\(\int \csc ^5(e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\) [109]

Optimal result

Integrand size = 25, antiderivative size = 223 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {3 \left (a^2+6 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 \sqrt {a} f}+\frac {3 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {3 (a+3 b) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f} \] Output:

-3/8*(a^2+6*a*b+b^2)*arctanh(a^(1/2)*sec(f*x+e)/(a-b+b*sec(f*x+e)^2)^(1/2) 
)/a^(1/2)/f+3/2*b^(1/2)*(a+b)*arctanh(b^(1/2)*sec(f*x+e)/(a-b+b*sec(f*x+e) 
^2)^(1/2))/f+3/8*(a+3*b)*sec(f*x+e)*(a-b+b*sec(f*x+e)^2)^(1/2)/f-3/8*(a+b) 
*csc(f*x+e)^2*sec(f*x+e)*(a-b+b*sec(f*x+e)^2)^(1/2)/f-1/4*cot(f*x+e)*csc(f 
*x+e)^3*(a-b+b*sec(f*x+e)^2)^(3/2)/f
 

Mathematica [A] (warning: unable to verify)

Time = 3.56 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.86 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\cos (e+f x) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)} \left (-2 \csc ^2(e+f x) \left (3 a+5 b+2 a \csc ^2(e+f x)\right )+8 b \sec ^2(e+f x)+\frac {3 \left (16 \sqrt {a} \sqrt {b} (a+b) \text {arctanh}\left (\frac {-\sqrt {a} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{2 \sqrt {b}}\right )+\left (a^2+6 a b+b^2\right ) \left (2 \text {arctanh}\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-\frac {\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{\sqrt {a}}\right )+\log \left (a-2 b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}\right )\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\cos ^2(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )}}{\sqrt {a} \sqrt {\left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}\right )}{16 \sqrt {2} f} \] Input:

Integrate[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]
 

Output:

(Cos[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-2* 
Csc[e + f*x]^2*(3*a + 5*b + 2*a*Csc[e + f*x]^2) + 8*b*Sec[e + f*x]^2 + (3* 
(16*Sqrt[a]*Sqrt[b]*(a + b)*ArcTanh[(-(Sqrt[a]*(-1 + Tan[(e + f*x)/2]^2)) 
+ Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])/(2*Sqrt[b] 
)] + (a^2 + 6*a*b + b^2)*(2*ArcTanh[Tan[(e + f*x)/2]^2 - Sqrt[4*b*Tan[(e + 
 f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]/Sqrt[a]] + Log[a - 2*b - a*Tan 
[(e + f*x)/2]^2 + Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f 
*x)/2]^2)^2]]))*Sec[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]^2*Sec[(e + f*x)/2]^4] 
)/(Sqrt[a]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + 
 a*(-1 + Tan[(e + f*x)/2]^2)^2])))/(16*Sqrt[2]*f)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 4147, 25, 369, 27, 439, 25, 444, 27, 398, 224, 219, 291, 219}

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.

\(\displaystyle \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^{3/2}}{\sin (e+f x)^5}dx\)

\(\Big \downarrow \) 4147

\(\displaystyle \frac {\int -\frac {\sec ^4(e+f x) \left (b \sec ^2(e+f x)+a-b\right )^{3/2}}{\left (1-\sec ^2(e+f x)\right )^3}d\sec (e+f x)}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {\sec ^4(e+f x) \left (b \sec ^2(e+f x)+a-b\right )^{3/2}}{\left (1-\sec ^2(e+f x)\right )^3}d\sec (e+f x)}{f}\)

\(\Big \downarrow \) 369

\(\displaystyle \frac {\frac {1}{4} \int \frac {3 \sec ^2(e+f x) \sqrt {b \sec ^2(e+f x)+a-b} \left (2 b \sec ^2(e+f x)+a-b\right )}{\left (1-\sec ^2(e+f x)\right )^2}d\sec (e+f x)-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3}{4} \int \frac {\sec ^2(e+f x) \sqrt {b \sec ^2(e+f x)+a-b} \left (2 b \sec ^2(e+f x)+a-b\right )}{\left (1-\sec ^2(e+f x)\right )^2}d\sec (e+f x)-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 439

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \int -\frac {\sec ^2(e+f x) \left (2 b (a+3 b) \sec ^2(e+f x)+(a-b) (a+5 b)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3}{4} \left (\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}-\frac {1}{2} \int \frac {\sec ^2(e+f x) \left (2 b (a+3 b) \sec ^2(e+f x)+(a-b) (a+5 b)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left ((a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}-\frac {\int \frac {2 b \left (4 b (a+b) \sec ^2(e+f x)+(a-b) (a+3 b)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)}{2 b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left ((a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}-\int \frac {4 b (a+b) \sec ^2(e+f x)+(a-b) (a+3 b)}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+4 b (a+b) \int \frac {1}{\sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+4 b (a+b) \int \frac {1}{1-\frac {b \sec ^2(e+f x)}{b \sec ^2(e+f x)+a-b}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a-b}}+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)+4 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\left (a^2+6 a b+b^2\right ) \int \frac {1}{1-\frac {a \sec ^2(e+f x)}{b \sec ^2(e+f x)+a-b}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a-b}}+4 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\frac {\left (a^2+6 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{\sqrt {a}}+4 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )+(a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}\right )+\frac {(a+b) \sec ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{2 \left (1-\sec ^2(e+f x)\right )}\right )-\frac {\sec ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 \left (1-\sec ^2(e+f x)\right )^2}}{f}\)

Input:

Int[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]
 

Output:

(-1/4*(Sec[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(1 - Sec[e + f*x]^ 
2)^2 + (3*(((a + b)*Sec[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*(1 - 
 Sec[e + f*x]^2)) + (-(((a^2 + 6*a*b + b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x]) 
/Sqrt[a - b + b*Sec[e + f*x]^2]])/Sqrt[a]) + 4*Sqrt[b]*(a + b)*ArcTanh[(Sq 
rt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]] + (a + 3*b)*Sec[e + f* 
x]*Sqrt[a - b + b*Sec[e + f*x]^2])/2))/4)/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 369

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* 
b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1))   Int[(e*x)^(m - 2)*(a + b*x^2)^(p 
 + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] 
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 
] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 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 439

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
 + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p 
+ 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( 
p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 
))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G 
tQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
 

rule 444

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ 
(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ 
(b*d*(m + 2*(p + q + 1) + 1))   Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) 
^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( 
m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, 
q}, x] && GtQ[m, 1]
 

rule 3042

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

rule 4147

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ 
m)   Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 
)), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( 
m - 1)/2]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1385\) vs. \(2(195)=390\).

Time = 8.52 (sec) , antiderivative size = 1386, normalized size of antiderivative = 6.22

method	result	size
default	\(\text {Expression too large to display}\)	\(1386\)

Input:

int(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/16/f/a^(5/2)/b*(a+b*tan(f*x+e)^2)^(3/2)/((a*cos(f*x+e)^2+b*sin(f*x+e)^2) 
/(cos(f*x+e)+1)^2)^(1/2)/(a*cos(f*x+e)^2+b*sin(f*x+e)^2)*((-24*cos(f*x+e)+ 
24)*b^(5/2)*a^(5/2)*ln(4*(b*cot(f*x+e)^2-2*b*cot(f*x+e)*csc(f*x+e)+b*csc(f 
*x+e)^2+2*b^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2) 
+b)/(cot(f*x+e)^2-2*csc(f*x+e)*cot(f*x+e)+csc(f*x+e)^2-1))*cos(f*x+e)*cot( 
f*x+e)^2+(-24*cos(f*x+e)+24)*b^(3/2)*a^(7/2)*ln(4*(b*cot(f*x+e)^2-2*b*cot( 
f*x+e)*csc(f*x+e)+b*csc(f*x+e)^2+2*b^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2 
)/(cos(f*x+e)+1)^2)^(1/2)+b)/(cot(f*x+e)^2-2*csc(f*x+e)*cot(f*x+e)+csc(f*x 
+e)^2-1))*cos(f*x+e)*cot(f*x+e)^2+(6*cos(f*x+e)^2-10)*((a*cos(f*x+e)^2+b*s 
in(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(7/2)*b*cot(f*x+e)^3*csc(f*x+e)+(-1 
8*cos(f*x+e)^2+8)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2) 
*a^(5/2)*b^2*cot(f*x+e)*csc(f*x+e)+(3*cos(f*x+e)-3)*ln(2/a^(1/2)*(a^(1/2)* 
((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)+((a*co 
s(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+co 
s(f*x+e)*b+b)/(cos(f*x+e)+1))*a^4*b*cos(f*x+e)*cot(f*x+e)^2+(18*cos(f*x+e) 
-18)*ln(2/a^(1/2)*(a^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1) 
^2)^(1/2)*cos(f*x+e)+((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1 
/2)*a^(1/2)-a*cos(f*x+e)+cos(f*x+e)*b+b)/(cos(f*x+e)+1))*a^3*b^2*cos(f*x+e 
)*cot(f*x+e)^2+(3*cos(f*x+e)-3)*ln(2/a^(1/2)*(a^(1/2)*((a*cos(f*x+e)^2+b*s 
in(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)+((a*cos(f*x+e)^2+b*sin(...
 

Fricas [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 1431, normalized size of antiderivative = 6.42 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/16*(3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 6*a*b + b^2)*cos(f 
*x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos( 
f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*c 
os(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + 12*((a^2 + a*b)*cos(f*x + e)^ 
5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(b)*log(- 
((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos( 
f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(3*(a^2 + 3*a*b)*cos(f 
*x + e)^4 - (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt(((a - b)*cos(f*x 
 + e)^2 + b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + 
 a*f*cos(f*x + e)), -1/8*(3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 
 6*a*b + b^2)*cos(f*x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(-a)* 
arctan(-sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x 
 + e)/((a - b)*cos(f*x + e)^2 + b)) - 6*((a^2 + a*b)*cos(f*x + e)^5 - 2*(a 
^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(b)*log(-((a - b) 
*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e) 
^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) - (3*(a^2 + 3*a*b)*cos(f*x + e)^4 
- (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + 
b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f 
*x + e)), 1/16*(24*((a^2 + a*b)*cos(f*x + e)^5 - 2*(a^2 + a*b)*cos(f*x + e 
)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(-b)*arctan(-sqrt(-b)*sqrt(((a - b)...
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \] Input:

integrate(csc(f*x+e)**5*(a+b*tan(f*x+e)**2)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{5} \,d x } \] Input:

integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*tan(f*x + e)^2 + a)^(3/2)*csc(f*x + e)^5, x)
 

Giac [F(-2)]

Exception generated. \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^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. \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^5} \,d x \] Input:

int((a + b*tan(e + f*x)^2)^(3/2)/sin(e + f*x)^5,x)
 

Output:

int((a + b*tan(e + f*x)^2)^(3/2)/sin(e + f*x)^5, x)
 

Reduce [F]

\[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\tan \left (f x +e \right )^{2} b +a}\, \csc \left (f x +e \right )^{5} \tan \left (f x +e \right )^{2}d x \right ) b +\left (\int \sqrt {\tan \left (f x +e \right )^{2} b +a}\, \csc \left (f x +e \right )^{5}d x \right ) a \] Input:

int(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x)
 

Output:

int(sqrt(tan(e + f*x)**2*b + a)*csc(e + f*x)**5*tan(e + f*x)**2,x)*b + int 
(sqrt(tan(e + f*x)**2*b + a)*csc(e + f*x)**5,x)*a