3.26 \(\int \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx\)

Optimal. Leaf size=435 \[ \frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}+\frac{b \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt{a} e}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac{b \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt{c} e}-\frac{(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt{c} e} \]

[Out]

-(Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) +
(b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[a]*e)
+ (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2
+ c*Cot[d + e*x]^4])])/(2*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Co
t[d + e*x]^4])])/(4*Sqrt[c]*e) - ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x
]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - (Sqrt[c]*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot
[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*e
)

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Rubi [A]  time = 0.536773, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {3701, 1251, 960, 732, 843, 621, 206, 724, 734} \[ \frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}+\frac{b \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt{a} e}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac{b \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt{c} e}-\frac{(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt{c} e} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^3,x]

[Out]

-(Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) +
(b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[a]*e)
+ (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2
+ c*Cot[d + e*x]^4])])/(2*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Co
t[d + e*x]^4])])/(4*Sqrt[c]*e) - ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x
]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - (Sqrt[c]*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot
[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*e
)

Rule 3701

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(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*Cot[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]

Rule 1251

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 960

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

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 843

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 621

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 206

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

Rule 724

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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rubi steps

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

Mathematica [C]  time = 34.8468, size = 215131, normalized size = 494.55 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^3,x]

[Out]

Result too large to show

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Maple [F]  time = 0.408, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b \left ( \cot \left ( ex+d \right ) \right ) ^{2}+c \left ( \cot \left ( ex+d \right ) \right ) ^{4}} \left ( \tan \left ( ex+d \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a} \tan \left (e x + d\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 19.282, size = 3164, normalized size = 7.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x, algorithm="fricas")

[Out]

[1/8*(4*a*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2 - (2*a - b)*sqrt(a)*lo
g(8*a^2*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + b^2 + 4*a*c + 4*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(a
)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)) + 2*sqrt(a - b + c)*a*log(((8*a^2 - 8*a*b +
b^2 + 4*a*c)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + b^2 + 4*(a - 2*b)*c + 8*c^2 + 4
*((2*a - b)*tan(e*x + d)^4 + (b - 2*c)*tan(e*x + d)^2)*sqrt(a - b + c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)
^2 + c)/tan(e*x + d)^4))/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)))/(a*e), 1/8*(4*a*sqrt((a*tan(e*x + d)^4 + b*
tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2 + 4*a*sqrt(-a + b - c)*arctan(-1/2*((2*a - b)*tan(e*x + d)^
4 + (b - 2*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)/
((a^2 - a*b + a*c)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + (a - b)*c + c^2)) - (2*a - b)*sqrt(a)*l
og(8*a^2*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + b^2 + 4*a*c + 4*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(
a)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)))/(a*e), 1/4*(2*a*sqrt((a*tan(e*x + d)^4 + b
*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2 + sqrt(-a)*(2*a - b)*arctan(1/2*(2*a*tan(e*x + d)^4 + b*ta
n(e*x + d)^2)*sqrt(-a)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)/(a^2*tan(e*x + d)^4 + a*
b*tan(e*x + d)^2 + a*c)) + sqrt(a - b + c)*a*log(((8*a^2 - 8*a*b + b^2 + 4*a*c)*tan(e*x + d)^4 + 2*(4*a*b - 3*
b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + b^2 + 4*(a - 2*b)*c + 8*c^2 + 4*((2*a - b)*tan(e*x + d)^4 + (b - 2*c)*tan(
e*x + d)^2)*sqrt(a - b + c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4))/(tan(e*x + d)^4 +
2*tan(e*x + d)^2 + 1)))/(a*e), 1/4*(2*a*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x
+ d)^2 + sqrt(-a)*(2*a - b)*arctan(1/2*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(-a)*sqrt((a*tan(e*x + d)^
4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)/(a^2*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a*c)) + 2*a*sqrt(-a + b -
c)*arctan(-1/2*((2*a - b)*tan(e*x + d)^4 + (b - 2*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*sqrt((a*tan(e*x + d)^4
+ b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)/((a^2 - a*b + a*c)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 +
(a - b)*c + c^2)))/(a*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2)*tan(e*x+d)**3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x, algorithm="giac")

[Out]

Timed out