Integrand size = 24, antiderivative size = 355 \[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {\cot (c+d x)}{a^3 d}-\frac {b^2 \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {b \tan (c+d x) \left (2 a^2 (9 a-17 b)+(a-b) \left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)\right )}{32 a^3 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \] Output:
3/64*b^(1/2)*(20*a-34*a^(1/2)*b^(1/2)+15*b)*arctan((a^(1/2)-b^(1/2))^(1/2) *tan(d*x+c)/a^(1/4))/a^(13/4)/(a^(1/2)-b^(1/2))^(5/2)/d-3/64*b^(1/2)*(20*a +34*a^(1/2)*b^(1/2)+15*b)*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4 ))/a^(13/4)/(a^(1/2)+b^(1/2))^(5/2)/d-cot(d*x+c)/a^3/d-1/8*b^2*tan(d*x+c)* (a*(a+3*b)+(a^2+6*a*b+b^2)*tan(d*x+c)^2)/a^2/(a-b)^3/d/(a+2*a*tan(d*x+c)^2 +(a-b)*tan(d*x+c)^4)^2-1/32*b*tan(d*x+c)*(2*a^2*(9*a-17*b)+(a-b)*(18*a^2+1 5*a*b-13*b^2)*tan(d*x+c)^2)/a^3/(a-b)^3/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d* x+c)^4)
Time = 12.68 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.01 \[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {-a+\sqrt {a} \sqrt {b}}}+64 \cot (c+d x)+\frac {4 b \left (28 a^2+3 a b-13 b^2+b (-19 a+13 b) \cos (2 (c+d x))\right ) \sin (2 (c+d x))}{(a-b)^2 (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}+\frac {128 a b (2 a+b-b \cos (2 (c+d x))) \sin (2 (c+d x))}{(a-b) (-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}}{64 a^3 d} \] Input:
Integrate[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]
Output:
-1/64*((3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[((Sqrt[a] + Sq rt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])^2*Sq rt[a + Sqrt[a]*Sqrt[b]]) + (3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*A rcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((S qrt[a] - Sqrt[b])^2*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 64*Cot[c + d*x] + (4*b*( 28*a^2 + 3*a*b - 13*b^2 + b*(-19*a + 13*b)*Cos[2*(c + d*x)])*Sin[2*(c + d* x)])/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (128*a*b*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a - b)*(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2))/(a^3*d)
Time = 1.43 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3696, 1673, 27, 2198, 27, 2195, 2009}
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.
| \(\displaystyle \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^2 \left (a-b \sin (c+d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\cot ^2(c+d x) \left (\tan ^2(c+d x)+1\right )^6}{\left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^3}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1673 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \cot ^2(c+d x) \left (\frac {8 a^2 b \tan ^8(c+d x)}{a-b}+\frac {16 a^2 (2 a-3 b) b \tan ^6(c+d x)}{(a-b)^2}+\frac {b \left (48 a^4-136 b a^3+115 b^2 a^2-30 b^3 a-5 b^4\right ) \tan ^4(c+d x)}{(a-b)^3}+\frac {a b \left (32 a^3-96 b a^2+97 b^2 a-29 b^3\right ) \tan ^2(c+d x)}{(a-b)^3}+8 a b\right )}{\left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{16 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)}{(a-b)^3}+\frac {a (a+3 b)}{(a-b)^3}\right )}{8 a^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\cot ^2(c+d x) \left (\frac {8 a^2 b \tan ^8(c+d x)}{a-b}+\frac {16 a^2 (2 a-3 b) b \tan ^6(c+d x)}{(a-b)^2}+\frac {b \left (48 a^4-136 b a^3+115 b^2 a^2-30 b^3 a-5 b^4\right ) \tan ^4(c+d x)}{(a-b)^3}+\frac {a b \left (32 a^3-96 b a^2+97 b^2 a-29 b^3\right ) \tan ^2(c+d x)}{(a-b)^3}+8 a b\right )}{\left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{8 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)}{(a-b)^3}+\frac {a (a+3 b)}{(a-b)^3}\right )}{8 a^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \cot ^2(c+d x) \left (\frac {a b^2 \left (32 a^3-18 b a^2-15 b^2 a+13 b^3\right ) \tan ^4(c+d x)}{(a-b)^2}+\frac {2 a^2 b^2 \left (32 a^2-55 b a+26 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+32 a^2 b^2\right )}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{8 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a^2 (9 a-17 b)}{(a-b)^3}\right )}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{8 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)}{(a-b)^3}+\frac {a (a+3 b)}{(a-b)^3}\right )}{8 a^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cot ^2(c+d x) \left (\frac {a b^2 \left (32 a^3-18 b a^2-15 b^2 a+13 b^3\right ) \tan ^4(c+d x)}{(a-b)^2}+\frac {2 a^2 b^2 \left (32 a^2-55 b a+26 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+32 a^2 b^2\right )}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{4 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a^2 (9 a-17 b)}{(a-b)^3}\right )}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{8 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)}{(a-b)^3}+\frac {a (a+3 b)}{(a-b)^3}\right )}{8 a^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle \frac {\frac {\frac {\int \left (\frac {3 a \left (\left (26 a^2-37 b a+15 b^2\right ) \tan ^2(c+d x)+2 a (3 a-2 b)\right ) b^3}{(a-b)^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}+32 a \cot ^2(c+d x) b^2\right )d\tan (c+d x)}{4 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a^2 (9 a-17 b)}{(a-b)^3}\right )}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{8 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)}{(a-b)^3}+\frac {a (a+3 b)}{(a-b)^3}\right )}{8 a^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 a^{3/4} b^{5/2} \left (-34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 a^{3/4} b^{5/2} \left (34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-32 a b^2 \cot (c+d x)}{4 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a^2 (9 a-17 b)}{(a-b)^3}\right )}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{8 a^2 b}-\frac {b^2 \tan (c+d x) \left (\frac {\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)}{(a-b)^3}+\frac {a (a+3 b)}{(a-b)^3}\right )}{8 a^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}}{d}\) |
Input:
Int[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]
Output:
(-1/8*(b^2*Tan[c + d*x]*((a*(a + 3*b))/(a - b)^3 + ((a^2 + 6*a*b + b^2)*Ta n[c + d*x]^2)/(a - b)^3))/(a^2*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d *x]^4)^2) + (((3*a^(3/4)*b^(5/2)*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan [(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*(Sqrt[a] - Sqrt[b])^( 5/2)) - (3*a^(3/4)*b^(5/2)*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[(Sqrt [Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*(Sqrt[a] + Sqrt[b])^(5/2)) - 32*a*b^2*Cot[c + d*x])/(4*a^2*b) - (b^2*Tan[c + d*x]*((2*a^2*(9*a - 17*b ))/(a - b)^3 + ((18*a^2 + 15*a*b - 13*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(4* a*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)))/(8*a^2*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[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q , a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(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[x^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 ) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) ^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x ^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Time = 6.78 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {-\frac {\left (18 a^{2}+15 a b -13 b^{2}\right ) \tan \left (d x +c \right )^{7}}{32 \left (a -b \right )}-\frac {a \left (27 a^{2}-2 a b -13 b^{2}\right ) \tan \left (d x +c \right )^{5}}{16 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (54 a^{2}-13 a b -17 b^{2}\right ) a \tan \left (d x +c \right )^{3}}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a -2 b \right ) \tan \left (d x +c \right )}{16 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (d x +c \right )^{4} a -\tan \left (d x +c \right )^{4} b +2 a \tan \left (d x +c \right )^{2}+a \right )^{2}}+\frac {3 \left (a -b \right ) \left (\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}-20 a^{3}+27 a^{2} b -11 b^{2} a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}+20 a^{3}-27 a^{2} b +11 b^{2} a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}-\frac {1}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(437\) |
| default | \(\frac {\frac {b \left (\frac {-\frac {\left (18 a^{2}+15 a b -13 b^{2}\right ) \tan \left (d x +c \right )^{7}}{32 \left (a -b \right )}-\frac {a \left (27 a^{2}-2 a b -13 b^{2}\right ) \tan \left (d x +c \right )^{5}}{16 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (54 a^{2}-13 a b -17 b^{2}\right ) a \tan \left (d x +c \right )^{3}}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a -2 b \right ) \tan \left (d x +c \right )}{16 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (d x +c \right )^{4} a -\tan \left (d x +c \right )^{4} b +2 a \tan \left (d x +c \right )^{2}+a \right )^{2}}+\frac {3 \left (a -b \right ) \left (\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}-20 a^{3}+27 a^{2} b -11 b^{2} a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}+20 a^{3}-27 a^{2} b +11 b^{2} a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}-\frac {1}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(437\) |
| risch | \(\text {Expression too large to display}\) | \(2739\) |
Input:
int(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(b/a^3*((-1/32*(18*a^2+15*a*b-13*b^2)/(a-b)*tan(d*x+c)^7-1/16*a*(27*a^ 2-2*a*b-13*b^2)/(a^2-2*a*b+b^2)*tan(d*x+c)^5-1/32*(54*a^2-13*a*b-17*b^2)*a /(a^2-2*a*b+b^2)*tan(d*x+c)^3-3/16*a^2*(3*a-2*b)/(a^2-2*a*b+b^2)*tan(d*x+c ))/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2+3/32/(a^2-2*a*b+b^ 2)*(a-b)*(1/2*(26*a^2*(a*b)^(1/2)-37*a*b*(a*b)^(1/2)+15*b^2*(a*b)^(1/2)-20 *a^3+27*a^2*b-11*b^2*a)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*ar ctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+1/2*(26*a^2*(a*b)^( 1/2)-37*a*b*(a*b)^(1/2)+15*b^2*(a*b)^(1/2)+20*a^3-27*a^2*b+11*b^2*a)/(a*b) ^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b) ^(1/2)+a)*(a-b))^(1/2))))-1/a^3/tan(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 6323 vs. \(2 (303) = 606\).
Time = 2.38 (sec) , antiderivative size = 6323, normalized size of antiderivative = 17.81 \[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**2/(a-b*sin(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\csc \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")
Output:
1/16*(12*(160*a^3*b^3 - 57*a^2*b^4 - 195*a*b^5 + 135*b^6)*cos(4*d*x + 4*c) *sin(2*d*x + 2*c) + (3*(20*a^2*b^4 - 33*a*b^5 + 15*b^6)*sin(16*d*x + 16*c) - 12*(43*a^2*b^4 - 68*a*b^5 + 30*b^6)*sin(14*d*x + 14*c) - 4*(400*a^3*b^3 - 1137*a^2*b^4 + 1031*a*b^5 - 315*b^6)*sin(12*d*x + 12*c) + 12*(592*a^3*b ^3 - 1237*a^2*b^4 + 886*a*b^5 - 210*b^6)*sin(10*d*x + 10*c) + 2*(4096*a^4* b^2 - 12192*a^3*b^3 + 13634*a^2*b^4 - 7113*a*b^5 + 1575*b^6)*sin(8*d*x + 8 *c) + 4*(880*a^3*b^3 - 2855*a^2*b^4 + 2512*a*b^5 - 630*b^6)*sin(6*d*x + 6* c) - 4*(256*a^3*b^3 - 823*a^2*b^4 + 903*a*b^5 - 315*b^6)*sin(4*d*x + 4*c) - 12*(19*a^2*b^4 - 54*a*b^5 + 30*b^6)*sin(2*d*x + 2*c))*cos(18*d*x + 18*c) + 3*(4*(160*a^3*b^3 - 57*a^2*b^4 - 195*a*b^5 + 135*b^6)*sin(14*d*x + 14*c ) + 4*(400*a^3*b^3 - 1671*a^2*b^4 + 1800*a*b^5 - 630*b^6)*sin(12*d*x + 12* c) - 2*(2560*a^4*b^2 + 3232*a^3*b^3 - 13806*a^2*b^4 + 11469*a*b^5 - 2835*b ^6)*sin(10*d*x + 10*c) - 4*(4864*a^4*b^2 - 14576*a^3*b^3 + 16221*a^2*b^4 - 8430*a*b^5 + 1890*b^6)*sin(8*d*x + 8*c) - 4*(1840*a^3*b^3 - 6825*a^2*b^4 + 6243*a*b^5 - 1575*b^6)*sin(6*d*x + 6*c) + 4*(608*a^3*b^3 - 2025*a^2*b^4 + 2292*a*b^5 - 810*b^6)*sin(4*d*x + 4*c) + 9*(56*a^2*b^4 - 183*a*b^5 + 105 *b^6)*sin(2*d*x + 2*c))*cos(16*d*x + 16*c) + 4*(4*(3200*a^4*b^2 - 7536*a^3 *b^3 + 7612*a^2*b^4 - 3915*a*b^5 + 945*b^6)*sin(12*d*x + 12*c) - 6*(3968*a ^4*b^2 - 14864*a^3*b^3 + 19013*a^2*b^4 - 10224*a*b^5 + 1890*b^6)*sin(10*d* x + 10*c) - 2*(32768*a^5*b - 117888*a^4*b^2 + 172048*a^3*b^3 - 127323*a...
Leaf count of result is larger than twice the leaf count of optimal. 2203 vs. \(2 (303) = 606\).
Time = 1.26 (sec) , antiderivative size = 2203, normalized size of antiderivative = 6.21 \[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")
Output:
1/64*(3*((78*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - 267*sqr t(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 241*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 53*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^5 )*(a^5 - 2*a^4*b + a^3*b^2)^2*abs(-a + b) + 2*(9*sqrt(a^2 - a*b + sqrt(a*b )*(a - b))*a^10*b - 51*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^9*b^2 + 108*s qrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^8*b^3 - 106*sqrt(a^2 - a*b + sqrt(a*b )*(a - b))*a^7*b^4 + 45*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^5 - 3*sq rt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^6 - 2*sqrt(a^2 - a*b + sqrt(a*b)*( a - b))*a^4*b^7)*abs(a^5 - 2*a^4*b + a^3*b^2)*abs(-a + b) - (60*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^15 - 441*sqrt(a^2 - a*b + sqrt(a*b)* (a - b))*sqrt(a*b)*a^14*b + 1339*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt( a*b)*a^13*b^2 - 2185*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^ 3 + 2059*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b^4 - 1091*sqr t(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 265*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 - 11*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a ^7*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/s qrt((a^6 - 2*a^5*b + a^4*b^2 + sqrt((a^6 - 2*a^5*b + a^4*b^2)^2 - (a^6 - 2 *a^5*b + a^4*b^2)*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)))/(a^6 - 3*a^5*...
Time = 41.63 (sec) , antiderivative size = 7364, normalized size of antiderivative = 20.74 \[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/(sin(c + d*x)^2*(a - b*sin(c + d*x)^4)^3),x)
Output:
(atan((((-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^ 11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2 *b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3 )^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2315255808*a^15*b^12 - 201326592*a^14*b^13 - 1207959 5520*a^16*b^11 + 37748736000*a^17*b^10 - 78517370880*a^18*b^9 + 1141521776 64*a^19*b^8 - 118380036096*a^20*b^7 + 87577067520*a^21*b^6 - 45298483200*a ^22*b^5 + 15602810880*a^23*b^4 - 3221225472*a^24*b^3 + 301989888*a^25*b^2 + tan(c + d*x)*(-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2 085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a ^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15 *b^3 - 10*a^16*b^2)))^(1/2)*(2147483648*a^29*b + 2147483648*a^17*b^13 - 25 769803776*a^18*b^12 + 141733920768*a^19*b^11 - 472446402560*a^20*b^10 + 10 63004405760*a^21*b^9 - 1700807049216*a^22*b^8 + 1984274890752*a^23*b^7 - 1 700807049216*a^24*b^6 + 1063004405760*a^25*b^5 - 472446402560*a^26*b^4 + 1 41733920768*a^27*b^3 - 25769803776*a^28*b^2)) + tan(c + d*x)*(3024617472*a ^11*b^13 - 265420800*a^10*b^14 - 15574892544*a^12*b^12 + 47520940032*a^13* b^11 - 94402510848*a^14*b^10 + 125505110016*a^15*b^9 - 108421447680*a^16*b ^8 + 51536461824*a^17*b^7 + 484835328*a^18*b^6 - 18454413312*a^19*b^5 +...
\[ \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x)
Output:
( - 47488*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**17*a**8*b**2 + 22 11224*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**17*a**7*b**3 - 772166 17*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**17*a**6*b**4 - 132806313 28*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**17*a**5*b**5 - 139842104 768*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**17*a**4*b**6 + 17415370 7520*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**17*a**3*b**7 + 5504075 3664*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**17*a**2*b**8 - 379904* cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**15*a**8*b**2 + 17689792*cos (c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**15*a**7*b**3 - 617732936*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**15*a**6*b**4 - 106245050624*cos( c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**15*a**5*b**5 - 1118736838144*co s(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**15*a**4*b**6 + 1393229660160* cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**15*a**3*b**7 + 440326029312 *cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**15*a**2*b**8 - 1329664*cos (c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**13*a**8*b**2 + 63433888*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**13*a**7*b**3 - 2232824444*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**13*a**6*b**4 - 369386745440*cos(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**13*a**5*b**5 - 3490598731008*cos( c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**13*a**4*b**6 + 9351251163136*co s(c + d*x)*sin(c + d*x)**8*tan((c + d*x)/2)**13*a**3*b**7 - 40317775380...