\(\int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx\) [71]

Optimal result

Integrand size = 19, antiderivative size = 76 \[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {2 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {2 \cos (a+b x) \left (-3+2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) (d \tan (a+b x))^{3/2}}{3 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3073, 3042, 3081, 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.

\(\displaystyle \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3073

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3081

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3052

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

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

Output:

(-2*d^2*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x 
]]*Sqrt[d*Tan[a + b*x]]) + (2*d*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/b
 

Defintions of rubi rules used

rule 3042

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

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 3073

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*(n - 1))), x] - Simp[b^2*((m + 2)/(a^2*(n - 1)))   Int[(a*Sin[e + f* 
x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && G 
tQ[n, 1] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, 3/2])) && 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. \(266\) vs. \(2(71)=142\).

Time = 0.97 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.51

method	result	size
default	\(\frac {\csc \left (b x +a \right ) \left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \left (-\cos \left (b x +a \right )-1\right ) \operatorname {EllipticF}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \left (\cos \left (b x +a \right )+1\right ) \operatorname {EllipticE}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (b x +a \right )+2\right ) \sqrt {d \tan \left (b x +a \right )}\, d \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}}{2 b \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\)	\(267\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.79 \[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {-i \, \sqrt {i \, d} d E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + i \, \sqrt {-i \, d} d E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) + i \, \sqrt {i \, d} d F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) - i \, \sqrt {-i \, d} d F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) + 2 \, d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \sin \left (b x + a\right )}{b} \] Input:

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

Output:

(-I*sqrt(I*d)*d*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1) + I* 
sqrt(-I*d)*d*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) + I*sqr 
t(I*d)*d*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1) - I*sqrt(-I 
*d)*d*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) + 2*d*sqrt(d*s 
in(b*x + a)/cos(b*x + a))*sin(b*x + a))/b
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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