Integrand size = 16, antiderivative size = 119 \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} (3 a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b+b \cot ^2(c+d x)}}{2 d} \]
-a^(3/2)*arctan(cot(d*x+c)*a^(1/2)/(a+b+b*cot(d*x+c)^2)^(1/2))/d-1/2*(3*a+ b)*arctanh(cot(d*x+c)*b^(1/2)/(a+b+b*cot(d*x+c)^2)^(1/2))*b^(1/2)/d-1/2*b* cot(d*x+c)*(a+b+b*cot(d*x+c)^2)^(1/2)/d
Time = 1.21 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.75 \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\frac {b \left (a+b \csc ^2(c+d x)\right )^{3/2} \left (\sqrt {2} b (3 a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-b} \cos (c+d x)}{\sqrt {-a-2 b+a \cos (2 (c+d x))}}\right )+\sqrt {-b} \left (-b \sqrt {-a-2 b+a \cos (2 (c+d x))} \cot (c+d x) \csc (c+d x)+2 \sqrt {2} a^{3/2} \log \left (\sqrt {2} \sqrt {a} \cos (c+d x)+\sqrt {-a-2 b+a \cos (2 (c+d x))}\right )\right )\right ) \sin ^3(c+d x)}{(-b)^{3/2} d (-a-2 b+a \cos (2 (c+d x)))^{3/2}} \]
(b*(a + b*Csc[c + d*x]^2)^(3/2)*(Sqrt[2]*b*(3*a + b)*ArcTanh[(Sqrt[2]*Sqrt [-b]*Cos[c + d*x])/Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]] + Sqrt[-b]*(-(b*Sq rt[-a - 2*b + a*Cos[2*(c + d*x)]]*Cot[c + d*x]*Csc[c + d*x]) + 2*Sqrt[2]*a ^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x )]]]))*Sin[c + d*x]^3)/((-b)^(3/2)*d*(-a - 2*b + a*Cos[2*(c + d*x)])^(3/2) )
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4616, 318, 398, 224, 219, 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.
| \(\displaystyle \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sec \left (c+d x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4616 |
| \(\displaystyle -\frac {\int \frac {\left (b \cot ^2(c+d x)+a+b\right )^{3/2}}{\cot ^2(c+d x)+1}d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {b (3 a+b) \cot ^2(c+d x)+(a+b) (2 a+b)}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)+b}}{d}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle -\frac {\frac {1}{2} \left (2 a^2 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)+b (3 a+b) \int \frac {1}{\sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)+b}}{d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {\frac {1}{2} \left (2 a^2 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)+b (3 a+b) \int \frac {1}{1-\frac {b \cot ^2(c+d x)}{b \cot ^2(c+d x)+a+b}}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a+b}}\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)+b}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {1}{2} \left (2 a^2 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a+b}}d\cot (c+d x)+\sqrt {b} (3 a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)+b}}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle -\frac {\frac {1}{2} \left (2 a^2 \int \frac {1}{\frac {a \cot ^2(c+d x)}{b \cot ^2(c+d x)+a+b}+1}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a+b}}+\sqrt {b} (3 a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)+b}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\frac {1}{2} \left (2 a^{3/2} \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )+\sqrt {b} (3 a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)+b}}{d}\) |
-(((2*a^(3/2)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2] ] + Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/2 + (b*Cot[c + d*x]*Sqrt[a + b + b*Cot[c + d*x]^2])/2)/d)
3.1.10.3.1 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[((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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 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[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
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. \(1132\) vs. \(2(101)=202\).
Time = 2.51 (sec) , antiderivative size = 1133, normalized size of antiderivative = 9.52
1/16/d*csc(d*x+c)*(1/(1-cos(d*x+c))^2*(csc(d*x+c)^2*b*(1-cos(d*x+c))^4+4*a *(1-cos(d*x+c))^2+2*b*(1-cos(d*x+c))^2+b*sin(d*x+c)^2))^(3/2)*(1-cos(d*x+c ))*(2*csc(d*x+c)^2*b^(3/2)*ln((b*(1-cos(d*x+c))^2*csc(d*x+c)^2+b^(1/2)*(b* (1-cos(d*x+c))^4*csc(d*x+c)^4+4*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+2*b*(1-cos (d*x+c))^2*csc(d*x+c)^2+b)^(1/2)+2*a+b)/b^(1/2))*(1-cos(d*x+c))^2*(-a)^(1/ 2)-2*csc(d*x+c)^2*b^(3/2)*ln(2/(1-cos(d*x+c))^2*(2*a*(1-cos(d*x+c))^2+b*(1 -cos(d*x+c))^2+sin(d*x+c)^2*b^(1/2)*(b*(1-cos(d*x+c))^4*csc(d*x+c)^4+4*a*( 1-cos(d*x+c))^2*csc(d*x+c)^2+2*b*(1-cos(d*x+c))^2*csc(d*x+c)^2+b)^(1/2)+b* sin(d*x+c)^2))*(1-cos(d*x+c))^2*(-a)^(1/2)+6*csc(d*x+c)^2*a*b^(1/2)*ln((b* (1-cos(d*x+c))^2*csc(d*x+c)^2+b^(1/2)*(b*(1-cos(d*x+c))^4*csc(d*x+c)^4+4*a *(1-cos(d*x+c))^2*csc(d*x+c)^2+2*b*(1-cos(d*x+c))^2*csc(d*x+c)^2+b)^(1/2)+ 2*a+b)/b^(1/2))*(1-cos(d*x+c))^2*(-a)^(1/2)-6*csc(d*x+c)^2*b^(1/2)*ln(2/(1 -cos(d*x+c))^2*(2*a*(1-cos(d*x+c))^2+b*(1-cos(d*x+c))^2+sin(d*x+c)^2*b^(1/ 2)*(b*(1-cos(d*x+c))^4*csc(d*x+c)^4+4*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+2*b* (1-cos(d*x+c))^2*csc(d*x+c)^2+b)^(1/2)+b*sin(d*x+c)^2))*a*(1-cos(d*x+c))^2 *(-a)^(1/2)+csc(d*x+c)^2*b*(b*(1-cos(d*x+c))^4*csc(d*x+c)^4+4*a*(1-cos(d*x +c))^2*csc(d*x+c)^2+2*b*(1-cos(d*x+c))^2*csc(d*x+c)^2+b)^(1/2)*(1-cos(d*x+ c))^2*(-a)^(1/2)-8*csc(d*x+c)^2*a^2*ln(4*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2+ (-a)^(1/2)*(b*(1-cos(d*x+c))^4*csc(d*x+c)^4+4*a*(1-cos(d*x+c))^2*csc(d*x+c )^2+2*b*(1-cos(d*x+c))^2*csc(d*x+c)^2+b)^(1/2)-a)/((1-cos(d*x+c))^2*csc...
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (101) = 202\).
Time = 0.55 (sec) , antiderivative size = 1607, normalized size of antiderivative = 13.50 \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\text {Too large to display} \]
[1/8*(sqrt(-a)*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))*si n(d*x + c))*sin(d*x + c) + (3*a + b)*sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*co s(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(c os(d*x + c)^2 - 1))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2* cos(d*x + c)^2 + 1))*sin(d*x + c) - 4*b*sqrt((a*cos(d*x + c)^2 - a - b)/(c os(d*x + c)^2 - 1))*cos(d*x + c))/(d*sin(d*x + c)), -1/8*(2*(3*a + b)*sqrt (-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c)))*sin(d*x + c) - sqrt(-a)*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)*c os(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 +...
\[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\int \left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\int { {\left (b \csc \left (d x + c\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \]
Exception generated. \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\int {\left (a+\frac {b}{{\sin \left (c+d\,x\right )}^2}\right )}^{3/2} \,d x \]