\(\int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [513]

Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {i (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d} \]

[Out]

I*(a-I*b)^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-I*(a+I*b)^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)
/(a+I*b)^(1/2))/d-2*b*(a+b*tan(d*x+c))^(1/2)/d+2/5*(a+b*tan(d*x+c))^(5/2)/b/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3624, 3563, 3620, 3618, 65, 214} \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {i (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

Int[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2),x]

[Out]

(I*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*
Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*b*Sqrt[a + b*Tan[c + d*x]])/d + (2*(a + b*Tan[c + d*x])^(5/2))/(5*b*d)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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]

Rule 3563

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[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]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(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]

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 (a+b \tan (c+d x))^{5/2}}{5 b d}-\int (a+b \tan (c+d x))^{3/2} \, dx \\ & = -\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d}-\int \frac {a^2-b^2+2 a b \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d}-\frac {1}{2} (a-i b)^2 \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d}-\frac {\left (i (a-i b)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac {\left (i (a+i b)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d} \\ & = -\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d}+\frac {(a-i b)^2 \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}+\frac {(a+i b)^2 \text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d} \\ & = \frac {i (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.17 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {\frac {2 (a+b \tan (c+d x))^{5/2}}{b}+5 (i a+b) \left (\sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-\sqrt {a+b \tan (c+d x)}\right )+5 i (a+i b) \left (-\sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\sqrt {a+b \tan (c+d x)}\right )}{5 d} \]

[In]

Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2),x]

[Out]

((2*(a + b*Tan[c + d*x])^(5/2))/b + 5*(I*a + b)*(Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]
- Sqrt[a + b*Tan[c + d*x]]) + (5*I)*(a + I*b)*(-(Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]
]) + Sqrt[a + b*Tan[c + d*x]]))/(5*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(841\) vs. \(2(111)=222\).

Time = 0.32 (sec) , antiderivative size = 842, normalized size of antiderivative = 6.24

method	result	size
derivativedivides	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b d}-\frac {2 b \sqrt {a +b \tan \left (d x +c \right )}}{d}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a}{4 d b}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}+\frac {b \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a}{4 d b}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {2 b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(842\)
default	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b d}-\frac {2 b \sqrt {a +b \tan \left (d x +c \right )}}{d}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a}{4 d b}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}+\frac {b \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a}{4 d b}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {2 b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(842\)

[In]

int(tan(d*x+c)^2*(a+b*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (105) = 210\).

Time = 0.26 (sec) , antiderivative size = 806, normalized size of antiderivative = 5.97 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {5 \, b d \sqrt {-\frac {a^{3} - 3 \, a b^{2} + d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left (-{\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} + {\left (a d^{3} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}} + {\left (3 \, a^{2} b^{2} - b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} - 3 \, a b^{2} + d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) - 5 \, b d \sqrt {-\frac {a^{3} - 3 \, a b^{2} + d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left (-{\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} - {\left (a d^{3} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}} + {\left (3 \, a^{2} b^{2} - b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} - 3 \, a b^{2} + d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) - 5 \, b d \sqrt {-\frac {a^{3} - 3 \, a b^{2} - d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left (-{\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} + {\left (a d^{3} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}} - {\left (3 \, a^{2} b^{2} - b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} - 3 \, a b^{2} - d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) + 5 \, b d \sqrt {-\frac {a^{3} - 3 \, a b^{2} - d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left (-{\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} - {\left (a d^{3} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}} - {\left (3 \, a^{2} b^{2} - b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} - 3 \, a b^{2} - d^{2} \sqrt {-\frac {9 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) - 4 \, {\left (b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2} - 5 \, b^{2}\right )} \sqrt {b \tan \left (d x + c\right ) + a}}{10 \, b d} \]

[In]

integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/10*(5*b*d*sqrt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4*b + 2*a^2*b
^3 - b^5)*sqrt(b*tan(d*x + c) + a) + (a*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^2*b^2 - b^4)*d)*sq
rt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - 5*b*d*sqrt(-(a^3 - 3*a*b^2 + d^2*sq
rt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) - (a*d^
3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^2*b^2 - b^4)*d)*sqrt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2
- 6*a^2*b^4 + b^6)/d^4))/d^2)) - 5*b*d*sqrt(-(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^
2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) + (a*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4)
- (3*a^2*b^2 - b^4)*d)*sqrt(-(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) + 5*b*d*sqr
t(-(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b
*tan(d*x + c) + a) - (a*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^2*b^2 - b^4)*d)*sqrt(-(a^3 - 3*a*b
^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - 4*(b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2 -
5*b^2)*sqrt(b*tan(d*x + c) + a))/(b*d)

Sympy [F]

\[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(tan(d*x+c)**2*(a+b*tan(d*x+c))**(3/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(3/2)*tan(c + d*x)**2, x)

Maxima [F]

\[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^(3/2)*tan(d*x + c)^2, x)

Giac [F(-1)]

Timed out. \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 10.71 (sec) , antiderivative size = 1141, normalized size of antiderivative = 8.45 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]

[In]

int(tan(c + d*x)^2*(a + b*tan(c + d*x))^(3/2),x)

[Out]

atan((b^6*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2) - a^3/(4*d^2) + (a^2*b*3i)/(4*d^2))
^(1/2)*32i)/((b^8*16i)/d - (16*a*b^7)/d - (a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/d)
+ (32*a*b^5*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2) - a^3/(4*d^2) + (a^2*b*3i)/(4*d^
2))^(1/2))/((b^8*16i)/d - (16*a*b^7)/d - (a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/d)
- (a^2*b^4*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2) - a^3/(4*d^2) + (a^2*b*3i)/(4*d^2)
)^(1/2)*96i)/((b^8*16i)/d - (16*a*b^7)/d - (a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/d
) - (96*a^3*b^3*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2) - a^3/(4*d^2) + (a^2*b*3i)/(4
*d^2))^(1/2))/((b^8*16i)/d - (16*a*b^7)/d - (a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/
d))*((3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2)*2i + atan((b^6*(a + b*tan(c + d*x))^(1/2)*((b^3*1i)/(4
*d^2) - a^3/(4*d^2) + (3*a*b^2)/(4*d^2) - (a^2*b*3i)/(4*d^2))^(1/2)*32i)/((a^2*b^6*32i)/d - (16*a*b^7)/d - (b^
8*16i)/d + (32*a^3*b^5)/d + (a^4*b^4*48i)/d + (48*a^5*b^3)/d) - (32*a*b^5*(a + b*tan(c + d*x))^(1/2)*((b^3*1i)
/(4*d^2) - a^3/(4*d^2) + (3*a*b^2)/(4*d^2) - (a^2*b*3i)/(4*d^2))^(1/2))/((a^2*b^6*32i)/d - (16*a*b^7)/d - (b^8
*16i)/d + (32*a^3*b^5)/d + (a^4*b^4*48i)/d + (48*a^5*b^3)/d) - (a^2*b^4*(a + b*tan(c + d*x))^(1/2)*((b^3*1i)/(
4*d^2) - a^3/(4*d^2) + (3*a*b^2)/(4*d^2) - (a^2*b*3i)/(4*d^2))^(1/2)*96i)/((a^2*b^6*32i)/d - (16*a*b^7)/d - (b
^8*16i)/d + (32*a^3*b^5)/d + (a^4*b^4*48i)/d + (48*a^5*b^3)/d) + (96*a^3*b^3*(a + b*tan(c + d*x))^(1/2)*((b^3*
1i)/(4*d^2) - a^3/(4*d^2) + (3*a*b^2)/(4*d^2) - (a^2*b*3i)/(4*d^2))^(1/2))/((a^2*b^6*32i)/d - (16*a*b^7)/d - (
b^8*16i)/d + (32*a^3*b^5)/d + (a^4*b^4*48i)/d + (48*a^5*b^3)/d))*((3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))
^(1/2)*2i + ((2*a^2)/(b*d) - (2*(a^2 + b^2))/(b*d))*(a + b*tan(c + d*x))^(1/2) + (2*(a + b*tan(c + d*x))^(5/2)
)/(5*b*d)