\(\int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx\) [27]

Optimal result

Integrand size = 35, antiderivative size = 270 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{32 c^{5/2} e}-\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^2 e}+\frac {\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}}{6 c e} \]

[Out]

1/32*(b^3+2*b^2*c-4*b*(a-2*c)*c-8*c^2*(a+2*c))*arctanh(1/2*(b+2*c*tan(e*x+d)^2)/c^(1/2)/(a+b*tan(e*x+d)^2+c*ta
n(e*x+d)^4)^(1/2))/c^(5/2)/e-1/2*arctanh(1/2*(2*a-b+(b-2*c)*tan(e*x+d)^2)/(a-b+c)^(1/2)/(a+b*tan(e*x+d)^2+c*ta
n(e*x+d)^4)^(1/2))*(a-b+c)^(1/2)/e-1/16*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*((b-2*c)*(b+4*c)+2*c*(b+2*c)*t
an(e*x+d)^2)/c^2/e+1/6*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2)/c/e

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3781, 1265, 1667, 828, 857, 635, 212, 738} \[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\frac {\left (-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right ) \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{32 c^{5/2} e}-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\left (2 c (b+2 c) \tan ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^2 e}+\frac {\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}}{6 c e} \]

[In]

Int[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]

[Out]

-1/2*(Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^
2 + c*Tan[d + e*x]^4])])/e + ((b^3 + 2*b^2*c - 4*b*(a - 2*c)*c - 8*c^2*(a + 2*c))*ArcTanh[(b + 2*c*Tan[d + e*x
]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(32*c^(5/2)*e) - (((b - 2*c)*(b + 4*c) + 2*c*
(b + 2*c)*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(16*c^2*e) + (a + b*Tan[d + e*x]^2 +
c*Tan[d + e*x]^4)^(3/2)/(6*c*e)

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1667

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m
+ q + 2*p + 1))), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 3781

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)
), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^5 \sqrt {a+b x^2+c x^4}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {x^2 \sqrt {a+b x+c x^2}}{1+x} \, dx,x,\tan ^2(d+e x)\right )}{2 e} \\ & = \frac {\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}}{6 c e}+\frac {\text {Subst}\left (\int \frac {\left (-\frac {3 b}{2}-\frac {3}{2} (b+2 c) x\right ) \sqrt {a+b x+c x^2}}{1+x} \, dx,x,\tan ^2(d+e x)\right )}{6 c e} \\ & = -\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^2 e}+\frac {\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}}{6 c e}-\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} (b-2 c) \left (b^2-4 a c+4 b c\right )-\frac {3}{4} \left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{24 c^2 e} \\ & = -\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^2 e}+\frac {\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}}{6 c e}+\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}+\frac {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{32 c^2 e} \\ & = -\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^2 e}+\frac {\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}}{6 c e}-\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{4 a-4 b+4 c-x^2} \, dx,x,\frac {2 a-b-(-b+2 c) \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}+\frac {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{16 c^2 e} \\ & = -\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{32 c^{5/2} e}-\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^2 e}+\frac {\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}}{6 c e} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.06 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.07 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\frac {-48 c^{5/2} \sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )+3 \left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )+\frac {1}{4} \sqrt {c} \left (-9 b^2+24 a c-16 b c+84 c^2-4 \left (3 b^2+6 b c-8 c (a+2 c)\right ) \cos (2 (d+e x))+\left (-3 b^2+8 a c-8 b c+44 c^2\right ) \cos (4 (d+e x))\right ) \sec ^4(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{96 c^{5/2} e} \]

[In]

Integrate[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4],x]

[Out]

(-48*c^(5/2)*Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d
+ e*x]^2 + c*Tan[d + e*x]^4])] + 3*(b^3 + 2*b^2*c - 4*b*(a - 2*c)*c - 8*c^2*(a + 2*c))*ArcTanh[(b + 2*c*Tan[d
+ e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])] + (Sqrt[c]*(-9*b^2 + 24*a*c - 16*b*c + 84
*c^2 - 4*(3*b^2 + 6*b*c - 8*c*(a + 2*c))*Cos[2*(d + e*x)] + (-3*b^2 + 8*a*c - 8*b*c + 44*c^2)*Cos[4*(d + e*x)]
)*Sec[d + e*x]^4*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/4)/(96*c^(5/2)*e)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.69

method	result	size
derivativedivides	\(\frac {\frac {\left (a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}\right )^{\frac {3}{2}}}{6 c}-\frac {b \left (\frac {\left (b +2 c \tan \left (e x +d \right )^{2}\right ) \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \tan \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}-\frac {\left (b +2 c \tan \left (e x +d \right )^{2}\right ) \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}{8 c}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \tan \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}\right )}{16 c^{\frac {3}{2}}}+\frac {\sqrt {c \left (1+\tan \left (e x +d \right )^{2}\right )^{2}+\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+a -b +c}}{2}+\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (1+\tan \left (e x +d \right )^{2}\right ) c}{\sqrt {c}}+\sqrt {c \left (1+\tan \left (e x +d \right )^{2}\right )^{2}+\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+a -b +c}\right )}{4 \sqrt {c}}-\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (1+\tan \left (e x +d \right )^{2}\right )^{2}+\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+a -b +c}}{1+\tan \left (e x +d \right )^{2}}\right )}{2}}{e}\)	\(455\)
default	\(\frac {\frac {\left (a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}\right )^{\frac {3}{2}}}{6 c}-\frac {b \left (\frac {\left (b +2 c \tan \left (e x +d \right )^{2}\right ) \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \tan \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}-\frac {\left (b +2 c \tan \left (e x +d \right )^{2}\right ) \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}}{8 c}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \tan \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}\right )}{16 c^{\frac {3}{2}}}+\frac {\sqrt {c \left (1+\tan \left (e x +d \right )^{2}\right )^{2}+\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+a -b +c}}{2}+\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (1+\tan \left (e x +d \right )^{2}\right ) c}{\sqrt {c}}+\sqrt {c \left (1+\tan \left (e x +d \right )^{2}\right )^{2}+\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+a -b +c}\right )}{4 \sqrt {c}}-\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (1+\tan \left (e x +d \right )^{2}\right )^{2}+\left (b -2 c \right ) \left (1+\tan \left (e x +d \right )^{2}\right )+a -b +c}}{1+\tan \left (e x +d \right )^{2}}\right )}{2}}{e}\)	\(455\)

[In]

int((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

1/e*(1/6*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2)/c-1/4*b/c*(1/4*(b+2*c*tan(e*x+d)^2)/c*(a+b*tan(e*x+d)^2+c*tan
(e*x+d)^4)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*tan(e*x+d)^2)/c^(1/2)+(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(
1/2)))-1/8*(b+2*c*tan(e*x+d)^2)/c*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)-1/16*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c
*tan(e*x+d)^2)/c^(1/2)+(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2))+1/2*(c*(1+tan(e*x+d)^2)^2+(b-2*c)*(1+tan(e*x+d
)^2)+a-b+c)^(1/2)+1/4*(b-2*c)*ln((1/2*b-c+(1+tan(e*x+d)^2)*c)/c^(1/2)+(c*(1+tan(e*x+d)^2)^2+(b-2*c)*(1+tan(e*x
+d)^2)+a-b+c)^(1/2))/c^(1/2)-1/2*(a-b+c)^(1/2)*ln((2*a-2*b+2*c+(b-2*c)*(1+tan(e*x+d)^2)+2*(a-b+c)^(1/2)*(c*(1+
tan(e*x+d)^2)^2+(b-2*c)*(1+tan(e*x+d)^2)+a-b+c)^(1/2))/(1+tan(e*x+d)^2)))

Fricas [A] (verification not implemented)

none

Time = 2.89 (sec) , antiderivative size = 1405, normalized size of antiderivative = 5.20 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)^5,x, algorithm="fricas")

[Out]

[1/192*(48*sqrt(a - b + c)*c^3*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b
)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqr
t(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 3*(b^3 - 8*(a - b)*c^2
- 16*c^3 - 2*(2*a*b - b^2)*c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x
+ d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + 4*(8*c^3*tan(e*x + d)^4 - 3*b^2*c
+ 2*(4*a - 3*b)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x + d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)
)/(c^3*e), 1/96*(24*sqrt(a - b + c)*c^3*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 -
4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a
- b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 3*(b^3 - 8*(a
- b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*
tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) + 2*(8*c^3*tan(e*x + d)^4 - 3*b^
2*c + 2*(4*a - 3*b)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x + d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2
+ a))/(c^3*e), -1/192*(96*sqrt(-a + b - c)*c^3*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b -
2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x
+ d)^2 + a^2 - a*b + a*c)) + 3*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*sqrt(c)*log(8*c^2*tan(e*x +
d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*
sqrt(c) + 4*a*c) - 4*(8*c^3*tan(e*x + d)^4 - 3*b^2*c + 2*(4*a - 3*b)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x
+ d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/(c^3*e), -1/96*(48*sqrt(-a + b - c)*c^3*arctan(-1/2*sqr
t(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c +
c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + 3*(b^3 - 8*(a - b)*c^2 - 16*c^3
- 2*(2*a*b - b^2)*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b
)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - 2*(8*c^3*tan(e*x + d)^4 - 3*b^2*c + 2*(4*a - 3*b
)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x + d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/(c^3*e)]

Sympy [F]

\[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}} \tan ^{5}{\left (d + e x \right )}\, dx \]

[In]

integrate((a+b*tan(e*x+d)**2+c*tan(e*x+d)**4)**(1/2)*tan(e*x+d)**5,x)

[Out]

Integral(sqrt(a + b*tan(d + e*x)**2 + c*tan(d + e*x)**4)*tan(d + e*x)**5, x)

Maxima [F]

\[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \tan \left (e x + d\right )^{5} \,d x } \]

[In]

integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)^5,x, algorithm="maxima")

[Out]

integrate(sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*tan(e*x + d)^5, x)

Giac [F(-1)]

Timed out. \[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\text {Timed out} \]

[In]

integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)^5,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \tan ^5(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int {\mathrm {tan}\left (d+e\,x\right )}^5\,\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a} \,d x \]

[In]

int(tan(d + e*x)^5*(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(1/2),x)

[Out]

int(tan(d + e*x)^5*(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(1/2), x)