Integrand size = 26, antiderivative size = 145 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {7 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 d}-\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d} \]
7/4*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))*a^(1/2)/d-arctanh(1/2*(a+I*a *tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*a^(1/2)/d-1/4*I*cot(d*x+c)*(a+ I*a*tan(d*x+c))^(1/2)/d-1/2*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d
Time = 0.92 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.80 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {-7 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+4 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+\cot (c+d x) (i+2 \cot (c+d x)) \sqrt {a+i a \tan (c+d x)}}{4 d} \]
-1/4*(-7*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + 4*Sqrt[2]*S qrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] + Cot[c + d*x ]*(I + 2*Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/d
Time = 0.98 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3042, 4044, 27, 3042, 4081, 27, 3042, 4083, 3042, 3961, 219, 4082, 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.
\(\displaystyle \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+i a \tan (c+d x)}}{\tan (c+d x)^3}dx\) |
\(\Big \downarrow \) 4044 |
\(\displaystyle \frac {\int \frac {1}{2} \cot ^2(c+d x) (i a-3 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \cot ^2(c+d x) (i a-3 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(i a-3 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)^2}dx}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 4081 |
\(\displaystyle \frac {\frac {\int -\frac {1}{2} \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+7 a^2\right )dx}{a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\int \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+7 a^2\right )dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+7 a^2\right )}{\tan (c+d x)}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 4083 |
\(\displaystyle \frac {-\frac {8 i a^2 \int \sqrt {i \tan (c+d x) a+a}dx+7 a \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {8 i a^2 \int \sqrt {i \tan (c+d x) a+a}dx+7 a \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle \frac {-\frac {\frac {16 a^3 \int \frac {1}{a-i a \tan (c+d x)}d\sqrt {i \tan (c+d x) a+a}}{d}+7 a \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {-\frac {7 a \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx+\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 4082 |
\(\displaystyle \frac {-\frac {\frac {7 a^3 \int \frac {\cot (c+d x)}{\sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}+\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {14 i a^2 \int \frac {1}{i-\frac {i (i \tan (c+d x) a+a)}{a}}d\sqrt {i \tan (c+d x) a+a}}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {14 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\) |
-1/2*(Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/d + (-1/2*((-14*a^(5/2)*A rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (8*Sqrt[2]*a^(5/2)*ArcTanh [Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d)/a - (I*a*Cot[c + d*x]*S qrt[a + I*a*Tan[c + d*x]])/d)/(4*a)
3.1.91.3.1 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[(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[((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[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a , b, c, d}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(c^2 + d^2)*(n + 1)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*d*m - a*c*(n + 1) + a*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d , e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c* m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Fr eeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[( A*b + a*B)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[(B*c - A *d)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*T an[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (115 ) = 230\).
Time = 7.42 (sec) , antiderivative size = 669, normalized size of antiderivative = 4.61
method | result | size |
default | \(-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (7 i \sqrt {2}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+16 i \left (\sin ^{2}\left (d x +c \right )\right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-7 \sqrt {2}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\csc ^{2}\left (d x +c \right )-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot ^{2}\left (d x +c \right )-1}\right ) \cos \left (d x +c \right )+7 i \sqrt {2}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+16 \left (\sin ^{2}\left (d x +c \right )\right ) \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+16 i \left (\sin ^{2}\left (d x +c \right )\right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-7 \sqrt {2}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\csc ^{2}\left (d x +c \right )-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot ^{2}\left (d x +c \right )-1}\right )+16 \left (\sin ^{2}\left (d x +c \right )\right ) \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-6 i \left (\cos ^{3}\left (d x +c \right )\right )+6 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4 i \left (\cos ^{2}\left (d x +c \right )\right )+2 \sin \left (d x +c \right ) \cos \left (d x +c \right )+2 i \cos \left (d x +c \right )\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (i \cos \left (d x +c \right )+i+\sin \left (d x +c \right )\right )}{16 d \left (\cos \left (d x +c \right )+1\right )}\) | \(669\) |
-1/16/d*csc(d*x+c)^2*(7*I*2^(1/2)*sin(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+c)+1) )^(1/2)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)+ 16*I*sin(d*x+c)^2*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/ (cos(d*x+c)+1))^(1/2)*cos(d*x+c)-7*2^(1/2)*sin(d*x+c)^2*(-cos(d*x+c)/(cos( d*x+c)+1))^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(csc(d*x+c)^2-2*csc(d*x+c)*cot(d *x+c)+cot(d*x+c)^2-1)^(1/2))*cos(d*x+c)+7*I*2^(1/2)*sin(d*x+c)^2*(-cos(d*x +c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^ (1/2))+16*sin(d*x+c)^2*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos (d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+16*I*sin( d*x+c)^2*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+ c)+1))^(1/2)-7*2^(1/2)*sin(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln( csc(d*x+c)-cot(d*x+c)+(csc(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+cot(d*x+c)^2-1 )^(1/2))+16*sin(d*x+c)^2*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(c os(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-6*I*cos(d*x+c)^3+6 *cos(d*x+c)^2*sin(d*x+c)-4*I*cos(d*x+c)^2+2*sin(d*x+c)*cos(d*x+c)+2*I*cos( d*x+c))*(a*(1+I*tan(d*x+c)))^(1/2)*(I*cos(d*x+c)+I+sin(d*x+c))/(cos(d*x+c) +1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (110) = 220\).
Time = 0.25 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.58 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {8 \, \sqrt {2} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{d^{2}}} \log \left (4 \, {\left ({\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {a}{d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 8 \, \sqrt {2} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{d^{2}}} \log \left (-4 \, {\left ({\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {a}{d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 7 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, \sqrt {2} {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {a}{d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 7 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, \sqrt {2} {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {a}{d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 4 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (3 \, e^{\left (5 i \, d x + 5 i \, c\right )} + 4 \, e^{\left (3 i \, d x + 3 i \, c\right )} + e^{\left (i \, d x + i \, c\right )}\right )}}{16 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-1/16*(8*sqrt(2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqr t(a/d^2)*log(4*((d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(a/d^2) + a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 8*sqrt(2)*(d*e^(4 *I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/d^2)*log(-4*((d*e^(2 *I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(a/d^2) - a*e^( I*d*x + I*c))*e^(-I*d*x - I*c)) - 7*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d* x + 2*I*c) + d)*sqrt(a/d^2)*log(16*(3*a^2*e^(2*I*d*x + 2*I*c) + 2*sqrt(2)* (a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c ) + 1))*sqrt(a/d^2) + a^2)*e^(-2*I*d*x - 2*I*c)) + 7*(d*e^(4*I*d*x + 4*I*c ) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/d^2)*log(16*(3*a^2*e^(2*I*d*x + 2* I*c) - 2*sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))*sqrt(a/(e ^(2*I*d*x + 2*I*c) + 1))*sqrt(a/d^2) + a^2)*e^(-2*I*d*x - 2*I*c)) - 4*sqrt (2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(3*e^(5*I*d*x + 5*I*c) + 4*e^(3*I*d* x + 3*I*c) + e^(I*d*x + I*c)))/(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2 *I*c) + d)
\[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{3}{\left (c + d x \right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.23 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {a^{2} {\left (\frac {2 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} + \sqrt {i \, a \tan \left (d x + c\right ) + a} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} + a^{3}} + \frac {4 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} - \frac {7 \, \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )}}{8 \, d} \]
1/8*a^2*(2*((I*a*tan(d*x + c) + a)^(3/2) + sqrt(I*a*tan(d*x + c) + a)*a)/( (I*a*tan(d*x + c) + a)^2*a - 2*(I*a*tan(d*x + c) + a)*a^2 + a^3) + 4*sqrt( 2)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a)))/a^(3/2) - 7*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a)))/a^(3/2))/d
\[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{3} \,d x } \]
Time = 5.04 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.86 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{4\,d\,{\mathrm {tan}\left (c+d\,x\right )}^2}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,7{}\mathrm {i}}{4\,d}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{4\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^2}+\frac {\sqrt {2}\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {a}}\right )\,1{}\mathrm {i}}{d} \]