3.1.48 \(\int \frac {\tan (d+e x)}{(a+b \tan ^2(d+e x)+c \tan ^4(d+e x))^{3/2}} \, dx\) [48]

Optimal. Leaf size=155 \[ -\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}+\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3781, 1261, 754, 12, 738, 212} \begin {gather*} \frac {-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\tanh ^{-1}\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 (a-b+c)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[d + e*x]/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]

[Out]

-1/2*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])]/((a - b + c)^(3/2)*e) + (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*S
qrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(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]

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*} \int \frac {\tan (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {1}{(1+x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\text {Subst}\left (\int \frac {-\frac {b^2}{2}+2 a c}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e}\\ &=\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 (a-b+c) e}\\ &=\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\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 )}{(a-b+c) e}\\ &=-\frac {\tanh ^{-1}\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 (a-b+c)^{3/2} e}+\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\\ \end {align*}

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Mathematica [A]
time = 3.11, size = 156, normalized size = 1.01 \begin {gather*} -\frac {\frac {\tanh ^{-1}\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 )}{(a-b+c)^{3/2}}+\frac {2 \left (-b^2+2 a c+b c-(b-2 c) c \tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[d + e*x]/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]

[Out]

-1/2*(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])]/(a - b + c)^(3/2) + (2*(-b^2 + 2*a*c + b*c - (b - 2*c)*c*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*S
qrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]))/e

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(405\) vs. \(2(143)=286\).
time = 0.19, size = 406, normalized size = 2.62

method	result	size
derivativedivides	\(\frac {\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (1+\tan ^{2}\left (e x +d \right )\right )^{2}+\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}}{1+\tan ^{2}\left (e x +d \right )}\right )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}+\frac {2 c \sqrt {c \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\sqrt {-4 a c +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {2 c \sqrt {c \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}+\sqrt {-4 a c +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\)	\(406\)
default	\(\frac {\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (1+\tan ^{2}\left (e x +d \right )\right )^{2}+\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}}{1+\tan ^{2}\left (e x +d \right )}\right )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}+\frac {2 c \sqrt {c \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}-\sqrt {-4 a c +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {2 c \sqrt {c \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}+\sqrt {-4 a c +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\)	\(406\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="maxima")

[Out]

integrate(tan(x*e + d)/(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (147) = 294\).
time = 6.06, size = 1131, normalized size = 7.30 \begin {gather*} \left [-\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \tan \left (x e + d\right )^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} \tan \left (x e + d\right )^{2}\right )} \sqrt {a - b + c} \log \left (\frac {{\left (b^{2} + 4 \, {\left (a - 2 \, b\right )} c + 8 \, c^{2}\right )} \tan \left (x e + d\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2} - 4 \, {\left (a - b\right )} c\right )} \tan \left (x e + d\right )^{2} - 4 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2} + 2 \, a - b\right )} \sqrt {a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan \left (x e + d\right )^{4} + 2 \, \tan \left (x e + d\right )^{2} + 1}\right ) + 4 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (a b^{2} - b^{3} - {\left (2 \, a + b\right )} c^{2} - {\left ({\left (2 \, a - 3 \, b\right )} c^{2} + 2 \, c^{3} - {\left (a b - b^{2}\right )} c\right )} \tan \left (x e + d\right )^{2} - {\left (2 \, a^{2} - a b - 2 \, b^{2}\right )} c\right )}}{4 \, {\left ({\left (4 \, a c^{4} + {\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} c^{3} + 2 \, {\left (2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right )} c^{2} - {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} c\right )} e \tan \left (x e + d\right )^{4} - {\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left (8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right )} c^{2} - 2 \, {\left (2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right )} c\right )} e \tan \left (x e + d\right )^{2} - {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left (8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right )} c^{2} - 2 \, {\left (2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right )} c\right )} e\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \tan \left (x e + d\right )^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} \tan \left (x e + d\right )^{2}\right )} \sqrt {-a + b - c} \arctan \left (-\frac {\sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2} + 2 \, a - b\right )} \sqrt {-a + b - c}}{2 \, {\left ({\left ({\left (a - b\right )} c + c^{2}\right )} \tan \left (x e + d\right )^{4} + {\left (a b - b^{2} + b c\right )} \tan \left (x e + d\right )^{2} + a^{2} - a b + a c\right )}}\right ) - 2 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (a b^{2} - b^{3} - {\left (2 \, a + b\right )} c^{2} - {\left ({\left (2 \, a - 3 \, b\right )} c^{2} + 2 \, c^{3} - {\left (a b - b^{2}\right )} c\right )} \tan \left (x e + d\right )^{2} - {\left (2 \, a^{2} - a b - 2 \, b^{2}\right )} c\right )}}{2 \, {\left ({\left (4 \, a c^{4} + {\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} c^{3} + 2 \, {\left (2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right )} c^{2} - {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} c\right )} e \tan \left (x e + d\right )^{4} - {\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left (8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right )} c^{2} - 2 \, {\left (2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right )} c\right )} e \tan \left (x e + d\right )^{2} - {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left (8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right )} c^{2} - 2 \, {\left (2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right )} c\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="fricas")

[Out]

[-1/4*(((b^2*c - 4*a*c^2)*tan(x*e + d)^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*tan(x*e + d)^2)*sqrt(a - b + c)*l
og(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(x*e + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(x*e + d)^2 - 4*sqrt(c*t
an(x*e + d)^4 + b*tan(x*e + d)^2 + a)*((b - 2*c)*tan(x*e + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b
^2 + 4*a*c)/(tan(x*e + d)^4 + 2*tan(x*e + d)^2 + 1)) + 4*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a)*(a*b^2
- b^3 - (2*a + b)*c^2 - ((2*a - 3*b)*c^2 + 2*c^3 - (a*b - b^2)*c)*tan(x*e + d)^2 - (2*a^2 - a*b - 2*b^2)*c))/(
(4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*
tan(x*e + d)^4 - (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2
+ a*b^3 + b^4)*c)*e*tan(x*e + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2
- 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*e), 1/2*(((b^2*c - 4*a*c^2)*tan(x*e + d)^4 + a*b^2 - 4*a^2*c + (b^
3 - 4*a*b*c)*tan(x*e + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a)*((b -
2*c)*tan(x*e + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(x*e + d)^4 + (a*b - b^2 + b*c)*tan(x*e
+ d)^2 + a^2 - a*b + a*c)) - 2*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a)*(a*b^2 - b^3 - (2*a + b)*c^2 - ((
2*a - 3*b)*c^2 + 2*c^3 - (a*b - b^2)*c)*tan(x*e + d)^2 - (2*a^2 - a*b - 2*b^2)*c))/((4*a*c^4 + (8*a^2 - 8*a*b
- b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*tan(x*e + d)^4 - (a^2*b^3
- 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*e*tan(x
*e + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^
2*b^2 + a*b^3)*c)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan {\left (d + e x \right )}}{\left (a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(e*x+d)/(a+b*tan(e*x+d)**2+c*tan(e*x+d)**4)**(3/2),x)

[Out]

Integral(tan(d + e*x)/(a + b*tan(d + e*x)**2 + c*tan(d + e*x)**4)**(3/2), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {tan}\left (d+e\,x\right )}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d + e*x)/(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(3/2),x)

[Out]

int(tan(d + e*x)/(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(3/2), x)

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