3.1.9 \(\int \sqrt {b \tan ^3(e+f x)} \, dx\) [9]

3.1.9.1 Optimal result

Integrand size = 14, antiderivative size = 255 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \]

output

2*cot(f*x+e)*(b*tan(f*x+e)^3)^(1/2)/f-1/2*arctan(-1+2^(1/2)*tan(f*x+e)^(1/ 
2))*(b*tan(f*x+e)^3)^(1/2)/f*2^(1/2)/tan(f*x+e)^(3/2)-1/2*arctan(1+2^(1/2) 
*tan(f*x+e)^(1/2))*(b*tan(f*x+e)^3)^(1/2)/f*2^(1/2)/tan(f*x+e)^(3/2)+1/4*l 
n(1-2^(1/2)*tan(f*x+e)^(1/2)+tan(f*x+e))*(b*tan(f*x+e)^3)^(1/2)/f*2^(1/2)/ 
tan(f*x+e)^(3/2)-1/4*ln(1+2^(1/2)*tan(f*x+e)^(1/2)+tan(f*x+e))*(b*tan(f*x+ 
e)^3)^(1/2)/f*2^(1/2)/tan(f*x+e)^(3/2)
 

3.1.9.2 Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.64 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\frac {\left (\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )}{2 \sqrt {2}}+2 \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{f \tan ^{\frac {3}{2}}(e+f x)} \]

input

Integrate[Sqrt[b*Tan[e + f*x]^3],x]
 

output

((ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]/Sqrt[2] - ArcTan[1 + Sqrt[2]*Sqrt 
[Tan[e + f*x]]]/Sqrt[2] + Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x 
]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]/(2*Sqr 
t[2]) + 2*Sqrt[Tan[e + f*x]])*Sqrt[b*Tan[e + f*x]^3])/(f*Tan[e + f*x]^(3/2 
))
 

3.1.9.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.70, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {3042, 4141, 3042, 3954, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \sqrt {b \tan ^3(e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {b \tan (e+f x)^3}dx\)

\(\Big \downarrow \) 4141

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \int \tan ^{\frac {3}{2}}(e+f x)dx}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \int \tan (e+f x)^{3/2}dx}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \left (\frac {2 \sqrt {\tan (e+f x)}}{f}-\int \frac {1}{\sqrt {\tan (e+f x)}}dx\right )}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \left (\frac {2 \sqrt {\tan (e+f x)}}{f}-\int \frac {1}{\sqrt {\tan (e+f x)}}dx\right )}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \left (\frac {2 \sqrt {\tan (e+f x)}}{f}-\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}\right )}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \left (\frac {2 \sqrt {\tan (e+f x)}}{f}-\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}\right )}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \left (\frac {2 \sqrt {\tan (e+f x)}}{f}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}\right )}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \left (\frac {2 \sqrt {\tan (e+f x)}}{f}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (e+f x)}+1}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}\right )}{\tan ^{\frac {3}{2}}(e+f x)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt {b \tan ^3(e+f x)} \left (\frac {2 \sqrt {\tan (e+f x)}}{f}-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}\right )\right )}{f}\right )}{\tan ^{\frac {3}{2}}(e+f x)}\)

input

Int[Sqrt[b*Tan[e + f*x]^3],x]
 

output

(((-2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]/Sqrt[2]) + ArcTan[1 + Sqr 
t[2]*Sqrt[Tan[e + f*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f 
*x]] + Tan[e + f*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e 
+ f*x]]/(2*Sqrt[2]))/2))/f + (2*Sqrt[Tan[e + f*x]])/f)*Sqrt[b*Tan[e + f*x] 
^3])/Tan[e + f*x]^(3/2)
 

3.1.9.3.1 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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

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

rule 3954

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d 
*x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2   Int[(b*Tan[c + d*x])^(n - 2), x] 
, x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4141

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff 
= FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ 
n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p]))   Int[ActivateTrig[u]*(Ta 
n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] 
 && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / 
; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
 

3.1.9.4 Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(-\frac {\sqrt {b \tan \left (f x +e \right )^{3}}\, \left (\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {b \tan \left (f x +e \right )}\right )}{4 f \tan \left (f x +e \right ) \sqrt {b \tan \left (f x +e \right )}}\)	\(208\)
default	\(-\frac {\sqrt {b \tan \left (f x +e \right )^{3}}\, \left (\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {b \tan \left (f x +e \right )}\right )}{4 f \tan \left (f x +e \right ) \sqrt {b \tan \left (f x +e \right )}}\)	\(208\)

input

int((b*tan(f*x+e)^3)^(1/2),x,method=_RETURNVERBOSE)
 

output

-1/4/f*(b*tan(f*x+e)^3)^(1/2)*((b^2)^(1/4)*2^(1/2)*ln(-(b*tan(f*x+e)+(b^2) 
^(1/4)*(b*tan(f*x+e))^(1/2)*2^(1/2)+(b^2)^(1/2))/((b^2)^(1/4)*(b*tan(f*x+e 
))^(1/2)*2^(1/2)-b*tan(f*x+e)-(b^2)^(1/2)))+2*(b^2)^(1/4)*2^(1/2)*arctan(( 
2^(1/2)*(b*tan(f*x+e))^(1/2)+(b^2)^(1/4))/(b^2)^(1/4))+2*(b^2)^(1/4)*2^(1/ 
2)*arctan((2^(1/2)*(b*tan(f*x+e))^(1/2)-(b^2)^(1/4))/(b^2)^(1/4))-8*(b*tan 
(f*x+e))^(1/2))/tan(f*x+e)/(b*tan(f*x+e))^(1/2)
 

3.1.9.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.07 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=-\frac {f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) - f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) - \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) + i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) - i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) - 4 \, \sqrt {b \tan \left (f x + e\right )^{3}}}{2 \, f \tan \left (f x + e\right )} \]

input

integrate((b*tan(f*x+e)^3)^(1/2),x, algorithm="fricas")
 

output

-1/2*(f*(-b^2/f^4)^(1/4)*log((f*(-b^2/f^4)^(1/4)*tan(f*x + e) + sqrt(b*tan 
(f*x + e)^3))/tan(f*x + e))*tan(f*x + e) - f*(-b^2/f^4)^(1/4)*log(-(f*(-b^ 
2/f^4)^(1/4)*tan(f*x + e) - sqrt(b*tan(f*x + e)^3))/tan(f*x + e))*tan(f*x 
+ e) + I*f*(-b^2/f^4)^(1/4)*log((I*f*(-b^2/f^4)^(1/4)*tan(f*x + e) + sqrt( 
b*tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e) - I*f*(-b^2/f^4)^(1/4)*log(( 
-I*f*(-b^2/f^4)^(1/4)*tan(f*x + e) + sqrt(b*tan(f*x + e)^3))/tan(f*x + e)) 
*tan(f*x + e) - 4*sqrt(b*tan(f*x + e)^3))/(f*tan(f*x + e))
 

3.1.9.6 Sympy [F]

\[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\int \sqrt {b \tan ^{3}{\left (e + f x \right )}}\, dx \]

input

integrate((b*tan(f*x+e)**3)**(1/2),x)
 

output

Integral(sqrt(b*tan(e + f*x)**3), x)
 

3.1.9.7 Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.52 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=-\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + \sqrt {2} \sqrt {b} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) - \sqrt {2} \sqrt {b} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) - 8 \, \sqrt {b} \sqrt {\tan \left (f x + e\right )}}{4 \, f} \]

input

integrate((b*tan(f*x+e)^3)^(1/2),x, algorithm="maxima")
 

output

-1/4*(2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(f*x + e)) 
)) + 2*sqrt(2)*sqrt(b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(f*x + e)) 
)) + sqrt(2)*sqrt(b)*log(sqrt(2)*sqrt(tan(f*x + e)) + tan(f*x + e) + 1) - 
sqrt(2)*sqrt(b)*log(-sqrt(2)*sqrt(tan(f*x + e)) + tan(f*x + e) + 1) - 8*sq 
rt(b)*sqrt(tan(f*x + e)))/f
 

3.1.9.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.76 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{f} + \frac {2 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{f} + \frac {\sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{f} - \frac {\sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{f} - \frac {8 \, \sqrt {b \tan \left (f x + e\right )}}{f}\right )} \mathrm {sgn}\left (\tan \left (f x + e\right )\right ) \]

input

integrate((b*tan(f*x+e)^3)^(1/2),x, algorithm="giac")
 

output

-1/4*(2*sqrt(2)*sqrt(abs(b))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) + 2* 
sqrt(b*tan(f*x + e)))/sqrt(abs(b)))/f + 2*sqrt(2)*sqrt(abs(b))*arctan(-1/2 
*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sqrt(b*tan(f*x + e)))/sqrt(abs(b)))/f + 
 sqrt(2)*sqrt(abs(b))*log(b*tan(f*x + e) + sqrt(2)*sqrt(b*tan(f*x + e))*sq 
rt(abs(b)) + abs(b))/f - sqrt(2)*sqrt(abs(b))*log(b*tan(f*x + e) - sqrt(2) 
*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs(b))/f - 8*sqrt(b*tan(f*x + e))/f) 
*sgn(tan(f*x + e))
 

3.1.9.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\int \sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^3} \,d x \]

input

int((b*tan(e + f*x)^3)^(1/2),x)
 

output

int((b*tan(e + f*x)^3)^(1/2), x)