Integrand size = 16, antiderivative size = 126 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{a^{5/2} d}+\frac {b \cot (c+d x)}{3 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b (5 a+3 b) \cot (c+d x)}{3 a^2 (a+b)^2 d \sqrt {a+b+b \cot ^2(c+d x)}} \]
-arctan(cot(d*x+c)*a^(1/2)/(a+b+b*cot(d*x+c)^2)^(1/2))/a^(5/2)/d+1/3*b*cot (d*x+c)/a/(a+b)/d/(a+b+b*cot(d*x+c)^2)^(3/2)+1/3*b*(5*a+3*b)*cot(d*x+c)/a^ 2/(a+b)^2/d/(a+b+b*cot(d*x+c)^2)^(1/2)
Time = 1.97 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx=\frac {\csc ^5(c+d x) \left (\frac {4 b \cos (c+d x) (a+2 b-a \cos (2 (c+d x))) \left (3 a^2+7 a b+3 b^2-a (3 a+2 b) \cos (2 (c+d x))\right )}{3 a^2 (a+b)^2}-\frac {\sqrt {2} (-a-2 b+a \cos (2 (c+d x)))^{5/2} \log \left (\sqrt {2} \sqrt {a} \cos (c+d x)+\sqrt {-a-2 b+a \cos (2 (c+d x))}\right )}{a^{5/2}}\right )}{8 d \left (a+b \csc ^2(c+d x)\right )^{5/2}} \]
(Csc[c + d*x]^5*((4*b*Cos[c + d*x]*(a + 2*b - a*Cos[2*(c + d*x)])*(3*a^2 + 7*a*b + 3*b^2 - a*(3*a + 2*b)*Cos[2*(c + d*x)]))/(3*a^2*(a + b)^2) - (Sqr t[2]*(-a - 2*b + a*Cos[2*(c + d*x)])^(5/2)*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x ] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])/a^(5/2)))/(8*d*(a + b*Csc[c + d* x]^2)^(5/2))
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 4616, 316, 402, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sec \left (c+d x+\frac {\pi }{2}\right )^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4616 |
| \(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )^{5/2}}d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle -\frac {\frac {\int \frac {-2 b \cot ^2(c+d x)+3 a+b}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a+b\right )^{3/2}}d\cot (c+d x)}{3 a (a+b)}-\frac {b \cot (c+d x)}{3 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {3 (a+b)^2}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)}{a (a+b)}-\frac {b (5 a+3 b) \cot (c+d x)}{a (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}}{3 a (a+b)}-\frac {b \cot (c+d x)}{3 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {3 (a+b) \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)}{a}-\frac {b (5 a+3 b) \cot (c+d x)}{a (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}}{3 a (a+b)}-\frac {b \cot (c+d x)}{3 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle -\frac {\frac {\frac {3 (a+b) \int \frac {1}{\frac {a \cot ^2(c+d x)}{b \cot ^2(c+d x)+a+b}+1}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a+b}}}{a}-\frac {b (5 a+3 b) \cot (c+d x)}{a (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}}{3 a (a+b)}-\frac {b \cot (c+d x)}{3 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\frac {\frac {3 (a+b) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )}{a^{3/2}}-\frac {b (5 a+3 b) \cot (c+d x)}{a (a+b) \sqrt {a+b \cot ^2(c+d x)+b}}}{3 a (a+b)}-\frac {b \cot (c+d x)}{3 a (a+b) \left (a+b \cot ^2(c+d x)+b\right )^{3/2}}}{d}\) |
-((-1/3*(b*Cot[c + d*x])/(a*(a + b)*(a + b + b*Cot[c + d*x]^2)^(3/2)) + (( 3*(a + b)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/a ^(3/2) - (b*(5*a + 3*b)*Cot[c + d*x])/(a*(a + b)*Sqrt[a + b + b*Cot[c + d* x]^2]))/(3*a*(a + b)))/d)
3.1.14.3.1 Defintions of rubi rules used
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]))*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[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[((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[((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. \(2709\) vs. \(2(112)=224\).
Time = 7.28 (sec) , antiderivative size = 2710, normalized size of antiderivative = 21.51
-1/6/d*csc(d*x+c)*(3*cos(d*x+c)*a^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^ 2)^(1/2)*sin(d*x+c)^4*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)-6*a^3*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/ 2)*cos(d*x+c)^3*sin(d*x+c)^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+3*b^2*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+ 1)^2)^(1/2)*cos(d*x+c)^5*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^2-6*sin(d*x+c)^4*cos(d*x+c)*(-a)^(1/2)*a^3*b+ 3*a^4*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)^4*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)+4*sin( d*x+c)^2*cos(d*x+c)^3*(-a)^(1/2)*a^2*b^2-6*a^3*(-(a*cos(d*x+c)^2-a-b)/(cos (d*x+c)+1)^2)^(1/2)*cos(d*x+c)^2*sin(d*x+c)^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+3*b^2*(-(a*cos(d*x+c)^2- a-b)/(cos(d*x+c)+1)^2)^(1/2)*cos(d*x+c)^4*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+12*cos(d*x+c)*a^3*(-(a*...
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (112) = 224\).
Time = 0.69 (sec) , antiderivative size = 973, normalized size of antiderivative = 7.72 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx=\text {Too large to display} \]
[-1/24*(3*((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 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c) ^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))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d* x + c)) + 8*(2*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - 3*(2*a^3*b + 3*a^2*b ^2 + a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cos(d*x + c)^4 - 2*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 4*a^6*b + 6*a^5* b^2 + 4*a^4*b^3 + a^3*b^4)*d), 1/12*(3*((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 - 2*(a^4 + 3*a^3*b + 3* a^2*b^2 + a*b^3)*cos(d*x + c)^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))) - 4*(2*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - 3*(2*a^3*b + 3*a^2*b^2 + a* b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1)...
\[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (112) = 224\).
Time = 0.58 (sec) , antiderivative size = 407, normalized size of antiderivative = 3.23 \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx=-\frac {\frac {{\left ({\left (\frac {{\left (5 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 3 \, a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{12} + 2 \, a^{11} b + a^{10} b^{2}} + \frac {3 \, {\left (8 \, a^{10} b \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 7 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )\right )}}{a^{12} + 2 \, a^{11} b + a^{10} b^{2}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {3 \, {\left (8 \, a^{10} b \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 7 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )\right )}}{a^{12} + 2 \, a^{11} b + a^{10} b^{2}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {5 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 3 \, a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )}{a^{12} + 2 \, a^{11} b + a^{10} b^{2}}}{{\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b\right )}^{\frac {3}{2}}} - \frac {6 \, \arctan \left (-\frac {\sqrt {b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b} + \sqrt {b}}{2 \, \sqrt {a}}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )}}{3 \, d} \]
-1/3*(((((5*a^9*b^2*sgn(sin(d*x + c)) + 3*a^8*b^3*sgn(sin(d*x + c)))*tan(1 /2*d*x + 1/2*c)^2/(a^12 + 2*a^11*b + a^10*b^2) + 3*(8*a^10*b*sgn(sin(d*x + c)) + 7*a^9*b^2*sgn(sin(d*x + c)) + a^8*b^3*sgn(sin(d*x + c)))/(a^12 + 2* a^11*b + a^10*b^2))*tan(1/2*d*x + 1/2*c)^2 - 3*(8*a^10*b*sgn(sin(d*x + c)) + 7*a^9*b^2*sgn(sin(d*x + c)) + a^8*b^3*sgn(sin(d*x + c)))/(a^12 + 2*a^11 *b + a^10*b^2))*tan(1/2*d*x + 1/2*c)^2 - (5*a^9*b^2*sgn(sin(d*x + c)) + 3* a^8*b^3*sgn(sin(d*x + c)))/(a^12 + 2*a^11*b + a^10*b^2))/(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)^(3 /2) - 6*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^(5/2)*sgn(sin(d*x + c))))/d
Timed out. \[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\sin \left (c+d\,x\right )}^2}\right )}^{5/2}} \,d x \]