\(\int (a+b \csc ^2(c+d x))^{3/2} \, dx\) [10]

Optimal result

Integrand size = 16, antiderivative size = 119 \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} (3 a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b+b \cot ^2(c+d x)}}{2 d} \] Output:

-a^(3/2)*arctan(a^(1/2)*cot(d*x+c)/(a+b+b*cot(d*x+c)^2)^(1/2))/d-1/2*b^(1/ 
2)*(3*a+b)*arctanh(b^(1/2)*cot(d*x+c)/(a+b+b*cot(d*x+c)^2)^(1/2))/d-1/2*b* 
cot(d*x+c)*(a+b+b*cot(d*x+c)^2)^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.75 \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\frac {b \left (a+b \csc ^2(c+d x)\right )^{3/2} \left (\sqrt {2} b (3 a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-b} \cos (c+d x)}{\sqrt {-a-2 b+a \cos (2 (c+d x))}}\right )+\sqrt {-b} \left (-b \sqrt {-a-2 b+a \cos (2 (c+d x))} \cot (c+d x) \csc (c+d x)+2 \sqrt {2} a^{3/2} \log \left (\sqrt {2} \sqrt {a} \cos (c+d x)+\sqrt {-a-2 b+a \cos (2 (c+d x))}\right )\right )\right ) \sin ^3(c+d x)}{(-b)^{3/2} d (-a-2 b+a \cos (2 (c+d x)))^{3/2}} \] Input:

Integrate[(a + b*Csc[c + d*x]^2)^(3/2),x]
 

Output:

(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) 
)
 

Rubi [A] (verified)

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

\(\displaystyle -\frac {\frac {1}{2} \left (2 a^{3/2} \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )+\sqrt {b} (3 a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)+b}}{d}\)

Input:

Int[(a + b*Csc[c + d*x]^2)^(3/2),x]
 

Output:

-(((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)
 

Defintions of rubi rules used

rule 216

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])
 

rule 219

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])
 

rule 224

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]
 

rule 291

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]
 

rule 318

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]
 

rule 398

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4616

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(675\) vs. \(2(101)=202\).

Time = 0.46 (sec) , antiderivative size = 676, normalized size of antiderivative = 5.68

method	result	size
default	\(-\frac {\sqrt {4}\, \sin \left (d x +c \right ) \left (\left (-3 \cos \left (d x +c \right )+3\right ) \sqrt {b}\, \ln \left (\frac {4 \sqrt {b}\, \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sin \left (d x +c \right )^{2}+4 a \cos \left (d x +c \right )^{2}+2 b \cos \left (d x +c \right )^{2}+2 b \sin \left (d x +c \right )^{2}-8 \cos \left (d x +c \right ) a -4 \cos \left (d x +c \right ) b +4 a +2 b}{\left (-1+\cos \left (d x +c \right )\right )^{2}}\right ) a \sqrt {-a}+\left (3 \cos \left (d x +c \right )-3\right ) \sqrt {b}\, a \ln \left (\frac {2 \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sqrt {b}\, \cos \left (d x +c \right )+2 \sqrt {b}\, \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}+2 \cos \left (d x +c \right ) a +2 a +2 b}{\sqrt {b}\, \left (1+\cos \left (d x +c \right )\right )}\right ) \sqrt {-a}+\left (1-\cos \left (d x +c \right )\right ) b^{\frac {3}{2}} \ln \left (\frac {4 \sqrt {b}\, \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sin \left (d x +c \right )^{2}+4 a \cos \left (d x +c \right )^{2}+2 b \cos \left (d x +c \right )^{2}+2 b \sin \left (d x +c \right )^{2}-8 \cos \left (d x +c \right ) a -4 \cos \left (d x +c \right ) b +4 a +2 b}{\left (-1+\cos \left (d x +c \right )\right )^{2}}\right ) \sqrt {-a}+\left (-1+\cos \left (d x +c \right )\right ) b^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sqrt {b}\, \cos \left (d x +c \right )+2 \sqrt {b}\, \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}+2 \cos \left (d x +c \right ) a +2 a +2 b}{\sqrt {b}\, \left (1+\cos \left (d x +c \right )\right )}\right ) \sqrt {-a}+\left (-4 \cos \left (d x +c \right )+4\right ) a^{2} \ln \left (4 \sqrt {-a}\, \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \cos \left (d x +c \right )+4 \sqrt {-a}\, \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}-4 \cos \left (d x +c \right ) a \right )+2 \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \cos \left (d x +c \right ) \sqrt {-a}\, b \right ) \left (a +b \csc \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}{8 d \sqrt {-a}\, \left (a \sin \left (d x +c \right )^{2}+b \right ) \sqrt {\frac {a \sin \left (d x +c \right )^{2}+b}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\)	\(676\)

Input:

int((a+b*csc(d*x+c)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8/d*4^(1/2)/(-a)^(1/2)*sin(d*x+c)*((-3*cos(d*x+c)+3)*b^(1/2)*ln(2*(2*b^ 
(1/2)*((a*sin(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/2)*sin(d*x+c)^2+2*a*cos(d*x 
+c)^2+b*cos(d*x+c)^2+b*sin(d*x+c)^2-4*cos(d*x+c)*a-2*cos(d*x+c)*b+2*a+b)/( 
-1+cos(d*x+c))^2)*a*(-a)^(1/2)+(3*cos(d*x+c)-3)*b^(1/2)*a*ln(2/b^(1/2)*((( 
a*sin(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/2)*b^(1/2)*cos(d*x+c)+b^(1/2)*((a*s 
in(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/2)+cos(d*x+c)*a+a+b)/(1+cos(d*x+c)))*( 
-a)^(1/2)+(1-cos(d*x+c))*b^(3/2)*ln(2*(2*b^(1/2)*((a*sin(d*x+c)^2+b)/(1+co 
s(d*x+c))^2)^(1/2)*sin(d*x+c)^2+2*a*cos(d*x+c)^2+b*cos(d*x+c)^2+b*sin(d*x+ 
c)^2-4*cos(d*x+c)*a-2*cos(d*x+c)*b+2*a+b)/(-1+cos(d*x+c))^2)*(-a)^(1/2)+(- 
1+cos(d*x+c))*b^(3/2)*ln(2/b^(1/2)*(((a*sin(d*x+c)^2+b)/(1+cos(d*x+c))^2)^ 
(1/2)*b^(1/2)*cos(d*x+c)+b^(1/2)*((a*sin(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/ 
2)+cos(d*x+c)*a+a+b)/(1+cos(d*x+c)))*(-a)^(1/2)+(-4*cos(d*x+c)+4)*a^2*ln(4 
*(-a)^(1/2)*((a*sin(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/2)*cos(d*x+c)+4*(-a)^ 
(1/2)*((a*sin(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/2)-4*cos(d*x+c)*a)+2*((a*si 
n(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/2)*cos(d*x+c)*(-a)^(1/2)*b)*(a+b*csc(d* 
x+c)^2)^(3/2)/(a*sin(d*x+c)^2+b)/((a*sin(d*x+c)^2+b)/(1+cos(d*x+c))^2)^(1/ 
2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (101) = 202\).

Time = 0.38 (sec) , antiderivative size = 1607, normalized size of antiderivative = 13.50 \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((a+b*csc(d*x+c)^2)^(3/2),x, algorithm="fricas")
 

Output:

[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 +...
 

Sympy [F]

\[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\int \left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \] Input:

integrate((a+b*csc(d*x+c)**2)**(3/2),x)
 

Output:

Integral((a + b*csc(c + d*x)**2)**(3/2), x)
 

Maxima [F]

\[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\int { {\left (b \csc \left (d x + c\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((a+b*csc(d*x+c)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*csc(d*x + c)^2 + a)^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*csc(d*x+c)^2)^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\int {\left (a+\frac {b}{{\sin \left (c+d\,x\right )}^2}\right )}^{3/2} \,d x \] Input:

int((a + b/sin(c + d*x)^2)^(3/2),x)
 

Output:

int((a + b/sin(c + d*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \left (a+b \csc ^2(c+d x)\right )^{3/2} \, dx=\left (\int \sqrt {\csc \left (d x +c \right )^{2} b +a}d x \right ) a +\left (\int \sqrt {\csc \left (d x +c \right )^{2} b +a}\, \csc \left (d x +c \right )^{2}d x \right ) b \] Input:

int((a+b*csc(d*x+c)^2)^(3/2),x)
 

Output:

int(sqrt(csc(c + d*x)**2*b + a),x)*a + int(sqrt(csc(c + d*x)**2*b + a)*csc 
(c + d*x)**2,x)*b