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

Optimal. Leaf size=203 \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e} \]

[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) -
(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) - (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)

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Rubi [A]  time = 0.269807, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3701, 1251, 895, 724, 206, 843, 621} \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c \cot ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]

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

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 895

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist[(c
*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)), Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)
), Int[(Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*(a + b*x + c*x^2)^(p - 1))/(f + g*x), 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] && Fra
ctionQ[p] && GtQ[p, 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 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 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]

Rubi steps

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

Mathematica [A]  time = 12.737, size = 253, normalized size = 1.25 \[ -\frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \left (-\sqrt{a} \tanh ^{-1}\left (\frac{2 a \tan ^2(d+e x)+b}{2 \sqrt{a} \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}}\right )+\sqrt{a-b+c} \tanh ^{-1}\left (\frac{(2 a-b) \tan ^2(d+e x)+b-2 c}{2 \sqrt{a-b+c} \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}}\right )+\sqrt{c} \tanh ^{-1}\left (\frac{b \tan ^2(d+e x)+2 c}{2 \sqrt{c} \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}}\right )\right )}{2 e \sqrt{a \tan ^4(d+e x)+b \tan ^2(d+e x)+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-(Sqrt[a]*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])]) + Sq
rt[a - b + c]*ArcTanh[(b - 2*c + (2*a - b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Ta
n[d + e*x]^4])] + Sqrt[c]*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*
x]^4])])*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[
d + e*x]^4])

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Maple [F]  time = 0.364, 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}}\tan \left ( ex+d \right ) \, 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),x)

[Out]

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

[Out]

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

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Fricas [A]  time = 5.86361, size = 6334, normalized size = 31.2 \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),x, algorithm="fricas")

[Out]

[1/4*(sqrt(a)*log(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)) + sqrt(a - b + c)*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)) + sqrt(c)*log(((b^2 + 4*a*c)*tan
(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + 8*c^2 - 4*(b*tan(e*x + d)^4 + 2*c*tan(e*x + d)^2)*sqrt(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))/e, 1/4*(2*sqrt(-c)*arctan(2*sqrt(-c)*sqrt((a
*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2/(b*tan(e*x + d)^2 + 2*c)) + sqrt(a)*log
(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)) + sqrt(a - b + c)*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)))/e, -1/4*(2*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*ta
n(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2/(2*a*tan(e*x + d)^2 + b)) - sqrt(a - b + c
)*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)) - sqrt(c)*log(((b^2
+ 4*a*c)*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + 8*c^2 - 4*(b*tan(e*x + d)^4 + 2*c*tan(e*x + d)^2)*sqrt(c)*sqr
t((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4))/tan(e*x + d)^4))/e, -1/4*(2*sqrt(-a)*arctan(2*sqr
t(-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*tan(e*x + d)^2 + b))
- 2*sqrt(-c)*arctan(2*sqrt(-c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2/(
b*tan(e*x + d)^2 + 2*c)) - sqrt(a - b + c)*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)))/e, -1/4*(2*sqrt(-a + b - c)*arctan(-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)^2/((2*a - b)*tan(e*x + d)^2 + b - 2*c)) - sqrt(a)*log(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)) - sqrt(c)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*b*c*tan(e*x
+ d)^2 + 8*c^2 - 4*(b*tan(e*x + d)^4 + 2*c*tan(e*x + d)^2)*sqrt(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))/e, -1/4*(2*sqrt(-a + b - c)*arctan(-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)^2/((2*a - b)*tan(e*x + d)^2 + b - 2*c)) - 2*sqrt(-
c)*arctan(2*sqrt(-c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2/(b*tan(e*x
+ d)^2 + 2*c)) - sqrt(a)*log(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)))/e, -1/4*(2*sqrt(
-a)*arctan(2*sqrt(-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*tan(e
*x + d)^2 + b)) + 2*sqrt(-a + b - c)*arctan(-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)^2/((2*a - b)*tan(e*x + d)^2 + b - 2*c)) - sqrt(c)*log(((b^2 + 4*a*c)*tan(e*x + d
)^4 + 8*b*c*tan(e*x + d)^2 + 8*c^2 - 4*(b*tan(e*x + d)^4 + 2*c*tan(e*x + d)^2)*sqrt(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))/e, -1/2*(sqrt(-a)*arctan(2*sqrt(-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*tan(e*x + d)^2 + b)) + sqrt(-a + b - c)*arc
tan(-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)^2/((2*a -
b)*tan(e*x + d)^2 + b - 2*c)) - sqrt(-c)*arctan(2*sqrt(-c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(
e*x + d)^4)*tan(e*x + d)^2/(b*tan(e*x + d)^2 + 2*c)))/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{\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),x)

[Out]

Integral(sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)*tan(d + e*x), 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),x, algorithm="giac")

[Out]

Timed out