3.1.21 \(\int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\) [21]

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

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

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Rubi [A]
time = 0.22, antiderivative size = 249, normalized size of antiderivative = 1.00, 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 = {3782, 1265, 974, 744, 738, 212} \begin {gather*} -\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}} \end {gather*}

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]

-1/2*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])]/(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 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 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 744

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 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 \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx &=-\frac {\text {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 {\text {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 {\text {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 {\text {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 {\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 {\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 a e}-\frac {\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 {\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 {b \text {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 \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 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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 32.13, size = 37459, normalized size = 150.44 \begin {gather*} \text {Result too large to show} \end {gather*}

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 = 0.84, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{3}\left (e x +d \right )}{\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}\, dx\]

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)] 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(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 = 5.22, size = 1516, normalized size = 6.09 \begin {gather*} \left [\frac {{\left (2 \, \sqrt {a - b + c} a^{2} \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 ) + 4 \, {\left (a^{2} - a b + a c\right )} \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^{2} - a b - b^{2} + {\left (2 \, a + b\right )} c\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 \, {\left (a^{3} - a^{2} b + a^{2} c\right )}}, \frac {{\left (\sqrt {a - b + c} a^{2} \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 ) + 2 \, {\left (a^{2} - a b + a c\right )} \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^{2} - a b - b^{2} + {\left (2 \, a + b\right )} c\right )} \sqrt {-a} \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 )\right )} e^{\left (-1\right )}}{4 \, {\left (a^{3} - a^{2} b + a^{2} c\right )}}, \frac {{\left (4 \, a^{2} \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 ) + 4 \, {\left (a^{2} - a b + a c\right )} \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^{2} - a b - b^{2} + {\left (2 \, a + b\right )} c\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 \, {\left (a^{3} - a^{2} b + a^{2} c\right )}}, \frac {{\left (2 \, a^{2} \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 ) + 2 \, {\left (a^{2} - a b + a c\right )} \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^{2} - a b - b^{2} + {\left (2 \, a + b\right )} c\right )} \sqrt {-a} \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 )\right )} e^{\left (-1\right )}}{4 \, {\left (a^{3} - a^{2} b + a^{2} c\right )}}\right ] \end {gather*}

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(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)) + 4*(a^2 - a*b + a*c)*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^2 - a*b - b^2 + (2*a + b)*c)*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)*sqrt(a)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)))*e
^(-1)/(a^3 - a^2*b + a^2*c), 1/4*(sqrt(a - b + c)*a^2*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)) + 2*(a^2 - a*b + a*c)*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^2 - a*b - b^2 + (2*a + b)*c)*sqrt(-a)*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)))*e^(-1)/(a^3 - a^2*b + a^2*c), 1/8*(4*a^2*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)) + 4*(a^2 - a*b +
a*c)*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^2 - a*b - b^2 + (2*a
+ b)*c)*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)*sqrt(a)*sqrt((a*tan(x*e + d)^4 + b*tan(x*e + d)^2 + c)/tan(x*e + d)^4)))*e^(-1)/(a^3 - a^2*b + a^
2*c), 1/4*(2*a^2*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^2 - a*b + a*c)*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^2 - a*b - b^2 + (2*a + b)*c)*sqrt(-a)*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)))*e^(-1)/(a^3 - a^2*b + a^2*c)]

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

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)] 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(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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\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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