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

Optimal. Leaf size=249 \[ -\frac{b \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 a^{3/2} e}+\frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 a e}-\frac{\tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt{a} e}+\frac{\tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e \sqrt{a-b+c}} \]

[Out]

-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*Sqrt[a]*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*a^(3/2)*e) +
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*Sqrt[a - b + c]*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*a*e)

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Rubi [A]  time = 0.318505, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {3701, 1251, 960, 730, 724, 206} \[ -\frac{b \tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 a^{3/2} e}+\frac{\tan ^2(d+e x) \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 a e}-\frac{\tanh ^{-1}\left (\frac{2 a+b \cot ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt{a} e}+\frac{\tanh ^{-1}\left (\frac{2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e \sqrt{a-b+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-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*Sqrt[a]*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*a^(3/2)*e) +
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*Sqrt[a - b + c]*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*a*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 730

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

Rubi steps

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

Mathematica [C]  time = 31.4706, size = 37459, normalized size = 150.44 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.44, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tan \left ( ex+d \right ) \right ) ^{3}{\frac{1}{\sqrt{a+b \left ( \cot \left ( ex+d \right ) \right ) ^{2}+c \left ( \cot \left ( ex+d \right ) \right ) ^{4}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Fricas [A]  time = 13.6969, size = 3499, normalized size = 14.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*sqrt(a - b + c)*a^2*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)) + 4*(a^2 - a*b + a*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^2 - a*b - b^2 + (2*a + b)*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)))/(
(a^3 - a^2*b + a^2*c)*e), 1/4*(sqrt(a - b + c)*a^2*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)) + 2*(a^2 - a*b + a*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^2 - a*b - b^2 + (2*a + b)*c)*sqrt(-a)*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)))/((a^3 - a^2*b + a^2*c)*e), 1/8*(4*a^2*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)) + 4*(a^2 - a*b + a*c)*s
qrt((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^2 - a*b - b^2 + (2*a + b)*
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)))/((a^3 - a^2*b + a^2*c)*e), 1/4
*(2*a^2*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)*sq
rt((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^2 - a*b + a*c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/t
an(e*x + d)^4)*tan(e*x + d)^2 + (2*a^2 - a*b - b^2 + (2*a + b)*c)*sqrt(-a)*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)))/((a^3 - a^2*b + a^2*c)*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out