\(\int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) [570]

Optimal result

Integrand size = 23, antiderivative size = 299 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d} \]

[Out]

-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^
(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)+1/4*(a+b)*(a^2-4*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+b)*(a^2-4*a*b+b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)+2*a*(a^2-3*b^2)*tan(d*x+c)^(1/
2)/d+2/3*b*(3*a^2-b^2)*tan(d*x+c)^(3/2)/d+32/35*a*b^2*tan(d*x+c)^(5/2)/d+2/7*b^2*tan(d*x+c)^(5/2)*(a+b*tan(d*x
+c))/d

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3647, 3711, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d} \]

[In]

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

[Out]

((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2
)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[
c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + T
an[c + d*x]])/(2*Sqrt[2]*d) + (2*a*(a^2 - 3*b^2)*Sqrt[Tan[c + d*x]])/d + (2*b*(3*a^2 - b^2)*Tan[c + d*x]^(3/2)
)/(3*d) + (32*a*b^2*Tan[c + d*x]^(5/2))/(35*d) + (2*b^2*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]))/(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 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {2}{7} \int \tan ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (7 a^2-5 b^2\right )+\frac {7}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)+8 a b^2 \tan ^2(c+d x)\right ) \, dx \\ & = \frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {2}{7} \int \tan ^{\frac {3}{2}}(c+d x) \left (\frac {7}{2} a \left (a^2-3 b^2\right )+\frac {7}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {2}{7} \int \sqrt {\tan (c+d x)} \left (-\frac {7}{2} b \left (3 a^2-b^2\right )+\frac {7}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {2}{7} \int \frac {-\frac {7}{2} a \left (a^2-3 b^2\right )-\frac {7}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {4 \text {Subst}\left (\int \frac {-\frac {7}{2} a \left (a^2-3 b^2\right )-\frac {7}{2} b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{7 d} \\ & = \frac {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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+b) \left (a^2-4 a b+b^2\right )\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-b) \left (a^2+4 a b+b^2\right )\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-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d} \\ & = \frac {(a+b) \left (a^2-4 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 {(a+b) \left (a^2-4 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 {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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-b) \left (a^2+4 a b+b^2\right )\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 {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \left (a^2-4 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 {(a+b) \left (a^2-4 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 {2 a \left (a^2-3 b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {32 a b^2 \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.48 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.48 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {105 \sqrt [4]{-1} (a-i b)^3 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+105 \sqrt [4]{-1} (a+i b)^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (105 \left (a^3-3 a b^2\right )-35 b \left (-3 a^2+b^2\right ) \tan (c+d x)+63 a b^2 \tan ^2(c+d x)+15 b^3 \tan ^3(c+d x)\right )}{105 d} \]

[In]

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

[Out]

(105*(-1)^(1/4)*(a - I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(-1)^(1/4)*(a + I*b)^3*ArcTanh[(-1)^(3
/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(105*(a^3 - 3*a*b^2) - 35*b*(-3*a^2 + b^2)*Tan[c + d*x] + 63*a*
b^2*Tan[c + d*x]^2 + 15*b^3*Tan[c + d*x]^3))/(105*d)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {\frac {2 b^{3} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {6 a \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+2 a^{2} b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {2 b^{3} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{3} \left (\sqrt {\tan }\left (d x +c \right )\right )-6 a \,b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(279\)
default	\(\frac {\frac {2 b^{3} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {6 a \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+2 a^{2} b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {2 b^{3} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{3} \left (\sqrt {\tan }\left (d x +c \right )\right )-6 a \,b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(279\)
parts	\(\frac {a^{3} \left (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}+\frac {b^{3} \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 {3 a \,b^{2} \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}+\frac {3 a^{2} b \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}\)	\(438\)

[In]

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

[Out]

1/d*(2/7*b^3*tan(d*x+c)^(7/2)+6/5*a*b^2*tan(d*x+c)^(5/2)+2*a^2*b*tan(d*x+c)^(3/2)-2/3*b^3*tan(d*x+c)^(3/2)+2*a
^3*tan(d*x+c)^(1/2)-6*a*b^2*tan(d*x+c)^(1/2)+1/4*(-a^3+3*a*b^2)*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*(-3*a^2*b+b^3)*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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (257) = 514\).

Time = 0.30 (sec) , antiderivative size = 1427, normalized size of antiderivative = 4.77 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/210*(105*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6
+ 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(((3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b
^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*
b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 25
5*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^1
2)*sqrt(tan(d*x + c))) - 105*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8
*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(-((3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a
^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (a^9 - 18*a^7*b^2 + 60*a^5*b^4
- 46*a^3*b^6 + 3*a*b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4
- 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8
+ 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c))) - 105*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 - 30
*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(((3*a^2*b - b^3)*d^3*
sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (a^9 - 18*a^7
*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 - 30*a^10
*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8
*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c))) + 105*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^
2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(-((
3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d
^4) - (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sq
rt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 1
2*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c))) + 4*(15*b^3*tan(d*x + c)^3 + 63
*a*b^2*tan(d*x + c)^2 + 105*a^3 - 315*a*b^2 + 35*(3*a^2*b - b^3)*tan(d*x + c))*sqrt(tan(d*x + c)))/d

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.86 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {120 \, b^{3} \tan \left (d x + c\right )^{\frac {7}{2}} + 504 \, a b^{2} \tan \left (d x + c\right )^{\frac {5}{2}} - 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 280 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + 840 \, {\left (a^{3} - 3 \, a b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{420 \, d} \]

[In]

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

[Out]

1/420*(120*b^3*tan(d*x + c)^(7/2) + 504*a*b^2*tan(d*x + c)^(5/2) - 210*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)
*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) - 210*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/
2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - 105*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)*sqrt(tan
(d*x + c)) + tan(d*x + c) + 1) + 105*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)*sqrt(tan(d*x + c)) +
tan(d*x + c) + 1) + 280*(3*a^2*b - b^3)*tan(d*x + c)^(3/2) + 840*(a^3 - 3*a*b^2)*sqrt(tan(d*x + c)))/d

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 9.67 (sec) , antiderivative size = 1729, normalized size of antiderivative = 5.78 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

tan(c + d*x)^(1/2)*((2*a^3)/d - (6*a*b^2)/d) - tan(c + d*x)^(3/2)*((2*b^3)/(3*d) - (2*a^2*b)/d) - atan((((8*(4
*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d
^2))^(1/2))/d^3 - (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-(6*a*b^5 + 6*a^5*b - a^
6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2)*1i - ((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(
6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2))/d^3 + (16*tan(c
+ d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i
- 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2)*1i)/((16*(3*a^8*b - b^9 + 6*a^4*b^5 + 8*a^6*b^3))/d^3 + ((8*(4*a^3*
d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^
(1/2))/d^3 - (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i
+ b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2) + ((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5
+ 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2))/d^3 + (16*tan(c + d*x)^(
1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3
*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2)))*(-(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b
^2*15i)/(4*d^2))^(1/2)*2i - atan((((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b + a^6*1i - b^6*1i + a^2*
b^4*15i - 20*a^3*b^3 - a^4*b^2*15i)/(4*d^2))^(1/2))/d^3 - (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*
a^4*b^2))/d^2)*(-(6*a*b^5 + 6*a^5*b + a^6*1i - b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i)/(4*d^2))^(1/2)
*1i - ((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b + a^6*1i - b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b
^2*15i)/(4*d^2))^(1/2))/d^3 + (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-(6*a*b^5 +
6*a^5*b + a^6*1i - b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i)/(4*d^2))^(1/2)*1i)/((16*(3*a^8*b - b^9 + 6
*a^4*b^5 + 8*a^6*b^3))/d^3 + ((8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b + a^6*1i - b^6*1i + a^2*b^4*1
5i - 20*a^3*b^3 - a^4*b^2*15i)/(4*d^2))^(1/2))/d^3 - (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b
^2))/d^2)*(-(6*a*b^5 + 6*a^5*b + a^6*1i - b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i)/(4*d^2))^(1/2) + ((
8*(4*a^3*d^2 - 12*a*b^2*d^2)*(-(6*a*b^5 + 6*a^5*b + a^6*1i - b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i)/
(4*d^2))^(1/2))/d^3 + (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-(6*a*b^5 + 6*a^5*b
+ a^6*1i - b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i)/(4*d^2))^(1/2)))*(-(6*a*b^5 + 6*a^5*b + a^6*1i - b
^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i)/(4*d^2))^(1/2)*2i + (2*b^3*tan(c + d*x)^(7/2))/(7*d) + (6*a*b^
2*tan(c + d*x)^(5/2))/(5*d)