Integrand size = 24, antiderivative size = 290 \[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (5 a-17 b-3 (a-3 b) \cos ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \] Output:
3/64*(a^(1/2)-2*b^(1/2))*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2) )/a^(1/2)/(a^(1/2)-b^(1/2))^(5/2)/b^(7/4)/d-3/64*(a^(1/2)+2*b^(1/2))*arcta nh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(1/2)/(a^(1/2)+b^(1/2))^( 5/2)/b^(7/4)/d-1/8*a*cos(d*x+c)*(2-cos(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cos(d* x+c)^2-b*cos(d*x+c)^4)^2+1/32*cos(d*x+c)*(5*a-17*b-3*(a-3*b)*cos(d*x+c)^2) /(a-b)^2/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 4.60 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.17 \[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {-\frac {32 \cos (c+d x) (-7 a+25 b+3 (a-3 b) \cos (2 (c+d x)))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}+\frac {512 a (a-b) (-5 \cos (c+d x)+\cos (3 (c+d x)))}{(-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}-3 i \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {2 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-6 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+3 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-6 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+34 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+3 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-17 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+6 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-34 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-3 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+17 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-2 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+6 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6-3 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{256 (a-b)^2 b d} \] Input:
Integrate[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^3,x]
Output:
((-32*Cos[c + d*x]*(-7*a + 25*b + 3*(a - 3*b)*Cos[2*(c + d*x)]))/(8*a - 3* b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + (512*a*(a - b)*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*( c + d*x)])^2 - (3*I)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^ 6 + b*#1^8 & , (2*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 6*b*ArcTan[ Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 6*a*ArcTan[Sin[c + d*x]/(Cos [c + d*x] - #1)]*#1^2 + 34*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (17*I)*b*Log[1 - 2*Cos [c + d*x]*#1 + #1^2]*#1^2 + 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*# 1^4 - 34*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (17*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^ 2]*#1^4 - 2*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 6*b*ArcTan[S in[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1 ^2]*#1^6 - (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a* #1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(256*(a - b)^2*b*d)
Time = 0.56 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3694, 1517, 27, 1492, 27, 1480, 218, 221}
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.
| \(\displaystyle \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^7}{\left (a-b \sin (c+d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (1-\cos ^2(c+d x)\right )^3}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^3}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\int \frac {2 a \left (2 (a-4 b)-(3 a-8 b) \cos ^2(c+d x)\right )}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\int \frac {2 (a-4 b)-(3 a-8 b) \cos ^2(c+d x)}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {\cos (c+d x) \left (-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\int -\frac {6 a b \left (-\left ((a-3 b) \cos ^2(c+d x)\right )+a-5 b\right )}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{8 a b (a-b)}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {3 \int \frac {-\left ((a-3 b) \cos ^2(c+d x)\right )+a-5 b}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{4 (a-b)}+\frac {\cos (c+d x) \left (-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {3 \left (-\frac {1}{2} \left (-\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \int \frac {1}{-b \cos ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cos (c+d x)-\frac {1}{2} \left (\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)\right )}{4 (a-b)}+\frac {\cos (c+d x) \left (-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {3 \left (\frac {\left (-\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {1}{2} \left (\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)\right )}{4 (a-b)}+\frac {\cos (c+d x) \left (-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {3 \left (\frac {\left (-\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\left (\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{4 (a-b)}+\frac {\cos (c+d x) \left (-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
Input:
Int[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^3,x]
Output:
-(((a*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*b*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - ((3*(((a - 3*b - (2*b^(3/2))/Sqrt[a])*Arc Tan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqr t[b]]*b^(3/4)) - ((a - 3*b + (2*b^(3/2))/Sqrt[a])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/4))))/(4* (a - b)) + (Cos[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cos[c + d*x]^2))/(4*(a - b)*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)))/(8*(a - b)*b))/d)
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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 6.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {3 \left (a -3 b \right ) \cos \left (d x +c \right )^{7}}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (11 a -35 b \right ) \cos \left (d x +c \right )^{5}}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (a^{2}+18 a b -43 b^{2}\right ) \cos \left (d x +c \right )^{3}}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (3 a +17 b \right ) \cos \left (d x +c \right )}{32 \left (a -b \right ) b}}{\left (a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}\right )^{2}}+\frac {-\frac {3 \left (a \sqrt {a b}-3 \sqrt {a b}\, b +2 b^{2}\right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{64 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {3 \left (a \sqrt {a b}-3 \sqrt {a b}\, b -2 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{64 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}}{a^{2}-2 a b +b^{2}}}{d}\) | \(303\) |
| default | \(\frac {\frac {\frac {3 \left (a -3 b \right ) \cos \left (d x +c \right )^{7}}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (11 a -35 b \right ) \cos \left (d x +c \right )^{5}}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (a^{2}+18 a b -43 b^{2}\right ) \cos \left (d x +c \right )^{3}}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (3 a +17 b \right ) \cos \left (d x +c \right )}{32 \left (a -b \right ) b}}{\left (a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}\right )^{2}}+\frac {-\frac {3 \left (a \sqrt {a b}-3 \sqrt {a b}\, b +2 b^{2}\right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{64 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {3 \left (a \sqrt {a b}-3 \sqrt {a b}\, b -2 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{64 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}}{a^{2}-2 a b +b^{2}}}{d}\) | \(303\) |
| risch | \(\text {Expression too large to display}\) | \(1136\) |
Input:
int(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*((3/32*(a-3*b)/(a^2-2*a*b+b^2)*cos(d*x+c)^7-1/32*(11*a-35*b)/(a^2-2*a* b+b^2)*cos(d*x+c)^5+1/32*(a^2+18*a*b-43*b^2)/b/(a^2-2*a*b+b^2)*cos(d*x+c)^ 3-1/32*(3*a+17*b)/(a-b)/b*cos(d*x+c))/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4 )^2+3/32/(a^2-2*a*b+b^2)*(-1/2*(a*(a*b)^(1/2)-3*(a*b)^(1/2)*b+2*b^2)/(a*b) ^(1/2)/b/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(b*cos(d*x+c)/(((a*b)^(1/2)+b)*b )^(1/2))+1/2*(a*(a*b)^(1/2)-3*(a*b)^(1/2)*b-2*b^2)/(a*b)^(1/2)/b/(((a*b)^( 1/2)-b)*b)^(1/2)*arctan(b*cos(d*x+c)/(((a*b)^(1/2)-b)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 4185 vs. \(2 (234) = 468\).
Time = 0.51 (sec) , antiderivative size = 4185, normalized size of antiderivative = 14.43 \[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)**7/(a-b*sin(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{7}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")
Output:
-1/16*(24*(a*b^3 - 3*b^4)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(23*a*b^3 - 77 *b^4)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 24*(a*b^3 - 3*b^4)*sin(2*d*x + 2 *c)*sin(d*x + c) - (3*(a*b^3 - 3*b^4)*cos(15*d*x + 15*c) - (23*a*b^3 - 77* b^4)*cos(13*d*x + 13*c) + (16*a^2*b^2 + 131*a*b^3 - 177*b^4)*cos(11*d*x + 11*c) - (144*a^2*b^2 + 367*a*b^3 - 109*b^4)*cos(9*d*x + 9*c) - (144*a^2*b^ 2 + 367*a*b^3 - 109*b^4)*cos(7*d*x + 7*c) + (16*a^2*b^2 + 131*a*b^3 - 177* b^4)*cos(5*d*x + 5*c) - (23*a*b^3 - 77*b^4)*cos(3*d*x + 3*c) + 3*(a*b^3 - 3*b^4)*cos(d*x + c))*cos(16*d*x + 16*c) - 3*(a*b^3 - 3*b^4 - 8*(a*b^3 - 3* b^4)*cos(14*d*x + 14*c) - 4*(8*a^2*b^2 - 31*a*b^3 + 21*b^4)*cos(12*d*x + 1 2*c) + 8*(16*a^2*b^2 - 55*a*b^3 + 21*b^4)*cos(10*d*x + 10*c) + 2*(128*a^3* b - 480*a^2*b^2 + 323*a*b^3 - 105*b^4)*cos(8*d*x + 8*c) + 8*(16*a^2*b^2 - 55*a*b^3 + 21*b^4)*cos(6*d*x + 6*c) - 4*(8*a^2*b^2 - 31*a*b^3 + 21*b^4)*co s(4*d*x + 4*c) - 8*(a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((23*a*b^3 - 77*b^4)*cos(13*d*x + 13*c) - (16*a^2*b^2 + 131*a*b^3 - 177* b^4)*cos(11*d*x + 11*c) + (144*a^2*b^2 + 367*a*b^3 - 109*b^4)*cos(9*d*x + 9*c) + (144*a^2*b^2 + 367*a*b^3 - 109*b^4)*cos(7*d*x + 7*c) - (16*a^2*b^2 + 131*a*b^3 - 177*b^4)*cos(5*d*x + 5*c) + (23*a*b^3 - 77*b^4)*cos(3*d*x + 3*c) - 3*(a*b^3 - 3*b^4)*cos(d*x + c))*cos(14*d*x + 14*c) + (23*a*b^3 - 77 *b^4 - 4*(184*a^2*b^2 - 777*a*b^3 + 539*b^4)*cos(12*d*x + 12*c) + 8*(368*a ^2*b^2 - 1393*a*b^3 + 539*b^4)*cos(10*d*x + 10*c) + 2*(2944*a^3*b - 120...
\[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{7}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")
Output:
sage0*x
Time = 41.37 (sec) , antiderivative size = 5824, normalized size of antiderivative = 20.08 \[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(sin(c + d*x)^7/(a - b*sin(c + d*x)^4)^3,x)
Output:
((3*cos(c + d*x)^7*(a - 3*b))/(32*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)^5*( 11*a - 35*b))/(32*(a^2 - 2*a*b + b^2)) + (cos(c + d*x)^3*(18*a*b + a^2 - 4 3*b^2))/(32*b*(a - b)^2) - (cos(c + d*x)*(3*a + 17*b))/(32*b*(a - b)))/(d* (a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4*b^2) - cos(c + d*x)^4*(2*a* b - 6*b^2) - 4*b^2*cos(c + d*x)^6 + b^2*cos(c + d*x)^8)) + (atan(((((3*(81 920*a*b^7 - 180224*a^2*b^6 + 114688*a^3*b^5 - 16384*a^4*b^4))/(32768*(b^6 - 4*a*b^5 + 6*a^2*b^4 - 4*a^3*b^3 + a^4*b^2)) - (cos(c + d*x)*((9*(a^3*(a^ 3*b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^ 2*b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1 /2)*(16384*a*b^8 - 65536*a^2*b^7 + 98304*a^3*b^6 - 65536*a^4*b^5 + 16384*a ^5*b^4))/(256*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a^3*(a^3* b^7)^(1/2) + 16*b^3*(a^3*b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2* b^12 - 5*a^3*b^11 + 10*a^4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2 ) + (cos(c + d*x)*(81*a*b^2 - 54*a^2*b + 9*a^3 + 36*b^3))/(256*(a^4 - 4*a^ 3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*((9*(a^3*(a^3*b^7)^(1/2) + 16*b^3*(a^3* b^7)^(1/2) + 4*a*b^7 + 21*a^2*b^6 - 10*a^3*b^5 + a^4*b^4 + 5*a*b^2*(a^3*b^ 7)^(1/2) - 6*a^2*b*(a^3*b^7)^(1/2)))/(16384*(a^2*b^12 - 5*a^3*b^11 + 10*a^ 4*b^10 - 10*a^5*b^9 + 5*a^6*b^8 - a^7*b^7)))^(1/2)*1i - (((3*(81920*a*b...
\[ \int \frac {\sin ^7(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\left (\int \frac {\sin \left (d x +c \right )^{7}}{\sin \left (d x +c \right )^{12} b^{3}-3 \sin \left (d x +c \right )^{8} a \,b^{2}+3 \sin \left (d x +c \right )^{4} a^{2} b -a^{3}}d x \right ) \] Input:
int(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x)
Output:
- int(sin(c + d*x)**7/(sin(c + d*x)**12*b**3 - 3*sin(c + d*x)**8*a*b**2 + 3*sin(c + d*x)**4*a**2*b - a**3),x)