3.1.31 \(\int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx\) [31]

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

[Out]

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

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Rubi [A]
time = 0.36, antiderivative size = 435, normalized size of antiderivative = 1.00, 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 = {3781, 1265, 974, 746, 857, 635, 212, 738, 748} \begin {gather*} -\frac {b \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {a} e}+\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\sqrt {a-b+c} \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}+\frac {\sqrt {c} \tanh ^{-1}\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 )}{2 e}-\frac {b \tanh ^{-1}\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 )}{4 \sqrt {c} e}+\frac {(b-2 c) \tanh ^{-1}\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 )}{4 \sqrt {c} e}-\frac {\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e) -
(b*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*Sqrt[a]*e)
- (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])])/(2*e) - (b*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])])/(4*Sqrt[c]*e) + ((b - 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])])/(4*Sqrt[c]*e) + (Sqrt[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])])/(2*e) - (Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*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 746

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 748

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]

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 974

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

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Mathematica [A]
time = 1.33, size = 187, normalized size = 0.43 \begin {gather*} \frac {(2 a-b) \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )-2 \sqrt {a} \left (\sqrt {a-b+c} \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 )+\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}\right )}{4 \sqrt {a} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int \left (\cot ^{3}\left (e x +d \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 10.22, size = 1248, normalized size = 2.87 \begin {gather*} \left [\frac {{\left (2 \, \sqrt {a - b + c} a \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 ) \tan \left (x e + d\right )^{2} - {\left (2 \, a - b\right )} \sqrt {a} \log \left (\frac {{\left (b^{2} + 4 \, a c\right )} \tan \left (x e + d\right )^{4} + 8 \, a b \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 (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{\tan \left (x e + d\right )^{4}}\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} a\right )} e^{\left (-1\right )}}{8 \, a \tan \left (x e + d\right )^{2}}, -\frac {{\left (4 \, a \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 ) \tan \left (x e + d\right )^{2} + {\left (2 \, a - b\right )} \sqrt {a} \log \left (\frac {{\left (b^{2} + 4 \, a c\right )} \tan \left (x e + d\right )^{4} + 8 \, a b \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 (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{\tan \left (x e + d\right )^{4}}\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} a\right )} e^{\left (-1\right )}}{8 \, a \tan \left (x e + d\right )^{2}}, -\frac {{\left (\sqrt {-a} {\left (2 \, a - b\right )} \arctan \left (\frac {\sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c \tan \left (x e + d\right )^{4} + a b \tan \left (x e + d\right )^{2} + a^{2}\right )}}\right ) \tan \left (x e + d\right )^{2} - \sqrt {a - b + c} a \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 ) \tan \left (x e + d\right )^{2} + 2 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} a\right )} e^{\left (-1\right )}}{4 \, a \tan \left (x e + d\right )^{2}}, -\frac {{\left (\sqrt {-a} {\left (2 \, a - b\right )} \arctan \left (\frac {\sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c \tan \left (x e + d\right )^{4} + a b \tan \left (x e + d\right )^{2} + a^{2}\right )}}\right ) \tan \left (x e + d\right )^{2} + 2 \, a \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 ) \tan \left (x e + d\right )^{2} + 2 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} a\right )} e^{\left (-1\right )}}{4 \, a \tan \left (x e + d\right )^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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