Integrand size = 23, antiderivative size = 198 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}} \] Output:
1/2*(a^2-2*a*b-b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/d+1/2*(a^2 -2*a*b-b^2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/d-1/2*(a^2+2*a*b-b^ 2)*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1+tan(d*x+c)))*2^(1/2)/d-2/5*a^2/d/ta n(d*x+c)^(5/2)-4/3*a*b/d/tan(d*x+c)^(3/2)+2*(a^2-b^2)/d/tan(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.41 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\frac {-6 \left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\tan ^2(c+d x)\right )-2 b \left (3 b+10 a \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right ) \tan (c+d x)\right )}{15 d \tan ^{\frac {5}{2}}(c+d x)} \] Input:
Integrate[(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(7/2),x]
Output:
(-6*(a^2 - b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] - 2*b*(3 *b + 10*a*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]*Tan[c + d*x]))/ (15*d*Tan[c + d*x]^(5/2))
Time = 0.75 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.15, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 4025, 3042, 4012, 25, 3042, 4012, 3042, 4017, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^2}{\tan (c+d x)^{7/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)}dx-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^{5/2}}dx-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int -\frac {a^2+2 b \tan (c+d x) a-b^2}{\tan ^{\frac {3}{2}}(c+d x)}dx-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {a^2+2 b \tan (c+d x) a-b^2}{\tan ^{\frac {3}{2}}(c+d x)}dx-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {a^2+2 b \tan (c+d x) a-b^2}{\tan (c+d x)^{3/2}}dx-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle -\frac {2 \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{3 d \tan ^{\frac {3}{2}}(c+d x)}\) |
Input:
Int[(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(7/2),x]
Output:
(-2*(-1/2*((a^2 - 2*a*b - b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/S qrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])) + ((a^2 + 2*a*b - b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/d - (2*a^2)/(5*d*Tan[c + d*x]^(5/2)) - (4*a*b)/(3*d*Tan[c + d*x]^(3/2)) + (2*( a^2 - b^2))/(d*Sqrt[Tan[c + d*x]])
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_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.14 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(\frac {-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}-\frac {2 a^{2}}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 \left (-a^{2}+b^{2}\right )}{\sqrt {\tan \left (d x +c \right )}}-\frac {4 a b}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}}{d}\) | \(231\) |
| default | \(\frac {-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}-\frac {2 a^{2}}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 \left (-a^{2}+b^{2}\right )}{\sqrt {\tan \left (d x +c \right )}}-\frac {4 a b}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}}{d}\) | \(231\) |
| parts | \(\frac {a^{2} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}-\frac {2}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}+\frac {2}{\sqrt {\tan \left (d x +c \right )}}\right )}{d}+\frac {b^{2} \left (-\frac {2}{\sqrt {\tan \left (d x +c \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}\right )}{d}+\frac {2 a b \left (-\frac {2}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(321\) |
Input:
int((a+b*tan(d*x+c))^2/tan(d*x+c)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2*a*b*2^(1/2)*(ln((tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(tan(d*x +c)-2^(1/2)*tan(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*ar ctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/4*(a^2-b^2)*2^(1/2)*(ln((tan(d*x+c)-2 ^(1/2)*tan(d*x+c)^(1/2)+1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))+2*arct an(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))-2/5* a^2/tan(d*x+c)^(5/2)-2*(-a^2+b^2)/tan(d*x+c)^(1/2)-4/3*a*b/tan(d*x+c)^(3/2 ))
Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (170) = 340\).
Time = 0.10 (sec) , antiderivative size = 714, normalized size of antiderivative = 3.61 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((a+b*tan(d*x+c))^2/tan(d*x+c)^(7/2),x, algorithm="fricas")
Output:
-1/30*(30*sqrt(1/2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2 )*arctan(-(2*sqrt(1/2)*(a^2 + 2*a*b - b^2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b ^2 + 4*a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^4 + 4*a^3*b + 2* a^2*b^2 - 4*a*b^3 + b^4)/d^2)*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2))/(a^4 - 6*a^2*b^2 + b^4))*tan(d*x + c)^3 + 30*sqrt(1/2)*d*sqrt(( a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2)*arctan(-(2*sqrt(1/2)*(a^2 + 2*a*b - b^2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2)*sqr t(tan(d*x + c)) - d^2*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/d^2 )*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2))/(a^4 - 6*a^2*b^2 + b^4))*tan(d*x + c)^3 - 15*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/d^2)*log(2*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4* a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) - a^2 - 2*a*b + b^2 - (a^2 + 2*a*b - b^2)*tan(d*x + c))*tan(d*x + c)^3 + 15*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2 *a^2*b^2 - 4*a*b^3 + b^4)/d^2)*log(-2*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2* a^2*b^2 - 4*a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) - a^2 - 2*a*b + b^2 - (a^ 2 + 2*a*b - b^2)*tan(d*x + c))*tan(d*x + c)^3 + 4*(10*a*b*tan(d*x + c) - 1 5*(a^2 - b^2)*tan(d*x + c)^2 + 3*a^2)*sqrt(tan(d*x + c)))/(d*tan(d*x + c)^ 3)
\[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{2}}{\tan ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**2/tan(d*x+c)**(7/2),x)
Output:
Integral((a + b*tan(c + d*x))**2/tan(c + d*x)**(7/2), x)
Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\frac {30 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 30 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 15 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 15 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \frac {8 \, {\left (10 \, a b \tan \left (d x + c\right ) - 15 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, a^{2}\right )}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{60 \, d} \] Input:
integrate((a+b*tan(d*x+c))^2/tan(d*x+c)^(7/2),x, algorithm="maxima")
Output:
1/60*(30*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt( tan(d*x + c)))) + 30*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt (2) - 2*sqrt(tan(d*x + c)))) - 15*sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)* sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + 15*sqrt(2)*(a^2 + 2*a*b - b^2)*lo g(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 8*(10*a*b*tan(d*x + c) - 15*(a^2 - b^2)*tan(d*x + c)^2 + 3*a^2)/tan(d*x + c)^(5/2))/d
\[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}{\tan \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^2/tan(d*x+c)^(7/2),x, algorithm="giac")
Output:
undef
Time = 2.99 (sec) , antiderivative size = 983, normalized size of antiderivative = 4.96 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
int((a + b*tan(c + d*x))^2/tan(c + d*x)^(7/2),x)
Output:
2*atanh((32*a^4*d^3*tan(c + d*x)^(1/2)*((a^4*1i)/(4*d^2) + (b^4*1i)/(4*d^2 ) - (a*b^3)/d^2 + (a^3*b)/d^2 - (a^2*b^2*3i)/(2*d^2))^(1/2))/(16*b^6*d^2 - 16*a^6*d^2 + a*b^5*d^2*32i + a^5*b*d^2*32i - 112*a^2*b^4*d^2 - a^3*b^3*d^ 2*192i + 112*a^4*b^2*d^2) + (32*b^4*d^3*tan(c + d*x)^(1/2)*((a^4*1i)/(4*d^ 2) + (b^4*1i)/(4*d^2) - (a*b^3)/d^2 + (a^3*b)/d^2 - (a^2*b^2*3i)/(2*d^2))^ (1/2))/(16*b^6*d^2 - 16*a^6*d^2 + a*b^5*d^2*32i + a^5*b*d^2*32i - 112*a^2* b^4*d^2 - a^3*b^3*d^2*192i + 112*a^4*b^2*d^2) - (192*a^2*b^2*d^3*tan(c + d *x)^(1/2)*((a^4*1i)/(4*d^2) + (b^4*1i)/(4*d^2) - (a*b^3)/d^2 + (a^3*b)/d^2 - (a^2*b^2*3i)/(2*d^2))^(1/2))/(16*b^6*d^2 - 16*a^6*d^2 + a*b^5*d^2*32i + a^5*b*d^2*32i - 112*a^2*b^4*d^2 - a^3*b^3*d^2*192i + 112*a^4*b^2*d^2))*(( 4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*d^2))^(1/2) - 2*atanh ((32*a^4*d^3*tan(c + d*x)^(1/2)*((a^3*b)/d^2 - (b^4*1i)/(4*d^2) - (a*b^3)/ d^2 - (a^4*1i)/(4*d^2) + (a^2*b^2*3i)/(2*d^2))^(1/2))/(16*a^6*d^2 - 16*b^6 *d^2 + a*b^5*d^2*32i + a^5*b*d^2*32i + 112*a^2*b^4*d^2 - a^3*b^3*d^2*192i - 112*a^4*b^2*d^2) + (32*b^4*d^3*tan(c + d*x)^(1/2)*((a^3*b)/d^2 - (b^4*1i )/(4*d^2) - (a*b^3)/d^2 - (a^4*1i)/(4*d^2) + (a^2*b^2*3i)/(2*d^2))^(1/2))/ (16*a^6*d^2 - 16*b^6*d^2 + a*b^5*d^2*32i + a^5*b*d^2*32i + 112*a^2*b^4*d^2 - a^3*b^3*d^2*192i - 112*a^4*b^2*d^2) - (192*a^2*b^2*d^3*tan(c + d*x)^(1/ 2)*((a^3*b)/d^2 - (b^4*1i)/(4*d^2) - (a*b^3)/d^2 - (a^4*1i)/(4*d^2) + (a^2 *b^2*3i)/(2*d^2))^(1/2))/(16*a^6*d^2 - 16*b^6*d^2 + a*b^5*d^2*32i + a^5...
\[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{4}}d x \right ) a^{2}+2 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{3}}d x \right ) a b +\left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{2}}d x \right ) b^{2} \] Input:
int((a+b*tan(d*x+c))^2/tan(d*x+c)^(7/2),x)
Output:
int(sqrt(tan(c + d*x))/tan(c + d*x)**4,x)*a**2 + 2*int(sqrt(tan(c + d*x))/ tan(c + d*x)**3,x)*a*b + int(sqrt(tan(c + d*x))/tan(c + d*x)**2,x)*b**2