Integrand size = 23, antiderivative size = 189 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\frac {\left (8 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]
1/4*(8*a^2+b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-arctanh( (a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))*(a-I*b)^(1/2)/d-arctanh((a+b*tan(d*x +c))^(1/2)/(a+I*b)^(1/2))*(a+I*b)^(1/2)/d-1/4*b*cot(d*x+c)*(a+b*tan(d*x+c) )^(1/2)/a/d-1/2*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d
Time = 1.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.88 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\frac {\left (8 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )-\sqrt {a} \left (4 a \sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+4 a \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\cot (c+d x) (b+2 a \cot (c+d x)) \sqrt {a+b \tan (c+d x)}\right )}{4 a^{3/2} d} \]
((8*a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - Sqrt[a]*(4*a*Sq rt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*a*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(b + 2*a*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(4*a^(3/2)*d)
Time = 1.58 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4051, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
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.
\(\displaystyle \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+b \tan (c+d x)}}{\tan (c+d x)^3}dx\) |
\(\Big \downarrow \) 4051 |
\(\displaystyle -\frac {1}{2} \int -\frac {\cot ^2(c+d x) \left (-3 b \tan ^2(c+d x)-4 a \tan (c+d x)+b\right )}{2 \sqrt {a+b \tan (c+d x)}}dx-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {\cot ^2(c+d x) \left (-3 b \tan ^2(c+d x)-4 a \tan (c+d x)+b\right )}{\sqrt {a+b \tan (c+d x)}}dx-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \int \frac {-3 b \tan (c+d x)^2-4 a \tan (c+d x)+b}{\tan (c+d x)^2 \sqrt {a+b \tan (c+d x)}}dx-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {\cot (c+d x) \left (8 a^2+8 b \tan (c+d x) a+b^2+b^2 \tan ^2(c+d x)\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}\right )-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {\cot (c+d x) \left (8 a^2+8 b \tan (c+d x) a+b^2+b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}\right )-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {8 a^2+8 b \tan (c+d x) a+b^2+b^2 \tan (c+d x)^2}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}\right )-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {1}{4} \left (-\frac {\left (8 a^2+b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx+\int \frac {8 \left (a b-a^2 \tan (c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}\right )-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (-\frac {\left (8 a^2+b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx+8 \int \frac {a b-a^2 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}\right )-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (-\frac {\left (8 a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \int \frac {a b-a^2 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}\right )-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\left (8 a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {1}{2} a (b+i a) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a (-b+i a) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\left (8 a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {1}{2} a (b+i a) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a (-b+i a) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\left (8 a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {i a (b+i a) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {i a (-b+i a) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\left (8 a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (-\frac {i a (b+i a) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a (-b+i a) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\left (8 a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {a (b+i a) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}-\frac {a (-b+i a) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\left (8 a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {a (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {a (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\frac {\left (8 a^2+b^2\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+8 \left (\frac {a (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {a (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {\frac {2 \left (8 a^2+b^2\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+8 \left (\frac {a (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {a (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a d}-\frac {-\frac {2 \left (8 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+8 \left (\frac {a (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {a (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{2 a}\right )\) |
-1/2*(Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/d + (-1/2*(8*((a*(I*a + b)* ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - (a*(I*a - b)*ArcTa n[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)) - (2*(8*a^2 + b^2)*ArcTa nh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d))/a - (b*Cot[c + d*x]*Sqr t[a + b*Tan[c + d*x]])/(a*d))/4
3.6.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2 )) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c *(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && Int egerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 22.29 (sec) , antiderivative size = 1957605, normalized size of antiderivative = 10357.70
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (153) = 306\).
Time = 0.30 (sec) , antiderivative size = 818, normalized size of antiderivative = 4.33 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\left [-\frac {4 \, a^{2} d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} - 4 \, a^{2} d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (-d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} + 4 \, a^{2} d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} - 4 \, a^{2} d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (-d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} - {\left (8 \, a^{2} + b^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (d x + c\right ) + 2 \, \sqrt {b \tan \left (d x + c\right ) + a} \sqrt {a} + 2 \, a}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{2} + 2 \, {\left (a b \tan \left (d x + c\right ) + 2 \, a^{2}\right )} \sqrt {b \tan \left (d x + c\right ) + a}}{8 \, a^{2} d \tan \left (d x + c\right )^{2}}, -\frac {2 \, a^{2} d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} - 2 \, a^{2} d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (-d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} + 2 \, a^{2} d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} - 2 \, a^{2} d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (-d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) \tan \left (d x + c\right )^{2} + {\left (8 \, a^{2} + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right ) + a} \sqrt {-a}}{a}\right ) \tan \left (d x + c\right )^{2} + {\left (a b \tan \left (d x + c\right ) + 2 \, a^{2}\right )} \sqrt {b \tan \left (d x + c\right ) + a}}{4 \, a^{2} d \tan \left (d x + c\right )^{2}}\right ] \]
[-1/8*(4*a^2*d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d*sqrt((d^2*sqrt(-b^ 2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 - 4*a^2*d*sqrt ((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(-d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 + 4*a^2*d*sqrt(-(d^2*sqrt(-b^2/d ^4) - a)/d^2)*log(d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 - 4*a^2*d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*log (-d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d* x + c)^2 - (8*a^2 + b^2)*sqrt(a)*log((b*tan(d*x + c) + 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c))*tan(d*x + c)^2 + 2*(a*b*tan(d*x + c) + 2*a^2)*sqrt(b*tan(d*x + c) + a))/(a^2*d*tan(d*x + c)^2), -1/4*(2*a^2*d*s qrt((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 - 2*a^2*d*sqrt((d^2*sqrt(-b^2/ d^4) + a)/d^2)*log(-d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 + 2*a^2*d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*lo g(d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d* x + c)^2 - 2*a^2*d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*log(-d*sqrt(-(d^2*s qrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 + (8*a^ 2 + b^2)*sqrt(-a)*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a)*tan(d*x + c) ^2 + (a*b*tan(d*x + c) + 2*a^2)*sqrt(b*tan(d*x + c) + a))/(a^2*d*tan(d*x + c)^2)]
\[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int \sqrt {a + b \tan {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\, dx \]
\[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{3} \,d x } \]
Exception generated. \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Time = 6.18 (sec) , antiderivative size = 1910, normalized size of antiderivative = 10.11 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
atan((b^14*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*2 i)/((b^15*1i)/d + (a^2*b^13*17i)/d + (16*a^3*b^12)/d + (a^4*b^11*64i)/d + (16*a^5*b^10)/d + (a^6*b^9*48i)/d) - (2*b^13*(a/(4*d^2) + (b*1i)/(4*d^2))^ (1/2)*(a + b*tan(c + d*x))^(1/2))/((a*b^13*17i)/d + (16*a^2*b^12)/d + (a^3 *b^11*64i)/d + (16*a^4*b^10)/d + (a^5*b^9*48i)/d + (b^15*1i)/(a*d)) + (b^1 2*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((b^1 3*17i)/d + (16*a*b^12)/d + (a^2*b^11*64i)/d + (16*a^3*b^10)/d + (a^4*b^9*4 8i)/d + (b^15*1i)/(a^2*d)) + (a^2*b^10*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)* (a + b*tan(c + d*x))^(1/2)*128i)/((b^13*17i)/d + (16*a*b^12)/d + (a^2*b^11 *64i)/d + (16*a^3*b^10)/d + (a^4*b^9*48i)/d + (b^15*1i)/(a^2*d)) - (96*a^3 *b^9*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((b^13 *17i)/d + (16*a*b^12)/d + (a^2*b^11*64i)/d + (16*a^3*b^10)/d + (a^4*b^9*48 i)/d + (b^15*1i)/(a^2*d)))*((a + b*1i)/(4*d^2))^(1/2)*2i - atan((b^14*(a/( 4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*2i)/((b^15*1i)/d + (a^2*b^13*17i)/d - (16*a^3*b^12)/d + (a^4*b^11*64i)/d - (16*a^5*b^10)/d + (a^6*b^9*48i)/d) + (2*b^13*(a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*ta n(c + d*x))^(1/2))/((a*b^13*17i)/d - (16*a^2*b^12)/d + (a^3*b^11*64i)/d - (16*a^4*b^10)/d + (a^5*b^9*48i)/d + (b^15*1i)/(a*d)) + (b^12*(a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((b^13*17i)/d - (16* a*b^12)/d + (a^2*b^11*64i)/d - (16*a^3*b^10)/d + (a^4*b^9*48i)/d + (b^1...