3.386 \(\int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=360 \[ \frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (a^2 A-2 a b B-A b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b (9 a B+7 A b) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d} \]

[Out]

-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*(2*a*b*(A-B)+a^2*(A+B
)-b^2*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)+1/4*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*ln(1-2^(1/2)*t
an(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)-1/4*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(
d*x+c))/d*2^(1/2)+2*(A*a^2-A*b^2-2*B*a*b)*tan(d*x+c)^(1/2)/d+2/3*(2*A*a*b+B*a^2-B*b^2)*tan(d*x+c)^(3/2)/d+2/35
*b*(7*A*b+9*B*a)*tan(d*x+c)^(5/2)/d+2/7*b*B*tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))/d

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Rubi [A]  time = 0.58, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3607, 3630, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {2 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2 A-2 a b B-A b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b (9 a B+7 A b) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((2*a*b*(A
- B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^2*(A - B) - b^2*(A
- B) - 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^2*(A - B) - b^2
*(A - B) - 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(a^2*A - A*b^
2 - 2*a*b*B)*Sqrt[Tan[c + d*x]])/d + (2*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*b*(7*A*b + 9*
a*B)*Tan[c + d*x]^(5/2))/(35*d) + (2*b*B*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]))/(7*d)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

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 628

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 1162

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 1165

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 1168

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 3528

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 3534

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 3607

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

Rule 3630

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*} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=\frac {2 b B \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 (7 a A-5 b B)+\frac {7}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {1}{2} b (7 A b+9 a B) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \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} \left (a^2 A-A b^2-2 a b B\right )+\frac {7}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \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} \left (2 a A b+a^2 B-b^2 B\right )+\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {2}{7} \int \frac {-\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right )-\frac {7}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx\\ &=\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {4 \operatorname {Subst}\left (\int \frac {-\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right )-\frac {7}{2} \left (2 a A b+a^2 B-b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{7 d}\\ &=\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \operatorname {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 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \operatorname {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 (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \operatorname {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 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}\\ &=\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b (7 A b+9 a B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}\\ \end {align*}

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Mathematica [C]  time = 2.27, size = 178, normalized size = 0.49 \[ \frac {2 \sqrt {\tan (c+d x)} \left (35 \left (a^2 B+2 a A b-b^2 B\right ) \tan (c+d x)+105 \left (a^2 A-2 a b B-A b^2\right )+21 b (2 a B+A b) \tan ^2(c+d x)+15 b^2 B \tan ^3(c+d x)\right )+105 \sqrt [4]{-1} (a-i b)^2 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+105 \sqrt [4]{-1} (a+i b)^2 (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{105 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105*(-1)^(1/4)*(a - I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(-1)^(1/4)*(a + I*b)^2*(A +
I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(105*(a^2*A - A*b^2 - 2*a*b*B) + 35*(2*a*A*
b + a^2*B - b^2*B)*Tan[c + d*x] + 21*b*(A*b + 2*a*B)*Tan[c + d*x]^2 + 15*b^2*B*Tan[c + d*x]^3))/(105*d)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.11, size = 810, normalized size = 2.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*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)))*a*b-1/2/d*
A*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)))*a*b+1/4/d*A*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)))*b^2-1/2/d*a^2*A*2^(1/2
)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))-1/4/d*a^2*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)))-1/2/d*a^2*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))-1/2/d*a^2*B*2^(1/2)*arc
tan(1+2^(1/2)*tan(d*x+c)^(1/2))-1/2/d*a^2*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))+1/2/d*B*2^(1/2)*arctan
(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+1/2/d*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+1/2/d*A*2^(1/2)*arcta
n(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+1/2/d*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+4/5/d*B*tan(d*x+c)^(
5/2)*a*b+1/4/d*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)))*b
^2+4/3/d*A*tan(d*x+c)^(3/2)*a*b-4/d*B*tan(d*x+c)^(1/2)*a*b-2/d*A*tan(d*x+c)^(1/2)*b^2+2/d*a^2*A*tan(d*x+c)^(1/
2)-2/3/d*B*tan(d*x+c)^(3/2)*b^2+2/3/d*B*tan(d*x+c)^(3/2)*a^2+2/7/d*b^2*B*tan(d*x+c)^(7/2)+2/5/d*A*tan(d*x+c)^(
5/2)*b^2-1/4/d*a^2*A*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)
))-1/d*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b+1/d*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b-
1/d*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b+1/d*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b

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maxima [A]  time = 0.88, size = 302, normalized size = 0.84 \[ \frac {120 \, B b^{2} \tan \left (d x + c\right )^{\frac {7}{2}} + 168 \, {\left (2 \, B a b + A b^{2}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 280 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + 840 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(120*B*b^2*tan(d*x + c)^(7/2) + 168*(2*B*a*b + A*b^2)*tan(d*x + c)^(5/2) - 210*sqrt(2)*((A + B)*a^2 + 2*
(A - B)*a*b - (A + B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) - 210*sqrt(2)*((A + B)*a^2 + 2
*(A - B)*a*b - (A + B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - 105*sqrt(2)*((A - B)*a^2 -
2*(A + B)*a*b - (A - B)*b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + 105*sqrt(2)*((A - B)*a^2 -
2*(A + B)*a*b - (A - B)*b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + 280*(B*a^2 + 2*A*a*b - B*b^
2)*tan(d*x + c)^(3/2) + 840*(A*a^2 - 2*B*a*b - A*b^2)*sqrt(tan(d*x + c)))/d

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mupad [B]  time = 18.48, size = 3869, normalized size = 10.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((B^2*a^4*tan(c + d*x)^(1/2)*((12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^
4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (B^2*a*b^3)/d^2 + (B^2*a^3*b)/d^2)^(1/2)*32i)/((16*B^3*b^6)/d - (16*B^3*a^6)/d
- (112*B^3*a^2*b^4)/d + (112*B^3*a^4*b^2)/d + (32*B*a*b*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B
^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3) + (B^2*b^4*tan(c + d*x)^(1/2)*((12*B^4*a^2*b^6*d^4 - B^4*b^8*
d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (B^2*a*b^3)/d^2 + (B^2*a^3*b)/d^2
)^(1/2)*32i)/((16*B^3*b^6)/d - (16*B^3*a^6)/d - (112*B^3*a^2*b^4)/d + (112*B^3*a^4*b^2)/d + (32*B*a*b*(12*B^4*
a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3) - (B^2*a^2*b^2*
tan(c + d*x)^(1/2)*((12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)
^(1/2)/(4*d^4) - (B^2*a*b^3)/d^2 + (B^2*a^3*b)/d^2)^(1/2)*192i)/((16*B^3*b^6)/d - (16*B^3*a^6)/d - (112*B^3*a^
2*b^4)/d + (112*B^3*a^4*b^2)/d + (32*B*a*b*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^
4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3))*((12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 1
2*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (B^2*a*b^3)/d^2 + (B^2*a^3*b)/d^2)^(1/2)*2i - atan((B^2*a^4*tan(c + d*x)^(1
/2)*((B^2*a^3*b)/d^2 - (B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4
+ 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*32i)/((16*B^3*a^6)/d - (16*B^3*b^6)/d + (112*B^3*a^2*b^4)/d - (112*
B^3*a^4*b^2)/d + (32*B*a*b*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b
^2*d^4)^(1/2))/d^3) + (B^2*b^4*tan(c + d*x)^(1/2)*((B^2*a^3*b)/d^2 - (B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B
^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*32i)/((16*B^3*a^6)/d
- (16*B^3*b^6)/d + (112*B^3*a^2*b^4)/d - (112*B^3*a^4*b^2)/d + (32*B*a*b*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B
^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3) - (B^2*a^2*b^2*tan(c + d*x)^(1/2)*((B^2*a^3*
b)/d^2 - (B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b
^2*d^4)^(1/2)/(4*d^4))^(1/2)*192i)/((16*B^3*a^6)/d - (16*B^3*b^6)/d + (112*B^3*a^2*b^4)/d - (112*B^3*a^4*b^2)/
d + (32*B*a*b*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)
)/d^3))*((B^2*a^3*b)/d^2 - (B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*
d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*2i - atan((A^2*a^4*tan(c + d*x)^(1/2)*((A^2*a*b^3)/d^2 - (12*A^
4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a^3*
b)/d^2)^(1/2)*32i)/((16*A*a^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^
6*b^2*d^4)^(1/2))/d^3 - (192*A^3*a^3*b^3)/d - (16*A*b^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A
^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 + (32*A^3*a*b^5)/d + (32*A^3*a^5*b)/d) + (A^2*b^4*tan(c + d*x)
^(1/2)*((A^2*a*b^3)/d^2 - (12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^
2*d^4)^(1/2)/(4*d^4) - (A^2*a^3*b)/d^2)^(1/2)*32i)/((16*A*a^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4
- 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*A^3*a^3*b^3)/d - (16*A*b^2*(12*A^4*a^2*b^6*d^4 -
A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 + (32*A^3*a*b^5)/d + (32*A^3*a
^5*b)/d) - (A^2*a^2*b^2*tan(c + d*x)^(1/2)*((A^2*a*b^3)/d^2 - (12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4
- 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a^3*b)/d^2)^(1/2)*192i)/((16*A*a^2*(12*A^4*a^2
*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*A^3*a^3*b^3)
/d - (16*A*b^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2
))/d^3 + (32*A^3*a*b^5)/d + (32*A^3*a^5*b)/d))*((A^2*a*b^3)/d^2 - (12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*
d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a^3*b)/d^2)^(1/2)*2i - atan((A^2*a^4*tan(c
+ d*x)^(1/2)*((12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2
)/(4*d^4) + (A^2*a*b^3)/d^2 - (A^2*a^3*b)/d^2)^(1/2)*32i)/((16*A*b^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a
^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 - (16*A*a^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A
^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*A^3*a^3*b^3)/d + (32*A^3*a*b^5)/d + (3
2*A^3*a^5*b)/d) + (A^2*b^4*tan(c + d*x)^(1/2)*((12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^
4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) + (A^2*a*b^3)/d^2 - (A^2*a^3*b)/d^2)^(1/2)*32i)/((16*A*b^2*(12*A^4*a
^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 - (16*A*a^2*(12*A
^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*A^3*a^
3*b^3)/d + (32*A^3*a*b^5)/d + (32*A^3*a^5*b)/d) - (A^2*a^2*b^2*tan(c + d*x)^(1/2)*((12*A^4*a^2*b^6*d^4 - A^4*b
^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) + (A^2*a*b^3)/d^2 - (A^2*a^3*b)/
d^2)^(1/2)*192i)/((16*A*b^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*
b^2*d^4)^(1/2))/d^3 - (16*A*a^2*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*
a^6*b^2*d^4)^(1/2))/d^3 - (192*A^3*a^3*b^3)/d + (32*A^3*a*b^5)/d + (32*A^3*a^5*b)/d))*((12*A^4*a^2*b^6*d^4 - A
^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) + (A^2*a*b^3)/d^2 - (A^2*a^3
*b)/d^2)^(1/2)*2i + tan(c + d*x)^(1/2)*((2*A*a^2)/d - (2*A*b^2)/d) + tan(c + d*x)^(3/2)*((2*B*a^2)/(3*d) - (2*
B*b^2)/(3*d)) + (2*A*b^2*tan(c + d*x)^(5/2))/(5*d) + (2*B*b^2*tan(c + d*x)^(7/2))/(7*d) + (4*A*a*b*tan(c + d*x
)^(3/2))/(3*d) - (4*B*a*b*tan(c + d*x)^(1/2))/d + (4*B*a*b*tan(c + d*x)^(5/2))/(5*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{2} \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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