Integrand size = 16, antiderivative size = 77 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{a^{3/2} d}+\frac {b \cot (c+d x)}{a (a+b) d \sqrt {a+b+b \cot ^2(c+d x)}} \]
-arctan(cot(d*x+c)*a^(1/2)/(a+b+b*cot(d*x+c)^2)^(1/2))/a^(3/2)/d+b*cot(d*x +c)/a/(a+b)/d/(a+b+b*cot(d*x+c)^2)^(1/2)
Time = 0.54 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx=\frac {\csc ^2(c+d x) \left (-\frac {2 \sqrt {a} b (-a-2 b+a \cos (2 (c+d x))) \cot (c+d x)}{a+b}+\sqrt {2} (-a-2 b+a \cos (2 (c+d x)))^{3/2} \csc (c+d x) \log \left (\sqrt {2} \sqrt {a} \cos (c+d x)+\sqrt {-a-2 b+a \cos (2 (c+d x))}\right )\right )}{4 a^{3/2} d \left (a+b \csc ^2(c+d x)\right )^{3/2}} \]
(Csc[c + d*x]^2*((-2*Sqrt[a]*b*(-a - 2*b + a*Cos[2*(c + d*x)])*Cot[c + d*x ])/(a + b) + Sqrt[2]*(-a - 2*b + a*Cos[2*(c + d*x)])^(3/2)*Csc[c + d*x]*Lo g[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]]))/(4 *a^(3/2)*d*(a + b*Csc[c + d*x]^2)^(3/2))
Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3042, 4616, 296, 291, 216}
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.
| \(\displaystyle \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sec \left (c+d x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4616 |
| \(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )^{3/2}}d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)}{a}-\frac {b \cot (c+d x)}{a (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{\frac {a \cot ^2(c+d x)}{b \cot ^2(c+d x)+a+b}+1}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a+b}}}{a}-\frac {b \cot (c+d x)}{a (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )}{a^{3/2}}-\frac {b \cot (c+d x)}{a (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}}{d}\) |
-((ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/a^(3/2) - (b*Cot[c + d*x])/(a*(a + b)*Sqrt[a + b + b*Cot[c + d*x]^2]))/d)
3.1.13.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b + b*ff^2*x^2)^p /(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1179\) vs. \(2(69)=138\).
Time = 7.15 (sec) , antiderivative size = 1180, normalized size of antiderivative = 15.32
-1/2/d*csc(d*x+c)^3*(sin(d*x+c)^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2) ^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a) ^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d *x+c)*a)*a^2*cos(d*x+c)-cos(d*x+c)^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1) ^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*( -a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*co s(d*x+c)*a)*a*b-sin(d*x+c)^2*cos(d*x+c)*(-a)^(1/2)*a*b+sin(d*x+c)^2*(-(a*c os(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos( d*x+c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a- b)/(cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)*a^2-a*b*cos(d*x+c)^2*(-(a*cos(d *x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+ c)+1)^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/( cos(d*x+c)+1)^2)^(1/2)-4*cos(d*x+c)*a)+2*cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b) /(cos(d*x+c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/ 2)*cos(d*x+c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1 )^2)^(1/2)-4*cos(d*x+c)*a)*a*b+cos(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+ c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x +c)*(-a)^(1/2)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2) -4*cos(d*x+c)*a)*b^2-cos(d*x+c)*(-a)^(1/2)*b^2+2*a*b*(-(a*cos(d*x+c)^2-a-b )/(cos(d*x+c)+1)^2)^(1/2)*ln(4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)...
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (69) = 138\).
Time = 0.38 (sec) , antiderivative size = 652, normalized size of antiderivative = 8.47 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx=\left [-\frac {8 \, a b \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left ({\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sqrt {-a} \log \left (128 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \, {\left (a^{4} + a^{3} b\right )} \cos \left (d x + c\right )^{6} + 160 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (16 \, a^{3} \cos \left (d x + c\right )^{7} - 24 \, {\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )\right )}{8 \, {\left ({\left (a^{4} + a^{3} b\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d\right )}}, -\frac {4 \, a b \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left ({\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sqrt {a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (d x + c\right )^{4} - 8 \, {\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )}{4 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{5} - 3 \, {\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )\right )}}\right )}{4 \, {\left ({\left (a^{4} + a^{3} b\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d\right )}}\right ] \]
[-1/8*(8*a*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c)*sin(d*x + c) + ((a^2 + a*b)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*sqrt (-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*( a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a* b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 + 8*(16* a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqr t(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))) /((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d), -1/4*(4*a *b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c)*sin( d*x + c) - ((a^2 + a*b)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*sqrt(a)*arctan (1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^ 2*b + a*b^2)*cos(d*x + c))))/((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + 2*a^ 3*b + a^2*b^2)*d)]
\[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 2050 vs. \(2 (69) = 138\).
Time = 0.63 (sec) , antiderivative size = 2050, normalized size of antiderivative = 26.62 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx=\text {Too large to display} \]
-1/2*(2*a*b*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2 *c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))*sin(2*d*x + 2 *c) + 2*(a^2 + a*b)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2 *d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^3 - 2 *(a*b*cos(2*d*x + 2*c) - (a^2 + a*b)*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2 + a^2 + 2*a*b)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2 *b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a)) - (a^2*cos(4*d*x + 4*c)^2 + a^2*sin(4*d*x + 4*c)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(a^2 + 2*a*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2 *c) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*d*x + 2*c)^2 + a^2 + 2*(a^2 - 2*(a^2 + 2*a*b)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(a^2 + 2*a*b)*cos(2*d*x + 2 *c))^(1/4)*(((a + b)*cos(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin( 2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2 + (a + b)*sin(1/2*arctan2(a*sin(4*d*x + 4*c) - 2*(a + 2*b)*sin(2*d*x + 2*c), a*cos(4*d*x + 4*c) - 2*(a + 2*b)*cos(2*d*x + 2*c) + a))^2)*arctan2(2*a*si n(2*d*x + 2*c) + 2*(a^2*cos(4*d*x + 4*c)^2 + a^2*sin(4*d*x + 4*c)^2 + 4*(a ^2 + 4*a*b + 4*b^2)*cos(2*d*x + 2*c)^2 - 4*(a^2 + 2*a*b)*sin(4*d*x + 4*c)* sin(2*d*x + 2*c) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*d*x + 2*c)^2 + a^2 + 2*(a ^2 - 2*(a^2 + 2*a*b)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(a^2 + 2*a*...
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (69) = 138\).
Time = 0.43 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx=-\frac {\frac {\frac {a^{2} b \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{4} + a^{3} b} - \frac {a^{2} b \mathrm {sgn}\left (\sin \left (d x + c\right )\right )}{a^{4} + a^{3} b}}{\sqrt {b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b}} - \frac {2 \, \arctan \left (-\frac {\sqrt {b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b} + \sqrt {b}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )}}{d} \]
-((a^2*b*sgn(sin(d*x + c))*tan(1/2*d*x + 1/2*c)^2/(a^4 + a^3*b) - a^2*b*sg n(sin(d*x + c))/(a^4 + a^3*b))/sqrt(b*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2 *d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c)^2 + b) - 2*arctan(-1/2*(sqrt(b) *tan(1/2*d*x + 1/2*c)^2 - sqrt(b*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c)^2 + b) + sqrt(b))/sqrt(a))/(a^(3/2)* sgn(sin(d*x + c))))/d
Timed out. \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\sin \left (c+d\,x\right )}^2}\right )}^{3/2}} \,d x \]