\(\int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx\) [562]

Optimal result

Integrand size = 23, antiderivative size = 268 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d} \]

[Out]

-1/2*(a^2-2*a*b-b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*tan(d*
x+c)^(1/2))/d*2^(1/2)-1/4*(a^2+2*a*b-b^2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)+1/4*(a^2+2*a*b-b
^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)-4*a*b*tan(d*x+c)^(1/2)/d+2/3*(a^2-b^2)*tan(d*x+c)^(3/2
)/d+4/5*a*b*tan(d*x+c)^(5/2)/d+2/7*b^2*tan(d*x+c)^(7/2)/d

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3624, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d} \]

[In]

Int[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2,x]

[Out]

((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqr
t[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]
])/(2*Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (4
*a*b*Sqrt[Tan[c + d*x]])/d + (2*(a^2 - b^2)*Tan[c + d*x]^(3/2))/(3*d) + (4*a*b*Tan[c + d*x]^(5/2))/(5*d) + (2*
b^2*Tan[c + d*x]^(7/2))/(7*d)

Rule 210

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])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(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]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[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]

Rule 3624

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}+\int \tan ^{\frac {5}{2}}(c+d x) \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx \\ & = \frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}+\int \tan ^{\frac {3}{2}}(c+d x) \left (-2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}+\int \sqrt {\tan (c+d x)} \left (-a^2+b^2-2 a b \tan (c+d x)\right ) \, dx \\ & = -\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}+\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}+\frac {2 \text {Subst}\left (\int \frac {2 a b+\left (-a^2+b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d} \\ & = -\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,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 ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {4 a b \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a b \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.50 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {-105 (-1)^{3/4} (a-i b)^2 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+105 (-1)^{3/4} (a+i b)^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (-210 a b+35 \left (a^2-b^2\right ) \tan (c+d x)+42 a b \tan ^2(c+d x)+15 b^2 \tan ^3(c+d x)\right )}{105 d} \]

[In]

Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2,x]

[Out]

(-105*(-1)^(3/4)*(a - I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(-1)^(3/4)*(a + I*b)^2*ArcTanh[(-1)^(
3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(-210*a*b + 35*(a^2 - b^2)*Tan[c + d*x] + 42*a*b*Tan[c + d*x]^
2 + 15*b^2*Tan[c + d*x]^3))/(105*d)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 a b \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 a^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {2 b^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 a b \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(250\)
default	\(\frac {\frac {2 b^{2} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 a b \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 a^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {2 b^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 a b \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(250\)
parts	\(\frac {a^{2} \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {b^{2} \left (\frac {2 \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}-\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {2 a b \left (\frac {2 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-2 \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(331\)

[In]

int(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/7*b^2*tan(d*x+c)^(7/2)+4/5*a*b*tan(d*x+c)^(5/2)+2/3*a^2*tan(d*x+c)^(3/2)-2/3*b^2*tan(d*x+c)^(3/2)-4*a*b
*tan(d*x+c)^(1/2)+1/2*a*b*2^(1/2)*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(
d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/4*(-a^2+b^2)*2^(1/2)*(l
n((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+
c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1037 vs. \(2 (226) = 452\).

Time = 0.27 (sec) , antiderivative size = 1037, normalized size of antiderivative = 3.87 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {105 \, d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - 105 \, d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - 105 \, d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) + 105 \, d \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left ({\left (a^{2} - b^{2}\right )} d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + 2 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} d\right )} \sqrt {\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - 4 \, {\left (15 \, b^{2} \tan \left (d x + c\right )^{3} + 42 \, a b \tan \left (d x + c\right )^{2} - 210 \, a b + 35 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {\tan \left (d x + c\right )}}{210 \, d} \]

[In]

integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/210*(105*d*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2
)*log(((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4) - 2*(a^5*b - 6*a^3*b^3 +
a*b^5)*d)*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2) +
(a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt(tan(d*x + c))) - 105*d*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqr
t(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2)*log(-((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2
+ 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4) - 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d)*sqrt((4*a^3*b - 4*a*b^3 + d^2*sqrt(-
(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2) + (a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^
8)*sqrt(tan(d*x + c))) - 105*d*sqrt((4*a^3*b - 4*a*b^3 - d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6
+ b^8)/d^4))/d^2)*log(((a^2 - b^2)*d^3*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4) + 2*(a^5
*b - 6*a^3*b^3 + a*b^5)*d)*sqrt((4*a^3*b - 4*a*b^3 - d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b
^8)/d^4))/d^2) + (a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt(tan(d*x + c))) + 105*d*sqrt((4*a^3*b -
4*a*b^3 - d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2)*log(-((a^2 - b^2)*d^3*sqrt(-
(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4) + 2*(a^5*b - 6*a^3*b^3 + a*b^5)*d)*sqrt((4*a^3*b - 4*a
*b^3 - d^2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/d^4))/d^2) + (a^8 - 4*a^6*b^2 - 10*a^4*b^4
- 4*a^2*b^6 + b^8)*sqrt(tan(d*x + c))) - 4*(15*b^2*tan(d*x + c)^3 + 42*a*b*tan(d*x + c)^2 - 210*a*b + 35*(a^2
- b^2)*tan(d*x + c))*sqrt(tan(d*x + c)))/d

Sympy [F]

\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \tan ^{\frac {5}{2}}{\left (c + d x \right )}\, dx \]

[In]

integrate(tan(d*x+c)**(5/2)*(a+b*tan(d*x+c))**2,x)

[Out]

Integral((a + b*tan(c + d*x))**2*tan(c + d*x)**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.81 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {120 \, b^{2} \tan \left (d x + c\right )^{\frac {7}{2}} + 336 \, a b \tan \left (d x + c\right )^{\frac {5}{2}} - 210 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 210 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 105 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 105 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 1680 \, a b \sqrt {\tan \left (d x + c\right )} + 280 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{420 \, d} \]

[In]

integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/420*(120*b^2*tan(d*x + c)^(7/2) + 336*a*b*tan(d*x + c)^(5/2) - 210*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(1/2*sq
rt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) - 210*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqr
t(tan(d*x + c)))) + 105*sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 105*s
qrt(2)*(a^2 + 2*a*b - b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 1680*a*b*sqrt(tan(d*x + c)) +
280*(a^2 - b^2)*tan(d*x + c)^(3/2))/d

Giac [F(-1)]

Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 8.31 (sec) , antiderivative size = 995, normalized size of antiderivative = 3.71 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\text {Too large to display} \]

[In]

int(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^2,x)

[Out]

tan(c + d*x)^(3/2)*((2*a^2)/(3*d) - (2*b^2)/(3*d)) - atan((a^4*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)*32i)/((16*a^6)/d - (16*b^6)/d + (a*b^5*32i)
/d + (a^5*b*32i)/d + (112*a^2*b^4)/d - (a^3*b^3*192i)/d - (112*a^4*b^2)/d) + (b^4*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)*32i)/((16*a^6)/d - (16*b
^6)/d + (a*b^5*32i)/d + (a^5*b*32i)/d + (112*a^2*b^4)/d - (a^3*b^3*192i)/d - (112*a^4*b^2)/d) - (a^2*b^2*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)*1
92i)/((16*a^6)/d - (16*b^6)/d + (a*b^5*32i)/d + (a^5*b*32i)/d + (112*a^2*b^4)/d - (a^3*b^3*192i)/d - (112*a^4*
b^2)/d))*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*d^2))^(1/2)*2i + atan((a^4*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)*32i)/((16*b^6)
/d - (16*a^6)/d + (a*b^5*32i)/d + (a^5*b*32i)/d - (112*a^2*b^4)/d - (a^3*b^3*192i)/d + (112*a^4*b^2)/d) + (b^4
*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)*32i)/((16*b^6)/d - (16*a^6)/d + (a*b^5*32i)/d + (a^5*b*32i)/d - (112*a^2*b^4)/d - (a^3*b^3*192i)/d + (112
*a^4*b^2)/d) - (a^2*b^2*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)*192i)/((16*b^6)/d - (16*a^6)/d + (a*b^5*32i)/d + (a^5*b*32i)/d - (112*a^2*b^4)/d -
(a^3*b^3*192i)/d + (112*a^4*b^2)/d))*((4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*d^2))^(1/2)*2i +
(2*b^2*tan(c + d*x)^(7/2))/(7*d) - (4*a*b*tan(c + d*x)^(1/2))/d + (4*a*b*tan(c + d*x)^(5/2))/(5*d)