*x)/ 2)^3*(12*a^2*b + 8*b^3) + a^3 + 6*a^2*b*tan(c/2 + (d*x)/2) + 6*a^2*b*tan(c /2 + (d*x)/2)^5)) + (a*atan((((a^2*tan(c/2 + (d*x)/2)*(2*a^2 + 3*b^2))/((a + b)^(7/2)*(a - b)^(7/2)) + (a*(2*a^2 + 3*b^2)*(2*a^6*b - 2*b^7 + 6*a^2*b ^5 - 6*a^4*b^3))/(2*(a + b)^(7/2)*(a - b)^(7/2)*(a^6 - b^6 + 3*a^2*b^4 - 3 *a^4*b^2)))*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2))/(3*a*b^2 + 2*a^3))*(2*a^2 + 3*b^2))/(d*(a + b)^(7/2)*(a - b)^(7/2))
Time = 0.21 (sec) , antiderivative size = 997, normalized size of antiderivative = 5.48 \[ \int \frac {1}{(a+b \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(1/(a+b*sin(d*x+c))^4,x)
Output:
(12*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin (c + d*x)**3*a**4*b**3 + 18*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b )/sqrt(a**2 - b**2))*sin(c + d*x)**3*a**2*b**5 + 36*sqrt(a**2 - b**2)*atan ((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**5*b**2 + 5 4*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**3*b**4 + 36*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/ sqrt(a**2 - b**2))*sin(c + d*x)*a**6*b + 54*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**4*b**3 + 12*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**7 + 18*sqrt(a **2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**5*b**2 + 1 1*cos(c + d*x)*sin(c + d*x)**2*a**5*b**3 - 7*cos(c + d*x)*sin(c + d*x)**2* a**3*b**5 - 4*cos(c + d*x)*sin(c + d*x)**2*a*b**7 + 27*cos(c + d*x)*sin(c + d*x)*a**6*b**2 - 24*cos(c + d*x)*sin(c + d*x)*a**4*b**4 - 3*cos(c + d*x) *sin(c + d*x)*a**2*b**6 + 18*cos(c + d*x)*a**7*b - 23*cos(c + d*x)*a**5*b* *3 + 7*cos(c + d*x)*a**3*b**5 - 2*cos(c + d*x)*a*b**7 + 9*sin(c + d*x)**3* a**4*b**4 - 8*sin(c + d*x)**3*a**2*b**6 - sin(c + d*x)**3*b**8 + 27*sin(c + d*x)**2*a**5*b**3 - 24*sin(c + d*x)**2*a**3*b**5 - 3*sin(c + d*x)**2*a*b **7 + 27*sin(c + d*x)*a**6*b**2 - 24*sin(c + d*x)*a**4*b**4 - 3*sin(c + d* x)*a**2*b**6 + 9*a**7*b - 8*a**5*b**3 - a**3*b**5)/(6*a*d*(sin(c + d*x)**3 *a**8*b**3 - 4*sin(c + d*x)**3*a**6*b**5 + 6*sin(c + d*x)**3*a**4*b**7 ...