\(\int \sqrt {a+b \tan ^4(c+d x)} \, dx\) [387]

Optimal result

Integrand size = 16, antiderivative size = 516 \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a+b \tan ^4(c+d x)}}\right )}{2 d}+\frac {\sqrt {b} \tan (c+d x) \sqrt {a+b \tan ^4(c+d x)}}{d \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}-\frac {\sqrt [4]{a} \sqrt [4]{b} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{d \sqrt {a+b \tan ^4(c+d x)}}-\frac {\sqrt [4]{a} b^{3/4} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{\left (\sqrt {a}-\sqrt {b}\right ) d \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a+b) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} d \sqrt {a+b \tan ^4(c+d x)}} \] Output:

1/2*(a+b)^(1/2)*arctan((a+b)^(1/2)*tan(d*x+c)/(a+tan(d*x+c)^4*b)^(1/2))/d+ 
b^(1/2)*tan(d*x+c)*(a+tan(d*x+c)^4*b)^(1/2)/d/(a^(1/2)+b^(1/2)*tan(d*x+c)^ 
2)-a^(1/4)*b^(1/4)*EllipticE(sin(2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4))),1/2 
*2^(1/2))*(a^(1/2)+b^(1/2)*tan(d*x+c)^2)*((a+tan(d*x+c)^4*b)/(a^(1/2)+b^(1 
/2)*tan(d*x+c)^2)^2)^(1/2)/d/(a+tan(d*x+c)^4*b)^(1/2)-a^(1/4)*b^(3/4)*Inve 
rseJacobiAM(2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4)),1/2*2^(1/2))*(a^(1/2)+b^( 
1/2)*tan(d*x+c)^2)*((a+tan(d*x+c)^4*b)/(a^(1/2)+b^(1/2)*tan(d*x+c)^2)^2)^( 
1/2)/(a^(1/2)-b^(1/2))/d/(a+tan(d*x+c)^4*b)^(1/2)+1/4*(a^(1/2)+b^(1/2))*(a 
+b)*EllipticPi(sin(2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4))),-1/4*(a^(1/2)-b^( 
1/2))^2/a^(1/2)/b^(1/2),1/2*2^(1/2))*(a^(1/2)+b^(1/2)*tan(d*x+c)^2)*((a+ta 
n(d*x+c)^4*b)/(a^(1/2)+b^(1/2)*tan(d*x+c)^2)^2)^(1/2)/a^(1/4)/(a^(1/2)-b^( 
1/2))/b^(1/4)/d/(a+tan(d*x+c)^4*b)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.67 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.42 \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\frac {\left (\sqrt {a} \sqrt {b} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right )\right |-1\right )+\left (\sqrt {a}-i \sqrt {b}\right ) \left (-\sqrt {b} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right ),-1\right )+\left (-i \sqrt {a}+\sqrt {b}\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a}}{\sqrt {b}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right ),-1\right )\right )\right ) \sqrt {1+\frac {b \tan ^4(c+d x)}{a}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} d \sqrt {a+b \tan ^4(c+d x)}} \] Input:

Integrate[Sqrt[a + b*Tan[c + d*x]^4],x]
 

Output:

((Sqrt[a]*Sqrt[b]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[c + d* 
x]], -1] + (Sqrt[a] - I*Sqrt[b])*(-(Sqrt[b]*EllipticF[I*ArcSinh[Sqrt[(I*Sq 
rt[b])/Sqrt[a]]*Tan[c + d*x]], -1]) + ((-I)*Sqrt[a] + Sqrt[b])*EllipticPi[ 
((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], 
 -1]))*Sqrt[1 + (b*Tan[c + d*x]^4)/a])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*d*Sqrt[a 
 + b*Tan[c + d*x]^4])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 4144, 1524, 27, 1512, 27, 761, 1510, 2221}

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.

\(\displaystyle \int \sqrt {a+b \tan ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a+b \tan (c+d x)^4}dx\)

\(\Big \downarrow \) 4144

\(\displaystyle \frac {\int \frac {\sqrt {b \tan ^4(c+d x)+a}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 1524

\(\displaystyle \frac {\frac {(a+b) \int \frac {\sqrt {b} \tan ^2(c+d x)+\sqrt {a}}{\sqrt {a} \left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{1-\frac {\sqrt {b}}{\sqrt {a}}}-\frac {\int \frac {\sqrt {b} \left (-\left (\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \sqrt {b} \tan ^2(c+d x)\right )+\sqrt {a}+\sqrt {b}\right )}{\sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(a+b) \int \frac {\sqrt {b} \tan ^2(c+d x)+\sqrt {a}}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{\sqrt {a} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )}-\frac {\sqrt {b} \int \frac {-\left (\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \sqrt {b} \tan ^2(c+d x)\right )+\sqrt {a}+\sqrt {b}}{\sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\)

\(\Big \downarrow \) 1512

\(\displaystyle \frac {\frac {(a+b) \int \frac {\sqrt {b} \tan ^2(c+d x)+\sqrt {a}}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{\sqrt {a} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )}-\frac {\sqrt {b} \left (2 \sqrt {b} \int \frac {1}{\sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)+\left (\sqrt {a}-\sqrt {b}\right ) \int \frac {\sqrt {a}-\sqrt {b} \tan ^2(c+d x)}{\sqrt {a} \sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)\right )}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(a+b) \int \frac {\sqrt {b} \tan ^2(c+d x)+\sqrt {a}}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{\sqrt {a} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )}-\frac {\sqrt {b} \left (2 \sqrt {b} \int \frac {1}{\sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \int \frac {\sqrt {a}-\sqrt {b} \tan ^2(c+d x)}{\sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{\sqrt {a}}\right )}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {(a+b) \int \frac {\sqrt {b} \tan ^2(c+d x)+\sqrt {a}}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{\sqrt {a} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )}-\frac {\sqrt {b} \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \int \frac {\sqrt {a}-\sqrt {b} \tan ^2(c+d x)}{\sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{\sqrt {a}}+\frac {\sqrt [4]{b} \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b \tan ^4(c+d x)}}\right )}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {\frac {(a+b) \int \frac {\sqrt {b} \tan ^2(c+d x)+\sqrt {a}}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^4(c+d x)+a}}d\tan (c+d x)}{\sqrt {a} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )}-\frac {\sqrt {b} \left (\frac {\sqrt [4]{b} \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b \tan ^4(c+d x)}}-\frac {\tan (c+d x) \sqrt {a+b \tan ^4(c+d x)}}{\sqrt {a}+\sqrt {b} \tan ^2(c+d x)}\right )}{\sqrt {a}}\right )}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\)

\(\Big \downarrow \) 2221

\(\displaystyle \frac {\frac {(a+b) \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a+b \tan ^4(c+d x)}}\right )}{2 \sqrt {a+b}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b \tan ^4(c+d x)}}\right )}{\sqrt {a} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )}-\frac {\sqrt {b} \left (\frac {\sqrt [4]{b} \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b \tan ^4(c+d x)}}-\frac {\tan (c+d x) \sqrt {a+b \tan ^4(c+d x)}}{\sqrt {a}+\sqrt {b} \tan ^2(c+d x)}\right )}{\sqrt {a}}\right )}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\)

Input:

Int[Sqrt[a + b*Tan[c + d*x]^4],x]
 

Output:

(((a + b)*(((Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a + 
 b*Tan[c + d*x]^4]])/(2*Sqrt[a + b]) + ((Sqrt[a] + Sqrt[b])*EllipticPi[-1/ 
4*(Sqrt[a] - Sqrt[b])^2/(Sqrt[a]*Sqrt[b]), 2*ArcTan[(b^(1/4)*Tan[c + d*x]) 
/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x 
]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(4*a^(1/4)*b^(1/4)*Sqrt[a + b* 
Tan[c + d*x]^4])))/(Sqrt[a]*(1 - Sqrt[b]/Sqrt[a])) - (Sqrt[b]*((b^(1/4)*El 
lipticF[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]* 
Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x 
]^2)^2])/(a^(1/4)*Sqrt[a + b*Tan[c + d*x]^4]) + ((Sqrt[a] - Sqrt[b])*(-((T 
an[c + d*x]*Sqrt[a + b*Tan[c + d*x]^4])/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2) 
) + (a^(1/4)*EllipticE[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqr 
t[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt 
[b]*Tan[c + d*x]^2)^2])/(b^(1/4)*Sqrt[a + b*Tan[c + d*x]^4])))/Sqrt[a]))/( 
1 - Sqrt[b]/Sqrt[a]))/d
 

Defintions of rubi rules used

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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

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

rule 1512

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

rule 1524

Int[Sqrt[(a_) + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[(c*d^2 + a*e^2)/(e*(e - d*q))   Int[(1 + q*x^2)/((d + e* 
x^2)*Sqrt[a + c*x^4]), x], x] - Simp[1/(e*(e - d*q))   Int[(c*d + a*e*q - ( 
c*e - a*d*q^3)*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && N 
eQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2221

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

rule 3042

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

rule 4144

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> 
With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f)   Subst[Int[(a + b* 
(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, 
 b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || 
EqQ[n^2, 16])
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.80 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(\frac {-\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}+\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}-\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}+\frac {a \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}}{d}\)	\(531\)
default	\(\frac {-\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}+\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}-\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}+\frac {a \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (d x +c \right )^{4}}}}{d}\)	\(531\)

Input:

int((a+b*tan(d*x+c)^4)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(-b/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2) 
*(1+I/a^(1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2)/(a+b*tan(d*x+c)^4)^(1/2)*Ellipti 
cF(tan(d*x+c)*(I/a^(1/2)*b^(1/2))^(1/2),I)+I*b^(1/2)*a^(1/2)/(I/a^(1/2)*b^ 
(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2) 
*tan(d*x+c)^2)^(1/2)/(a+b*tan(d*x+c)^4)^(1/2)*EllipticF(tan(d*x+c)*(I/a^(1 
/2)*b^(1/2))^(1/2),I)-I*b^(1/2)*a^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^( 
1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2)/ 
(a+b*tan(d*x+c)^4)^(1/2)*EllipticE(tan(d*x+c)*(I/a^(1/2)*b^(1/2))^(1/2),I) 
+a/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2)*(1+I 
/a^(1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2)/(a+b*tan(d*x+c)^4)^(1/2)*EllipticPi(t 
an(d*x+c)*(I/a^(1/2)*b^(1/2))^(1/2),I*a^(1/2)/b^(1/2),(-I/a^(1/2)*b^(1/2)) 
^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2))+b/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2) 
*b^(1/2)*tan(d*x+c)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*tan(d*x+c)^2)^(1/2)/(a+b 
*tan(d*x+c)^4)^(1/2)*EllipticPi(tan(d*x+c)*(I/a^(1/2)*b^(1/2))^(1/2),I*a^( 
1/2)/b^(1/2),(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*tan(d*x+c)^4)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int \sqrt {a + b \tan ^{4}{\left (c + d x \right )}}\, dx \] Input:

integrate((a+b*tan(d*x+c)**4)**(1/2),x)
 

Output:

Integral(sqrt(a + b*tan(c + d*x)**4), x)
 

Maxima [F]

\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right )^{4} + a} \,d x } \] Input:

integrate((a+b*tan(d*x+c)^4)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*tan(d*x + c)^4 + a), x)
 

Giac [F]

\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right )^{4} + a} \,d x } \] Input:

integrate((a+b*tan(d*x+c)^4)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*tan(d*x + c)^4 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int \sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^4+a} \,d x \] Input:

int((a + b*tan(c + d*x)^4)^(1/2),x)
 

Output:

int((a + b*tan(c + d*x)^4)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int \sqrt {\tan \left (d x +c \right )^{4} b +a}d x \] Input:

int((a+b*tan(d*x+c)^4)^(1/2),x)
 

Output:

int(sqrt(tan(c + d*x)**4*b + a),x)