3.1.26 \(\int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx\) [26]

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

[Out]

1/4*b*arctanh(1/2*(2*a+b*cot(e*x+d)^2)/a^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/e/a^(1/2)-1/2*arctanh(
1/2*(2*a+b*cot(e*x+d)^2)/a^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))*a^(1/2)/e+1/4*b*arctanh(1/2*(b+2*c*c
ot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/e/c^(1/2)-1/4*(b-2*c)*arctanh(1/2*(b+2*c*cot(e*x
+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/e/c^(1/2)-1/2*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)
/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))*c^(1/2)/e+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(e*x+d)^2)/(a-b+c)^(1/2)
/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))*(a-b+c)^(1/2)/e+1/2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+
d)^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 = {3782, 1265, 974, 746, 857, 635, 212, 738, 748} \begin {gather*} \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} \end {gather*}

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]

-1/2*(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])])/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 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 3782

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]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 34.56, size = 215131, normalized size = 494.55 \begin {gather*} \text {Result too large to show} \end {gather*}

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 = 0.79, size = 0, normalized size = 0.00 \[\int \sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\, \left (\tan ^{3}\left (e x +d \right )\right )\, dx\]

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 = 0.00, 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((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(x*e + d)^4 + b*cot(x*e + d)^2 + a)*tan(x*e + d)^3, x)

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Fricas [A]
time = 6.50, size = 1354, normalized size = 3.11 \begin {gather*} \left [\frac {{\left (4 \, a \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}} \tan \left (x e + d\right )^{2} - {\left (2 \, a - b\right )} \sqrt {a} \log \left (8 \, a^{2} \tan \left (x e + d\right )^{4} + 8 \, a b \tan \left (x e + d\right )^{2} + b^{2} + 4 \, a c + 4 \, {\left (2 \, a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2}\right )} \sqrt {a} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}\right ) + 2 \, \sqrt {a - b + c} a \log \left (\frac {{\left (8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c\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} + b^{2} + 4 \, {\left (a - 2 \, b\right )} c + 8 \, c^{2} + 4 \, {\left ({\left (2 \, a - b\right )} \tan \left (x e + d\right )^{4} + {\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2}\right )} \sqrt {a - b + c} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}}{\tan \left (x e + d\right )^{4} + 2 \, \tan \left (x e + d\right )^{2} + 1}\right )\right )} e^{\left (-1\right )}}{8 \, a}, \frac {{\left (4 \, a \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}} \tan \left (x e + d\right )^{2} + 4 \, a \sqrt {-a + b - c} \arctan \left (-\frac {{\left ({\left (2 \, a - b\right )} \tan \left (x e + d\right )^{4} + {\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2}\right )} \sqrt {-a + b - c} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}}{2 \, {\left ({\left (a^{2} - a b + a c\right )} \tan \left (x e + d\right )^{4} + {\left (a b - b^{2} + b c\right )} \tan \left (x e + d\right )^{2} + {\left (a - b\right )} c + c^{2}\right )}}\right ) - {\left (2 \, a - b\right )} \sqrt {a} \log \left (8 \, a^{2} \tan \left (x e + d\right )^{4} + 8 \, a b \tan \left (x e + d\right )^{2} + b^{2} + 4 \, a c + 4 \, {\left (2 \, a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2}\right )} \sqrt {a} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}\right )\right )} e^{\left (-1\right )}}{8 \, a}, \frac {{\left (2 \, a \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}} \tan \left (x e + d\right )^{2} + \sqrt {-a} {\left (2 \, a - b\right )} \arctan \left (\frac {{\left (2 \, a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2}\right )} \sqrt {-a} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}}{2 \, {\left (a^{2} \tan \left (x e + d\right )^{4} + a b \tan \left (x e + d\right )^{2} + a c\right )}}\right ) + \sqrt {a - b + c} a \log \left (\frac {{\left (8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c\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} + b^{2} + 4 \, {\left (a - 2 \, b\right )} c + 8 \, c^{2} + 4 \, {\left ({\left (2 \, a - b\right )} \tan \left (x e + d\right )^{4} + {\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2}\right )} \sqrt {a - b + c} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}}{\tan \left (x e + d\right )^{4} + 2 \, \tan \left (x e + d\right )^{2} + 1}\right )\right )} e^{\left (-1\right )}}{4 \, a}, \frac {{\left (2 \, a \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}} \tan \left (x e + d\right )^{2} + \sqrt {-a} {\left (2 \, a - b\right )} \arctan \left (\frac {{\left (2 \, a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2}\right )} \sqrt {-a} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}}{2 \, {\left (a^{2} \tan \left (x e + d\right )^{4} + a b \tan \left (x e + d\right )^{2} + a c\right )}}\right ) + 2 \, a \sqrt {-a + b - c} \arctan \left (-\frac {{\left ({\left (2 \, a - b\right )} \tan \left (x e + d\right )^{4} + {\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2}\right )} \sqrt {-a + b - c} \sqrt {\frac {a \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + c}{\tan \left (x e + d\right )^{4}}}}{2 \, {\left ({\left (a^{2} - a b + a c\right )} \tan \left (x e + d\right )^{4} + {\left (a b - b^{2} + b c\right )} \tan \left (x e + d\right )^{2} + {\left (a - b\right )} c + c^{2}\right )}}\right )\right )} e^{\left (-1\right )}}{4 \, a}\right ] \end {gather*}

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(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)*tan(x*e + d)^2 - (2*a - b)*sqrt(a)*lo
g(8*a^2*tan(x*e + d)^4 + 8*a*b*tan(x*e + d)^2 + b^2 + 4*a*c + 4*(2*a*tan(x*e + d)^4 + b*tan(x*e + d)^2)*sqrt(a
)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)) + 2*sqrt(a - b + c)*a*log(((8*a^2 - 8*a*b +
b^2 + 4*a*c)*tan(x*e + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(x*e + d)^2 + b^2 + 4*(a - 2*b)*c + 8*c^2 + 4
*((2*a - b)*tan(x*e + d)^4 + (b - 2*c)*tan(x*e + d)^2)*sqrt(a - b + c)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)
^2 + c)/tan(x*e + d)^4))/(tan(x*e + d)^4 + 2*tan(x*e + d)^2 + 1)))*e^(-1)/a, 1/8*(4*a*sqrt((a*tan(x*e + d)^4 +
b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)*tan(x*e + d)^2 + 4*a*sqrt(-a + b - c)*arctan(-1/2*((2*a - b)*tan(x*e +
d)^4 + (b - 2*c)*tan(x*e + d)^2)*sqrt(-a + b - c)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^
4)/((a^2 - a*b + a*c)*tan(x*e + d)^4 + (a*b - b^2 + b*c)*tan(x*e + d)^2 + (a - b)*c + c^2)) - (2*a - b)*sqrt(a
)*log(8*a^2*tan(x*e + d)^4 + 8*a*b*tan(x*e + d)^2 + b^2 + 4*a*c + 4*(2*a*tan(x*e + d)^4 + b*tan(x*e + d)^2)*sq
rt(a)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)))*e^(-1)/a, 1/4*(2*a*sqrt((a*tan(x*e + d)
^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)*tan(x*e + d)^2 + sqrt(-a)*(2*a - b)*arctan(1/2*(2*a*tan(x*e + d)^4
+ b*tan(x*e + d)^2)*sqrt(-a)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)/(a^2*tan(x*e + d)^
4 + a*b*tan(x*e + d)^2 + a*c)) + sqrt(a - b + c)*a*log(((8*a^2 - 8*a*b + b^2 + 4*a*c)*tan(x*e + d)^4 + 2*(4*a*
b - 3*b^2 - 4*(a - b)*c)*tan(x*e + d)^2 + b^2 + 4*(a - 2*b)*c + 8*c^2 + 4*((2*a - b)*tan(x*e + d)^4 + (b - 2*c
)*tan(x*e + d)^2)*sqrt(a - b + c)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4))/(tan(x*e + d
)^4 + 2*tan(x*e + d)^2 + 1)))*e^(-1)/a, 1/4*(2*a*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4
)*tan(x*e + d)^2 + sqrt(-a)*(2*a - b)*arctan(1/2*(2*a*tan(x*e + d)^4 + b*tan(x*e + d)^2)*sqrt(-a)*sqrt((a*tan(
x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)/(a^2*tan(x*e + d)^4 + a*b*tan(x*e + d)^2 + a*c)) + 2*a*sqrt
(-a + b - c)*arctan(-1/2*((2*a - b)*tan(x*e + d)^4 + (b - 2*c)*tan(x*e + d)^2)*sqrt(-a + b - c)*sqrt((a*tan(x*
e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)/((a^2 - a*b + a*c)*tan(x*e + d)^4 + (a*b - b^2 + b*c)*tan(x*e
+ d)^2 + (a - b)*c + c^2)))*e^(-1)/a]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)] 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((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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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