Integrand size = 16, antiderivative size = 650 \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a+b \tan ^4(c+d x)}}\right )}{2 d}+\frac {\sqrt {b} \tan (c+d x) \sqrt {a+b \tan ^4(c+d x)}}{d \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}-\frac {\sqrt [4]{a} \sqrt [4]{b} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{d \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} d \sqrt {a+b \tan ^4(c+d x)}}-\frac {\sqrt [4]{b} (a+b) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) d \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a+b) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} d \sqrt {a+b \tan ^4(c+d x)}} \]
1/2*arctan((a+b)^(1/2)*tan(d*x+c)/(a+tan(d*x+c)^4*b)^(1/2))*(a+b)^(1/2)/d+ b^(1/2)*(a+tan(d*x+c)^4*b)^(1/2)*tan(d*x+c)/d/(a^(1/2)+b^(1/2)*tan(d*x+c)^ 2)-a^(1/4)*b^(1/4)*(cos(2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4)))^2)^(1/2)/cos (2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*tan( d*x+c)/a^(1/4))),1/2*2^(1/2))*((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)*tan(d*x+c)^2)/d/(a+tan(d*x+c)^4*b)^(1/2)- 1/2*b^(1/4)*(a+b)*(cos(2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4)))^2)^(1/2)/cos( 2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(d *x+c)/a^(1/4))),1/2*2^(1/2))*((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)*tan(d*x+c)^2)/a^(1/4)/d/(a^(1/2)-b^(1/2))/ (a+tan(d*x+c)^4*b)^(1/2)+1/2*b^(1/4)*(cos(2*arctan(b^(1/4)*tan(d*x+c)/a^(1 /4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4)))*EllipticF(sin(2*a rctan(b^(1/4)*tan(d*x+c)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)-b^(1/2))*((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)*tan(d *x+c)^2)/a^(1/4)/d/(a+tan(d*x+c)^4*b)^(1/2)+1/4*(a+b)*(cos(2*arctan(b^(1/4 )*tan(d*x+c)/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*tan(d*x+c)/a^(1/4)))* 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))*((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)*tan(d*x+c)^2)/a^(1/4 )/b^(1/4)/d/(a^(1/2)-b^(1/2))/(a+tan(d*x+c)^4*b)^(1/2)
Result contains complex when optimal does not.
Time = 11.05 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.34 \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\frac {\left (\sqrt {a} \sqrt {b} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right )\right |-1\right )+\left (\sqrt {a}-i \sqrt {b}\right ) \left (-\sqrt {b} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right ),-1\right )+\left (-i \sqrt {a}+\sqrt {b}\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a}}{\sqrt {b}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right ),-1\right )\right )\right ) \sqrt {1+\frac {b \tan ^4(c+d x)}{a}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} d \sqrt {a+b \tan ^4(c+d x)}} \]
((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])
Time = 0.70 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.84, 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 |
\(\displaystyle \frac {\frac {(a+b) \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a+b \tan ^4(c+d x)}}\right )}{2 \sqrt {a+b}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b \tan ^4(c+d x)}}\right )}{\sqrt {a} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )}-\frac {\sqrt {b} \left (\frac {\sqrt [4]{b} \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b \tan ^4(c+d x)}}-\frac {\tan (c+d x) \sqrt {a+b \tan ^4(c+d x)}}{\sqrt {a}+\sqrt {b} \tan ^2(c+d x)}\right )}{\sqrt {a}}\right )}{1-\frac {\sqrt {b}}{\sqrt {a}}}}{d}\) |
(((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
3.4.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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)]
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])
Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 531, normalized size of antiderivative = 0.82
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 +\tan \left (d x +c \right )^{4} b}}+\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 +\tan \left (d x +c \right )^{4} b}}-\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 +\tan \left (d x +c \right )^{4} b}}+\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 +\tan \left (d x +c \right )^{4} b}}+\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 +\tan \left (d x +c \right )^{4} b}}}{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 +\tan \left (d x +c \right )^{4} b}}+\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 +\tan \left (d x +c \right )^{4} b}}-\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 +\tan \left (d x +c \right )^{4} b}}+\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 +\tan \left (d x +c \right )^{4} b}}+\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 +\tan \left (d x +c \right )^{4} b}}}{d}\) | \(531\) |
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+tan(d*x+c)^4*b)^(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+tan(d*x+c)^4*b)^(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+tan(d*x+c)^4*b)^(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+tan(d*x+c)^4*b)^(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+t an(d*x+c)^4*b)^(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)))
Timed out. \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\text {Timed out} \]
\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int \sqrt {a + b \tan ^{4}{\left (c + d x \right )}}\, dx \]
\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right )^{4} + a} \,d x } \]
\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right )^{4} + a} \,d x } \]
Timed out. \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int \sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^4+a} \,d x \]