Integrand size = 23, antiderivative size = 153 \[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=-\frac {(a-4 b) \text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {b^{3/2} (5 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 a^3 (a+b)^{3/2} d}-\frac {b (a+2 b) \cos (c+d x)}{2 a^2 (a+b) d \left (a+b-b \cos ^2(c+d x)\right )}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \left (a+b-b \cos ^2(c+d x)\right )} \] Output:
-1/2*(a-4*b)*arctanh(cos(d*x+c))/a^3/d-1/2*b^(3/2)*(5*a+4*b)*arctanh(b^(1/ 2)*cos(d*x+c)/(a+b)^(1/2))/a^3/(a+b)^(3/2)/d-1/2*b*(a+2*b)*cos(d*x+c)/a^2/ (a+b)/d/(a+b-b*cos(d*x+c)^2)-1/2*cot(d*x+c)*csc(d*x+c)/a/d/(a+b-b*cos(d*x+ c)^2)
Result contains complex when optimal does not.
Time = 3.02 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.55 \[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {(-2 a-b+b \cos (2 (c+d x))) \csc ^3(c+d x) \left (\frac {8 a b^2 \cot (c+d x)}{a+b}+\frac {4 b^{3/2} (5 a+4 b) \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right ) (2 a+b-b \cos (2 (c+d x))) \csc (c+d x)}{(-a-b)^{3/2}}+\frac {4 b^{3/2} (5 a+4 b) \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right ) (2 a+b-b \cos (2 (c+d x))) \csc (c+d x)}{(-a-b)^{3/2}}+a (2 a+b-b \cos (2 (c+d x))) \csc ^2\left (\frac {1}{2} (c+d x)\right ) \csc (c+d x)+4 (a-4 b) (2 a+b-b \cos (2 (c+d x))) \csc (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 (a-4 b) (2 a+b-b \cos (2 (c+d x))) \csc (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-a (2 a+b-b \cos (2 (c+d x))) \csc (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )}{32 a^3 d \left (b+a \csc ^2(c+d x)\right )^2} \] Input:
Integrate[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2,x]
Output:
((-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^3*((8*a*b^2*Cot[c + d*x])/(a + b) + (4*b^(3/2)*(5*a + 4*b)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2 ])/Sqrt[-a - b]]*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x])/(-a - b)^(3/ 2) + (4*b^(3/2)*(5*a + 4*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/ Sqrt[-a - b]]*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x])/(-a - b)^(3/2) + a*(2*a + b - b*Cos[2*(c + d*x)])*Csc[(c + d*x)/2]^2*Csc[c + d*x] + 4*(a - 4*b)*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]*Log[Cos[(c + d*x)/2]] - 4*(a - 4*b)*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]*Log[Sin[(c + d*x) /2]] - a*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]*Sec[(c + d*x)/2]^2))/ (32*a^3*d*(b + a*Csc[c + d*x]^2)^2)
Time = 0.39 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3665, 316, 402, 27, 397, 219, 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.
| \(\displaystyle \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^3 \left (a+b \sin (c+d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right )^2 \left (-b \cos ^2(c+d x)+a+b\right )^2}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle -\frac {\frac {\int \frac {-3 b \cos ^2(c+d x)+a-b}{\left (1-\cos ^2(c+d x)\right ) \left (-b \cos ^2(c+d x)+a+b\right )^2}d\cos (c+d x)}{2 a}+\frac {\cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right ) \left (a-b \cos ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\frac {b (a+2 b) \cos (c+d x)}{a (a+b) \left (a-b \cos ^2(c+d x)+b\right )}-\frac {\int -\frac {2 \left (a^2-2 b a-2 b^2-b (a+2 b) \cos ^2(c+d x)\right )}{\left (1-\cos ^2(c+d x)\right ) \left (-b \cos ^2(c+d x)+a+b\right )}d\cos (c+d x)}{2 a (a+b)}}{2 a}+\frac {\cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right ) \left (a-b \cos ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {a^2-2 b a-2 b^2-b (a+2 b) \cos ^2(c+d x)}{\left (1-\cos ^2(c+d x)\right ) \left (-b \cos ^2(c+d x)+a+b\right )}d\cos (c+d x)}{a (a+b)}+\frac {b (a+2 b) \cos (c+d x)}{a (a+b) \left (a-b \cos ^2(c+d x)+b\right )}}{2 a}+\frac {\cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right ) \left (a-b \cos ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\frac {\frac {b^2 (5 a+4 b) \int \frac {1}{-b \cos ^2(c+d x)+a+b}d\cos (c+d x)}{a}+\frac {(a-4 b) (a+b) \int \frac {1}{1-\cos ^2(c+d x)}d\cos (c+d x)}{a}}{a (a+b)}+\frac {b (a+2 b) \cos (c+d x)}{a (a+b) \left (a-b \cos ^2(c+d x)+b\right )}}{2 a}+\frac {\cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right ) \left (a-b \cos ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {\frac {b^2 (5 a+4 b) \int \frac {1}{-b \cos ^2(c+d x)+a+b}d\cos (c+d x)}{a}+\frac {(a-4 b) (a+b) \text {arctanh}(\cos (c+d x))}{a}}{a (a+b)}+\frac {b (a+2 b) \cos (c+d x)}{a (a+b) \left (a-b \cos ^2(c+d x)+b\right )}}{2 a}+\frac {\cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right ) \left (a-b \cos ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {\frac {b^{3/2} (5 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}+\frac {(a-4 b) (a+b) \text {arctanh}(\cos (c+d x))}{a}}{a (a+b)}+\frac {b (a+2 b) \cos (c+d x)}{a (a+b) \left (a-b \cos ^2(c+d x)+b\right )}}{2 a}+\frac {\cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right ) \left (a-b \cos ^2(c+d x)+b\right )}}{d}\) |
Input:
Int[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2,x]
Output:
-((Cos[c + d*x]/(2*a*(1 - Cos[c + d*x]^2)*(a + b - b*Cos[c + d*x]^2)) + (( ((a - 4*b)*(a + b)*ArcTanh[Cos[c + d*x]])/a + (b^(3/2)*(5*a + 4*b)*ArcTanh [(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a*(a + b)) + (b*(a + 2*b)*Cos[c + d*x])/(a*(a + b)*(a + b - b*Cos[c + d*x]^2)))/(2*a))/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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (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 - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 1.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{2} \left (\frac {a \cos \left (d x +c \right )}{2 \left (a +b \right ) \left (a +b -b \cos \left (d x +c \right )^{2}\right )}+\frac {\left (5 a +4 b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (a -4 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{4 a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-a +4 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{4 a^{3}}}{d}\) | \(152\) |
| default | \(\frac {-\frac {b^{2} \left (\frac {a \cos \left (d x +c \right )}{2 \left (a +b \right ) \left (a +b -b \cos \left (d x +c \right )^{2}\right )}+\frac {\left (5 a +4 b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (a -4 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{4 a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-a +4 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{4 a^{3}}}{d}\) | \(152\) |
| risch | \(\frac {a b \,{\mathrm e}^{7 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-4 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-5 a b \,{\mathrm e}^{5 i \left (d x +c \right )}-2 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-4 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-5 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+a b \,{\mathrm e}^{i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (a +b \right ) \left (b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 a \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{2} d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{2} d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{a^{3} d}-\frac {5 i \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}\, {\mathrm e}^{i \left (d x +c \right )}}{b}+1\right ) b}{4 \left (a +b \right )^{2} d \,a^{2}}-\frac {i \sqrt {-\left (a +b \right ) b}\, b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}\, {\mathrm e}^{i \left (d x +c \right )}}{b}+1\right )}{\left (a +b \right )^{2} d \,a^{3}}+\frac {5 i \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}\, {\mathrm e}^{i \left (d x +c \right )}}{b}+1\right ) b}{4 \left (a +b \right )^{2} d \,a^{2}}+\frac {i \sqrt {-\left (a +b \right ) b}\, b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}\, {\mathrm e}^{i \left (d x +c \right )}}{b}+1\right )}{\left (a +b \right )^{2} d \,a^{3}}\) | \(519\) |
Input:
int(csc(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/a^3*b^2*(1/2*a/(a+b)*cos(d*x+c)/(a+b-b*cos(d*x+c)^2)+1/2*(5*a+4*b) /(a+b)/((a+b)*b)^(1/2)*arctanh(b*cos(d*x+c)/((a+b)*b)^(1/2)))+1/4/a^2/(cos (d*x+c)-1)+1/4*(a-4*b)/a^3*ln(cos(d*x+c)-1)+1/4/a^2/(cos(d*x+c)+1)+1/4/a^3 *(-a+4*b)*ln(cos(d*x+c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (143) = 286\).
Time = 0.16 (sec) , antiderivative size = 838, normalized size of antiderivative = 5.48 \[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x, algorithm="fricas")
Output:
[1/4*(2*(a^2*b + 2*a*b^2)*cos(d*x + c)^3 + ((5*a*b^2 + 4*b^3)*cos(d*x + c) ^4 + 5*a^2*b + 9*a*b^2 + 4*b^3 - (5*a^2*b + 14*a*b^2 + 8*b^3)*cos(d*x + c) ^2)*sqrt(b/(a + b))*log(-(b*cos(d*x + c)^2 - 2*(a + b)*sqrt(b/(a + b))*cos (d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - 2*(a^3 + 2*a^2*b + 2*a*b^ 2)*cos(d*x + c) - ((a^2*b - 3*a*b^2 - 4*b^3)*cos(d*x + c)^4 + a^3 - 2*a^2* b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*cos(d*x + c)^2)*log (1/2*cos(d*x + c) + 1/2) + ((a^2*b - 3*a*b^2 - 4*b^3)*cos(d*x + c)^4 + a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^4*b + a^3*b^2)*d*cos(d*x + c)^4 - (a^5 + 3*a^4*b + 2*a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + 2*a^4*b + a^3*b^2)*d ), 1/4*(2*(a^2*b + 2*a*b^2)*cos(d*x + c)^3 + 2*((5*a*b^2 + 4*b^3)*cos(d*x + c)^4 + 5*a^2*b + 9*a*b^2 + 4*b^3 - (5*a^2*b + 14*a*b^2 + 8*b^3)*cos(d*x + c)^2)*sqrt(-b/(a + b))*arctan(sqrt(-b/(a + b))*cos(d*x + c)) - 2*(a^3 + 2*a^2*b + 2*a*b^2)*cos(d*x + c) - ((a^2*b - 3*a*b^2 - 4*b^3)*cos(d*x + c)^ 4 + a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*cos (d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + ((a^2*b - 3*a*b^2 - 4*b^3)*cos( d*x + c)^4 + a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8 *b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^4*b + a^3*b^2)*d*c os(d*x + c)^4 - (a^5 + 3*a^4*b + 2*a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + 2*a^ 4*b + a^3*b^2)*d)]
\[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\left (a + b \sin ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(csc(d*x+c)**3/(a+b*sin(d*x+c)**2)**2,x)
Output:
Integral(csc(c + d*x)**3/(a + b*sin(c + d*x)**2)**2, x)
Time = 0.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.46 \[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {{\left (5 \, a b^{2} + 4 \, b^{3}\right )} \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} b}} + \frac {2 \, {\left ({\left (a b + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, a b + 2 \, b^{2}\right )} \cos \left (d x + c\right )\right )}}{{\left (a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - {\left (a^{4} + 3 \, a^{3} b + 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}} - \frac {{\left (a - 4 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {{\left (a - 4 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{4 \, d} \] Input:
integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/4*((5*a*b^2 + 4*b^3)*log((b*cos(d*x + c) - sqrt((a + b)*b))/(b*cos(d*x + c) + sqrt((a + b)*b)))/((a^4 + a^3*b)*sqrt((a + b)*b)) + 2*((a*b + 2*b^2) *cos(d*x + c)^3 - (a^2 + 2*a*b + 2*b^2)*cos(d*x + c))/((a^3*b + a^2*b^2)*c os(d*x + c)^4 + a^4 + 2*a^3*b + a^2*b^2 - (a^4 + 3*a^3*b + 2*a^2*b^2)*cos( d*x + c)^2) - (a - 4*b)*log(cos(d*x + c) + 1)/a^3 + (a - 4*b)*log(cos(d*x + c) - 1)/a^3)/d
Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (143) = 286\).
Time = 0.49 (sec) , antiderivative size = 512, normalized size of antiderivative = 3.35 \[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/24*(12*(5*a*b^2 + 4*b^3)*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^ 2)*cos(d*x + c) + sqrt(-a*b - b^2)))/((a^4 + a^3*b)*sqrt(-a*b - b^2)) + (3 *a^3 + 3*a^2*b - 8*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 12*a^2*b*(c os(d*x + c) - 1)/(cos(d*x + c) + 1) - 28*a*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 7*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - a^2*b*(cos( d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 16*a*b^2*(cos(d*x + c) - 1)^2/(cos( d*x + c) + 1)^2 + 16*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 2*a^3 *(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 6*a^2*b*(cos(d*x + c) - 1)^3/ (cos(d*x + c) + 1)^3 + 8*a*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)/ ((a^4 + a^3*b)*(a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*a*(cos(d*x + c ) - 1)^2/(cos(d*x + c) + 1)^2 - 4*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1 )^2 + a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)) + 6*(a - 4*b)*log(abs( -cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^3 - 3*(cos(d*x + c) - 1)/(a^2* (cos(d*x + c) + 1)))/d
Time = 38.97 (sec) , antiderivative size = 2338, normalized size of antiderivative = 15.28 \[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(sin(c + d*x)^3*(a + b*sin(c + d*x)^2)^2),x)
Output:
- ((cos(c + d*x)*(2*a*b + a^2 + 2*b^2))/(2*a^2*(a + b)) - (b*cos(c + d*x)^ 3*(a + 2*b))/(2*a^2*(a + b)))/(d*(a + b + b*cos(c + d*x)^4 - cos(c + d*x)^ 2*(a + 2*b))) - (atan((((a - 4*b)*((cos(c + d*x)*(64*a*b^6 + 32*b^7 + 26*a ^2*b^5 - 6*a^3*b^4 + a^4*b^3))/(2*(2*a^5*b + a^6 + a^4*b^2)) + (((4*a^6*b^ 5 + 8*a^7*b^4 + 2*a^8*b^3 - 2*a^9*b^2)/(2*a^7*b + a^8 + a^6*b^2) - (cos(c + d*x)*(a - 4*b)*(32*a^6*b^5 + 80*a^7*b^4 + 64*a^8*b^3 + 16*a^9*b^2))/(8*a ^3*(2*a^5*b + a^6 + a^4*b^2)))*(a - 4*b))/(4*a^3))*1i)/(4*a^3) + ((a - 4*b )*((cos(c + d*x)*(64*a*b^6 + 32*b^7 + 26*a^2*b^5 - 6*a^3*b^4 + a^4*b^3))/( 2*(2*a^5*b + a^6 + a^4*b^2)) - (((4*a^6*b^5 + 8*a^7*b^4 + 2*a^8*b^3 - 2*a^ 9*b^2)/(2*a^7*b + a^8 + a^6*b^2) + (cos(c + d*x)*(a - 4*b)*(32*a^6*b^5 + 8 0*a^7*b^4 + 64*a^8*b^3 + 16*a^9*b^2))/(8*a^3*(2*a^5*b + a^6 + a^4*b^2)))*( a - 4*b))/(4*a^3))*1i)/(4*a^3))/((12*a*b^6 + 8*b^7 + (3*a^2*b^5)/2 - (5*a^ 3*b^4)/4)/(2*a^7*b + a^8 + a^6*b^2) - ((a - 4*b)*((cos(c + d*x)*(64*a*b^6 + 32*b^7 + 26*a^2*b^5 - 6*a^3*b^4 + a^4*b^3))/(2*(2*a^5*b + a^6 + a^4*b^2) ) + (((4*a^6*b^5 + 8*a^7*b^4 + 2*a^8*b^3 - 2*a^9*b^2)/(2*a^7*b + a^8 + a^6 *b^2) - (cos(c + d*x)*(a - 4*b)*(32*a^6*b^5 + 80*a^7*b^4 + 64*a^8*b^3 + 16 *a^9*b^2))/(8*a^3*(2*a^5*b + a^6 + a^4*b^2)))*(a - 4*b))/(4*a^3)))/(4*a^3) + ((a - 4*b)*((cos(c + d*x)*(64*a*b^6 + 32*b^7 + 26*a^2*b^5 - 6*a^3*b^4 + a^4*b^3))/(2*(2*a^5*b + a^6 + a^4*b^2)) - (((4*a^6*b^5 + 8*a^7*b^4 + 2*a^ 8*b^3 - 2*a^9*b^2)/(2*a^7*b + a^8 + a^6*b^2) + (cos(c + d*x)*(a - 4*b)*...
Time = 0.25 (sec) , antiderivative size = 1071, normalized size of antiderivative = 7.00 \[ \int \frac {\csc ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(csc(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x)
Output:
( - 2*cos(c + d*x)*sin(c + d*x)**2*a**3*b - 6*cos(c + d*x)*sin(c + d*x)**2 *a**2*b**2 - 4*cos(c + d*x)*sin(c + d*x)**2*a*b**3 - 2*cos(c + d*x)*a**4 - 4*cos(c + d*x)*a**3*b - 2*cos(c + d*x)*a**2*b**2 + 5*sqrt(b)*sqrt(a + b)* log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*s in(c + d*x)**4*a*b**2 + 4*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)**4*b**3 + 5*sqrt (b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan ((c + d*x)/2))*sin(c + d*x)**2*a**2*b + 4*sqrt(b)*sqrt(a + b)*log( - sqrt( 2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)* *2*a*b**2 + 5*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b ) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)**4*a*b**2 + 4*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*s in(c + d*x)**4*b**3 + 5*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)**2*a**2*b + 4*sqrt(b) *sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)**2*a*b**2 - 5*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt (a + b) + tan((c + d*x)/2)**2*a + a + 2*b)*sin(c + d*x)**4*a*b**2 - 4*sqrt (b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + tan((c + d*x)/2)**2*a + a + 2* b)*sin(c + d*x)**4*b**3 - 5*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + tan((c + d*x)/2)**2*a + a + 2*b)*sin(c + d*x)**2*a**2*b - 4*sqrt(b)*s...