Integrand size = 45, antiderivative size = 287 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=-\frac {5 (2 i A-5 B) c^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {5 (2 i A-5 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {5 (2 i A-5 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 (2 i A-5 B) c (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \] Output:
-5*(2*I*A-5*B)*c^(7/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c- I*c*tan(f*x+e))^(1/2))/a^(3/2)/f-5/2*(2*I*A-5*B)*c^3*(a+I*a*tan(f*x+e))^(1 /2)*(c-I*c*tan(f*x+e))^(1/2)/a^2/f-5/6*(2*I*A-5*B)*c^2*(a+I*a*tan(f*x+e))^ (1/2)*(c-I*c*tan(f*x+e))^(3/2)/a^2/f-2/3*(2*I*A-5*B)*c*(c-I*c*tan(f*x+e))^ (5/2)/a/f/(a+I*a*tan(f*x+e))^(1/2)+1/3*(I*A-B)*(c-I*c*tan(f*x+e))^(7/2)/f/ (a+I*a*tan(f*x+e))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 7.15 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.56 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {c^3 \sec (e+f x) \sqrt {c-i c \tan (e+f x)} \left (20 i (2 A+5 i B) \cos ^3(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1+i \tan (e+f x))\right ) \sqrt {2-2 i \tan (e+f x)}+3 (\cos (3 (e+f x))-i \sin (3 (e+f x))) (-2 i A+6 B-i B \tan (e+f x))\right )}{6 a f \sqrt {a+i a \tan (e+f x)}} \] Input:
Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan [e + f*x])^(3/2),x]
Output:
(c^3*Sec[e + f*x]*Sqrt[c - I*c*Tan[e + f*x]]*((20*I)*(2*A + (5*I)*B)*Cos[e + f*x]^3*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 + I*Tan[e + f*x])/2]*Sqrt [2 - (2*I)*Tan[e + f*x]] + 3*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)])*((-2* I)*A + 6*B - I*B*Tan[e + f*x])))/(6*a*f*Sqrt[a + I*a*Tan[e + f*x]])
Time = 0.79 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {3042, 4071, 87, 57, 60, 60, 45, 218}
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.
| \(\displaystyle \int \frac {(c-i c \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(a+i a \tan (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-i c \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(a+i a \tan (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(i \tan (e+f x) a+a)^{5/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 a c (a+i a \tan (e+f x))^{3/2}}-\frac {(2 A+5 i B) \int \frac {(c-i c \tan (e+f x))^{5/2}}{(i \tan (e+f x) a+a)^{3/2}}d\tan (e+f x)}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 a c (a+i a \tan (e+f x))^{3/2}}-\frac {(2 A+5 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {5 c \int \frac {(c-i c \tan (e+f x))^{3/2}}{\sqrt {i \tan (e+f x) a+a}}d\tan (e+f x)}{a}\right )}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 a c (a+i a \tan (e+f x))^{3/2}}-\frac {(2 A+5 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {5 c \left (\frac {3}{2} c \int \frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}d\tan (e+f x)-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 a c (a+i a \tan (e+f x))^{3/2}}-\frac {(2 A+5 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {5 c \left (\frac {3}{2} c \left (c \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 a c (a+i a \tan (e+f x))^{3/2}}-\frac {(2 A+5 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {5 c \left (\frac {3}{2} c \left (2 c \int \frac {1}{i a+\frac {i c (i \tan (e+f x) a+a)}{c-i c \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 a c (a+i a \tan (e+f x))^{3/2}}-\frac {(2 A+5 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {5 c \left (\frac {3}{2} c \left (-\frac {2 i \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {a}}-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{3 a}\right )}{f}\) |
Input:
Int[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f *x])^(3/2),x]
Output:
(a*c*(((I*A - B)*(c - I*c*Tan[e + f*x])^(7/2))/(3*a*c*(a + I*a*Tan[e + f*x ])^(3/2)) - ((2*A + (5*I)*B)*(((2*I)*(c - I*c*Tan[e + f*x])^(5/2))/(a*Sqrt [a + I*a*Tan[e + f*x]]) - (5*c*(((-1/2*I)*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/a + (3*c*(((-2*I)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/Sqrt[a] - (I*S qrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/a))/2))/a))/(3*a)))/ f
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x , Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 732 vs. \(2 (236 ) = 472\).
Time = 0.87 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.55
| method | result | size |
| derivativedivides | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{3} \left (185 i B \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}+6 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-75 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}-118 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-30 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}+90 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-114 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-225 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-21 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+3 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+225 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+90 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+74 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}-30 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +75 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +279 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-46 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f \,a^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \left (i-\tan \left (f x +e \right )\right )^{3}}\) | \(733\) |
| default | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{3} \left (185 i B \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}+6 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-75 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}-118 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-30 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}+90 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-114 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-225 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}-21 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+3 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+225 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+90 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+74 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}-30 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +75 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +279 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-46 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f \,a^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \left (i-\tan \left (f x +e \right )\right )^{3}}\) | \(733\) |
| parts | \(\frac {A \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{3} \left (45 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{2} a c +3 i \tan \left (f x +e \right )^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-15 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{3} a c -15 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -57 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+45 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right ) a c +37 \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-23 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{3 f \,a^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{3} \sqrt {a c}}-\frac {B \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{3} \left (75 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{3} a c -3 i \tan \left (f x +e \right )^{4} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-225 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right ) a c -185 i \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+225 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) \tan \left (f x +e \right )^{2} a c +21 \tan \left (f x +e \right )^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+118 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-75 \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -279 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\right )}{6 f \,a^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{3} \sqrt {a c}}\) | \(795\) |
Input:
int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(3/2),x,m ethod=_RETURNVERBOSE)
Output:
1/6/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*c^3/a^2*(185* I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+6*I*A*(a*c)^(1/2 )*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^3-75*I*B*ln((a*c*tan(f*x+e)+(a*c )^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^3-118*I* B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)-30*A*ln((a*c*tan(f*x+e)+(a*c)^( 1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^3+90*I*A*ln ((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a* c*tan(f*x+e)^2-114*I*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e) -225*B*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^ (1/2))*a*c*tan(f*x+e)^2-21*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan( f*x+e)^3+3*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^4+225*I *B*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2 ))*a*c*tan(f*x+e)+90*A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2 ))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)+74*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a *c)^(1/2)*tan(f*x+e)^2-30*I*A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f *x+e)^2))^(1/2))/(a*c)^(1/2))*a*c+75*B*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c *(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c+279*B*(a*c)^(1/2)*(a*c*(1+tan(f *x+e)^2))^(1/2)*tan(f*x+e)-46*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/ (a*c*(1+tan(f*x+e)^2))^(1/2)/(a*c)^(1/2)/(I-tan(f*x+e))^3
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (219) = 438\).
Time = 0.14 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.05 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(3/ 2),x, algorithm="fricas")
Output:
1/12*(15*(a^2*f*e^(5*I*f*x + 5*I*e) + a^2*f*e^(3*I*f*x + 3*I*e))*sqrt((4*A ^2 + 20*I*A*B - 25*B^2)*c^7/(a^3*f^2))*log(4*(2*((2*I*A - 5*B)*c^3*e^(3*I* f*x + 3*I*e) + (2*I*A - 5*B)*c^3*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I *e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) + (a^2*f*e^(2*I*f*x + 2*I*e) - a^2*f)*sqrt((4*A^2 + 20*I*A*B - 25*B^2)*c^7/(a^3*f^2)))/((2*I*A - 5*B)*c^ 3*e^(2*I*f*x + 2*I*e) + (2*I*A - 5*B)*c^3)) - 15*(a^2*f*e^(5*I*f*x + 5*I*e ) + a^2*f*e^(3*I*f*x + 3*I*e))*sqrt((4*A^2 + 20*I*A*B - 25*B^2)*c^7/(a^3*f ^2))*log(4*(2*((2*I*A - 5*B)*c^3*e^(3*I*f*x + 3*I*e) + (2*I*A - 5*B)*c^3*e ^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I *e) + 1)) - (a^2*f*e^(2*I*f*x + 2*I*e) - a^2*f)*sqrt((4*A^2 + 20*I*A*B - 2 5*B^2)*c^7/(a^3*f^2)))/((2*I*A - 5*B)*c^3*e^(2*I*f*x + 2*I*e) + (2*I*A - 5 *B)*c^3)) - 4*(15*(2*I*A - 5*B)*c^3*e^(6*I*f*x + 6*I*e) + 25*(2*I*A - 5*B) *c^3*e^(4*I*f*x + 4*I*e) + 8*(2*I*A - 5*B)*c^3*e^(2*I*f*x + 2*I*e) + 4*(-I *A + B)*c^3)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)))/(a^2*f*e^(5*I*f*x + 5*I*e) + a^2*f*e^(3*I*f*x + 3*I*e))
Timed out. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(7/2)/(a+I*a*tan(f*x+e))**( 3/2),x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1373 vs. \(2 (219) = 438\).
Time = 0.60 (sec) , antiderivative size = 1373, normalized size of antiderivative = 4.78 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(3/ 2),x, algorithm="maxima")
Output:
-12*(60*(2*A + 5*I*B)*c^3*cos(6*f*x + 6*e) + 100*(2*A + 5*I*B)*c^3*cos(4*f *x + 4*e) + 32*(2*A + 5*I*B)*c^3*cos(2*f*x + 2*e) + 60*(2*I*A - 5*B)*c^3*s in(6*f*x + 6*e) + 100*(2*I*A - 5*B)*c^3*sin(4*f*x + 4*e) + 32*(2*I*A - 5*B )*c^3*sin(2*f*x + 2*e) - 16*(A + I*B)*c^3 + 30*((2*A + 5*I*B)*c^3*cos(7/2* arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*A + 5*I*B)*c^3*cos(5/2 *arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*A + 5*I*B)*c^3*cos(3/2* arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 5*B)*c^3*sin(7/2*a rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*I*A - 5*B)*c^3*sin(5/2* arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 5*B)*c^3*sin(3/2*a rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*arctan2(cos(1/2*arctan2(sin(2 *f*x + 2*e), cos(2*f*x + 2*e))), sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f *x + 2*e))) + 1) + 30*((2*A + 5*I*B)*c^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*A + 5*I*B)*c^3*cos(5/2*arctan2(sin(2*f*x + 2*e) , cos(2*f*x + 2*e))) + (2*A + 5*I*B)*c^3*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 5*B)*c^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*I*A - 5*B)*c^3*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 5*B)*c^3*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 15*(( 2*I*A - 5*B)*c^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) +...
Exception generated. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(3/ 2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:
int(((A + B*tan(e + f*x))*(c - c*tan(e + f*x)*1i)^(7/2))/(a + a*tan(e + f* x)*1i)^(3/2),x)
Output:
int(((A + B*tan(e + f*x))*(c - c*tan(e + f*x)*1i)^(7/2))/(a + a*tan(e + f* x)*1i)^(3/2), x)
\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:
int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(3/2),x)
Output:
(sqrt(c)*sqrt(a)*c**3*( - 10*sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x) *i + 1)*tan(e + f*x)*a - 5*sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)*b*i - 6*sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*a*i + 4*sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*b + 4*i nt(( - sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1))/(tan(e + f*x) **3 - tan(e + f*x)**2*i + tan(e + f*x) - i),x)*tan(e + f*x)**2*a*f*i - int (( - sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1))/(tan(e + f*x)** 3 - tan(e + f*x)**2*i + tan(e + f*x) - i),x)*tan(e + f*x)**2*b*f + 4*int(( - sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1))/(tan(e + f*x)**3 - tan(e + f*x)**2*i + tan(e + f*x) - i),x)*a*f*i - int(( - sqrt(tan(e + f* x)*i + 1)*sqrt( - tan(e + f*x)*i + 1))/(tan(e + f*x)**3 - tan(e + f*x)**2* i + tan(e + f*x) - i),x)*b*f + int((sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**5)/(tan(e + f*x)**3*i + tan(e + f*x)**2 + tan (e + f*x)*i + 1),x)*tan(e + f*x)**2*b*f + int((sqrt(tan(e + f*x)*i + 1)*sq rt( - tan(e + f*x)*i + 1)*tan(e + f*x)**5)/(tan(e + f*x)**3*i + tan(e + f* x)**2 + tan(e + f*x)*i + 1),x)*b*f + int((sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**4)/(tan(e + f*x)**3*i + tan(e + f*x)**2 + tan(e + f*x)*i + 1),x)*tan(e + f*x)**2*a*f + 4*int((sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**4)/(tan(e + f*x)**3*i + ta n(e + f*x)**2 + tan(e + f*x)*i + 1),x)*tan(e + f*x)**2*b*f*i + int((sqr...