Integrand size = 33, antiderivative size = 203 \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx=\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e} \]
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-1/2*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*co t(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
Time = 0.77 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.25 \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx=\frac {\left (\sqrt {a} \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )-\sqrt {a-b+c} \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )-\sqrt {c} \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{2 e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]
((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])] - Sqrt[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*Tan[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])
Time = 0.50 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4184, 1578, 1270, 1154, 219, 1269, 1092, 219, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \tan (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}}{\cot (d+e x)}dx\) |
| \(\Big \downarrow \) 4184 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a} \tan (d+e x)}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a} \tan (d+e x)}{\cot ^2(d+e x)+1}d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 1270 |
| \(\displaystyle -\frac {a \int \frac {\tan (d+e x)}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)-\int \frac {-c \cot ^2(d+e x)+a-b}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {-2 a \int \frac {1}{4 a-\cot ^4(d+e x)}d\frac {b \cot ^2(d+e x)+2 a}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}-\int \frac {-c \cot ^2(d+e x)+a-b}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\int \frac {-c \cot ^2(d+e x)+a-b}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)-\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {c \int \frac {1}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)-(a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)-\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {2 c \int \frac {1}{4 c-\cot ^4(d+e x)}d\frac {2 c \cot ^2(d+e x)+b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}-(a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)-\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-(a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)-\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {2 (a-b+c) \int \frac {1}{4 (a-b+c)-\cot ^4(d+e x)}d\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}-\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )+\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )+\sqrt {c} \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}\) |
-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])]) + 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*C ot[d + e*x]^4])] + Sqrt[c]*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqr t[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/e
3.1.25.3.1 Defintions of rubi rules used
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(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] && !IGtQ[m, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp[(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] - Simp[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] && FractionQ[p] && GtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[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] && Int egerQ[(m - 1)/2]
Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Simp[-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]
\[\int \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \tan \left (e x +d \right )d x\]
Time = 0.70 (sec) , antiderivative size = 2522, normalized size of antiderivative = 12.42 \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx=\text {Too large to display} \]
[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)*ta n(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*t an(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)*ta n(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)*arct an(2*sqrt(-a)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + ...
\[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx=\int \sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}} \tan {\left (d + e x \right )}\, dx \]
\[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a} \tan \left (e x + d\right ) \,d x } \]
Timed out. \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx=\text {Timed out} \]
Timed out. \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan (d+e x) \, dx=\int \mathrm {tan}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a} \,d x \]