\(\int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\) [93]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {4 \cos (a+b x)}{5 b \sqrt {d \tan (a+b x)}}-\frac {4 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{5 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \] Output:

-2/5*d*csc(b*x+a)/b/(d*tan(b*x+a))^(3/2)-4/5*cos(b*x+a)/b/(d*tan(b*x+a))^( 
1/2)+4/5*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*sin(b*x+a)/b/sin(2*b*x+2*a)^ 
(1/2)/(d*tan(b*x+a))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.59 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {6 (-2+\cos (2 (a+b x))) \cot (a+b x) \csc (a+b x) \sqrt {\sec ^2(a+b x)}-8 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sec (a+b x) \tan ^2(a+b x)}{15 b \sqrt {\sec ^2(a+b x)} \sqrt {d \tan (a+b x)}} \] Input:

Integrate[Csc[a + b*x]^3/Sqrt[d*Tan[a + b*x]],x]
 

Output:

(6*(-2 + Cos[2*(a + b*x)])*Cot[a + b*x]*Csc[a + b*x]*Sqrt[Sec[a + b*x]^2] 
- 8*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]*Tan[a + 
 b*x]^2)/(15*b*Sqrt[Sec[a + b*x]^2]*Sqrt[d*Tan[a + b*x]])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3079, 3042, 3081, 3042, 3050, 3042, 3052, 3042, 3119}

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.

\(\displaystyle \int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (a+b x)^3 \sqrt {d \tan (a+b x)}}dx\)

\(\Big \downarrow \) 3079

\(\displaystyle \frac {2}{5} \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} \int \frac {1}{\sin (a+b x) \sqrt {d \tan (a+b x)}}dx-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3081

\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin ^{\frac {3}{2}}(a+b x)}dx}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin (a+b x)^{3/2}}dx}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3050

\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3052

\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {2 \sqrt {\sin (a+b x)} \left (-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}\)

Input:

Int[Csc[a + b*x]^3/Sqrt[d*Tan[a + b*x]],x]
 

Output:

(-2*d*Csc[a + b*x])/(5*b*(d*Tan[a + b*x])^(3/2)) + (2*Sqrt[Sin[a + b*x]]*( 
(-2*Cos[a + b*x]^(3/2))/(b*Sqrt[Sin[a + b*x]]) - (2*Sqrt[Cos[a + b*x]]*Ell 
ipticE[a - Pi/4 + b*x, 2]*Sqrt[Sin[a + b*x]])/(b*Sqrt[Sin[2*a + 2*b*x]]))) 
/(5*Sqrt[Cos[a + b*x]]*Sqrt[d*Tan[a + b*x]])
 

Defintions of rubi rules used

rule 3042

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

rule 3050

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a 
*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1))   Int[(b*Cos[e + f*x])^ 
n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 
1] && IntegersQ[2*m, 2*n]
 

rule 3052

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] 
, x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e 
 + 2*f*x]])   Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
 

rule 3079

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n 
_.), x_Symbol] :> Simp[b*(a*Sin[e + f*x])^(m + 2)*((b*Tan[e + f*x])^(n - 1) 
/(a^2*f*(m + n + 1))), x] + Simp[(m + 2)/(a^2*(m + n + 1))   Int[(a*Sin[e + 
 f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && L 
tQ[m, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n]
 

rule 3081

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ 
n)   Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, 
 f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 
1)]) || IntegersQ[m - 1/2, n - 1/2])
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(89)=178\).

Time = 0.87 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.59

method	result	size
default	\(-\frac {\sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-2-2 \sec \left (b x +a \right )\right )+\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (1+\sec \left (b x +a \right )\right )+2+\cot \left (b x +a \right ) \csc \left (b x +a \right )\right ) \sqrt {2}}{5 b \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \tan \left (b x +a \right )}}\)	\(264\)

Input:

int(csc(b*x+a)^3/(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5/b*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/(-sin(b*x+a)*cos( 
b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/(d*tan(b*x+a))^(1/2)*((csc(b*x+a)-cot(b*x+a 
)+1)^(1/2)*(-2*csc(b*x+a)+2*cot(b*x+a)+2)^(1/2)*(-csc(b*x+a)+cot(b*x+a))^( 
1/2)*EllipticE((csc(b*x+a)-cot(b*x+a)+1)^(1/2),1/2*2^(1/2))*(-2-2*sec(b*x+ 
a))+(csc(b*x+a)-cot(b*x+a)+1)^(1/2)*(-2*csc(b*x+a)+2*cot(b*x+a)+2)^(1/2)*( 
-csc(b*x+a)+cot(b*x+a))^(1/2)*EllipticF((csc(b*x+a)-cot(b*x+a)+1)^(1/2),1/ 
2*2^(1/2))*(1+sec(b*x+a))+2+cot(b*x+a)*csc(b*x+a))*2^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.32 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 \, {\left ({\left (i \, \cos \left (b x + a\right )^{2} - i\right )} \sqrt {i \, d} E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + {\left (-i \, \cos \left (b x + a\right )^{2} + i\right )} \sqrt {-i \, d} E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + {\left (-i \, \cos \left (b x + a\right )^{2} + i\right )} \sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + {\left (i \, \cos \left (b x + a\right )^{2} - i\right )} \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + {\left (2 \, \cos \left (b x + a\right )^{4} - 3 \, \cos \left (b x + a\right )^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{5 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )} \sin \left (b x + a\right )} \] Input:

integrate(csc(b*x+a)^3/(d*tan(b*x+a))^(1/2),x, algorithm="fricas")
 

Output:

-2/5*((I*cos(b*x + a)^2 - I)*sqrt(I*d)*elliptic_e(arcsin(cos(b*x + a) + I* 
sin(b*x + a)), -1)*sin(b*x + a) + (-I*cos(b*x + a)^2 + I)*sqrt(-I*d)*ellip 
tic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + (-I*cos(b* 
x + a)^2 + I)*sqrt(I*d)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), 
-1)*sin(b*x + a) + (I*cos(b*x + a)^2 - I)*sqrt(-I*d)*elliptic_f(arcsin(cos 
(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + (2*cos(b*x + a)^4 - 3*cos( 
b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a)))/((b*d*cos(b*x + a)^2 - b*d) 
*sin(b*x + a))
 

Sympy [F]

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

integrate(csc(b*x+a)**3/(d*tan(b*x+a))**(1/2),x)
 

Output:

Integral(csc(a + b*x)**3/sqrt(d*tan(a + b*x)), x)
 

Maxima [F]

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

integrate(csc(b*x+a)^3/(d*tan(b*x+a))^(1/2),x, algorithm="maxima")
 

Output:

integrate(csc(b*x + a)^3/sqrt(d*tan(b*x + a)), x)
 

Giac [F]

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

integrate(csc(b*x+a)^3/(d*tan(b*x+a))^(1/2),x, algorithm="giac")
 

Output:

integrate(csc(b*x + a)^3/sqrt(d*tan(b*x + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^3\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}} \,d x \] Input:

int(1/(sin(a + b*x)^3*(d*tan(a + b*x))^(1/2)),x)
 

Output:

int(1/(sin(a + b*x)^3*(d*tan(a + b*x))^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

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

int(csc(b*x+a)^3/(d*tan(b*x+a))^(1/2),x)
 

Output:

(sqrt(d)*int((sqrt(tan(a + b*x))*csc(a + b*x)**3)/tan(a + b*x),x))/d