3.4.0 ∫ (a + b tan(c + dx))3∕2(A + B tan(c + dx)) dx [327]

Integrand size = 25, antiderivative size = 150 \[ \int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {(a-i b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 (A b+a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 d} \]

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

d*x+c))^(1/2)/(a+I*b)^(1/2))/d+2*(A*b+B*a)*(a+b*tan(d*x+c))^(1/2)/d+2/3*B*(a+b*tan(d*x+c))^(3/2)/d

Rubi [A] (veriﬁed)

Time = 0.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3609, 3620, 3618, 65, 214} \[ \int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {(a-i b)^{3/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 (a B+A b) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 d} \]

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

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

ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*(A*b + a*B)*Sqrt[a + b*Tan[c + d*x]])/d + (2*B*(a + b*

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,

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

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]

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]

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]

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

Mathematica [A] (veriﬁed)

Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {-3 i (a-i b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+3 i (a+i b)^{3/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 \sqrt {a+b \tan (c+d x)} (3 A b+4 a B+b B \tan (c+d x))}{3 d} \]

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

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

I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(3*A*b + 4*a*B + b*B*Tan[c

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1656\) vs. \(2(126)=252\).

Time = 0.09 (sec) , antiderivative size = 1657, normalized size of antiderivative = 11.05


method	result	size

parts	\(\text {Expression too large to display}\)	\(1657\)
derivativedivides	\(\text {Expression too large to display}\)	\(1665\)
default	\(\text {Expression too large to display}\)	\(1665\)

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

A*(2*b*(a+b*tan(d*x+c))^(1/2)/d-1/4/b/d*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))*(a^2+b^2)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/4/b/d*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*b/d*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)-b/d/(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*b/d/(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+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)+B*(2/3/d*(a+b*tan(d*x+c))^(3/2)+2/d*(a+

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

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3058 vs. \(2 (120) = 240\).

Time = 0.47 (sec) , antiderivative size = 3058, normalized size of antiderivative = 20.39 \[ \int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

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

1/6*(3*d*sqrt((6*A*B*a^2*b - 2*A*B*b^3 - (A^2 - B^2)*a^3 + 3*(A^2 - B^2)*a*b^2 + d^2*sqrt(-(4*A^2*B^2*a^6 + 12

*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*

B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4))/d^2)*log((2*(A^3*B + A*B^3)

*a^5 + 3*(A^4 - B^4)*a^4*b - 4*(A^3*B + A*B^3)*a^3*b^2 + 2*(A^4 - B^4)*a^2*b^3 - 6*(A^3*B + A*B^3)*a*b^4 - (A^

4 - B^4)*b^5)*sqrt(b*tan(d*x + c) + a) + ((A*a - B*b)*d^3*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*

(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^

3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4) - (2*A*B^2*a^4 + (5*A^2*B - 3*B^3)*a^3*b + 3*(A^3 - 3*A

*B^2)*a^2*b^2 - (7*A^2*B - B^3)*a*b^3 - (A^3 - A*B^2)*b^4)*d)*sqrt((6*A*B*a^2*b - 2*A*B*b^3 - (A^2 - B^2)*a^3

+ 3*(A^2 - B^2)*a*b^2 + d^2*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a

^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*

A^2*B^2 + B^4)*b^6)/d^4))/d^2)) - 3*d*sqrt((6*A*B*a^2*b - 2*A*B*b^3 - (A^2 - B^2)*a^3 + 3*(A^2 - B^2)*a*b^2 +

d^2*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*B

^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4)

)/d^2)*log((2*(A^3*B + A*B^3)*a^5 + 3*(A^4 - B^4)*a^4*b - 4*(A^3*B + A*B^3)*a^3*b^2 + 2*(A^4 - B^4)*a^2*b^3 -

6*(A^3*B + A*B^3)*a*b^4 - (A^4 - B^4)*b^5)*sqrt(b*tan(d*x + c) + a) - ((A*a - B*b)*d^3*sqrt(-(4*A^2*B^2*a^6 +

12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^

2*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4) - (2*A*B^2*a^4 + (5*A^2*B

- 3*B^3)*a^3*b + 3*(A^3 - 3*A*B^2)*a^2*b^2 - (7*A^2*B - B^3)*a*b^3 - (A^3 - A*B^2)*b^4)*d)*sqrt((6*A*B*a^2*b -

2*A*B*b^3 - (A^2 - B^2)*a^3 + 3*(A^2 - B^2)*a*b^2 + d^2*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*(

3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^3

*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4))/d^2)) - 3*d*sqrt((6*A*B*a^2*b - 2*A*B*b^3 - (A^2 - B^2)

*a^3 + 3*(A^2 - B^2)*a*b^2 - d^2*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B

^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4

- 2*A^2*B^2 + B^4)*b^6)/d^4))/d^2)*log((2*(A^3*B + A*B^3)*a^5 + 3*(A^4 - B^4)*a^4*b - 4*(A^3*B + A*B^3)*a^3*b

^2 + 2*(A^4 - B^4)*a^2*b^3 - 6*(A^3*B + A*B^3)*a*b^4 - (A^4 - B^4)*b^5)*sqrt(b*tan(d*x + c) + a) + ((A*a - B*b

)*d^3*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A

*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^

4) + (2*A*B^2*a^4 + (5*A^2*B - 3*B^3)*a^3*b + 3*(A^3 - 3*A*B^2)*a^2*b^2 - (7*A^2*B - B^3)*a*b^3 - (A^3 - A*B^2

)*b^4)*d)*sqrt((6*A*B*a^2*b - 2*A*B*b^3 - (A^2 - B^2)*a^3 + 3*(A^2 - B^2)*a*b^2 - d^2*sqrt(-(4*A^2*B^2*a^6 + 1

2*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2

*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4))/d^2)) + 3*d*sqrt((6*A*B*a^

2*b - 2*A*B*b^3 - (A^2 - B^2)*a^3 + 3*(A^2 - B^2)*a*b^2 - d^2*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b

+ 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12

*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4))/d^2)*log((2*(A^3*B + A*B^3)*a^5 + 3*(A^4 - B^4)*a^

4*b - 4*(A^3*B + A*B^3)*a^3*b^2 + 2*(A^4 - B^4)*a^2*b^3 - 6*(A^3*B + A*B^3)*a*b^4 - (A^4 - B^4)*b^5)*sqrt(b*ta

n(d*x + c) + a) - ((A*a - B*b)*d^3*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3

*B^4)*a^4*b^2 - 40*(A^3*B - A*B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A

^4 - 2*A^2*B^2 + B^4)*b^6)/d^4) + (2*A*B^2*a^4 + (5*A^2*B - 3*B^3)*a^3*b + 3*(A^3 - 3*A*B^2)*a^2*b^2 - (7*A^2*

B - B^3)*a*b^3 - (A^3 - A*B^2)*b^4)*d)*sqrt((6*A*B*a^2*b - 2*A*B*b^3 - (A^2 - B^2)*a^3 + 3*(A^2 - B^2)*a*b^2 -

d^2*sqrt(-(4*A^2*B^2*a^6 + 12*(A^3*B - A*B^3)*a^5*b + 3*(3*A^4 - 14*A^2*B^2 + 3*B^4)*a^4*b^2 - 40*(A^3*B - A*

B^3)*a^3*b^3 - 6*(A^4 - 8*A^2*B^2 + B^4)*a^2*b^4 + 12*(A^3*B - A*B^3)*a*b^5 + (A^4 - 2*A^2*B^2 + B^4)*b^6)/d^4

))/d^2)) + 4*(B*b*tan(d*x + c) + 4*B*a + 3*A*b)*sqrt(b*tan(d*x + c) + a))/d

Sympy [F]

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

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

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

Maxima [F(-2)]

Exception generated. \[ \int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Exception raised: ValueError} \]

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

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor

Giac [F(-1)]

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

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

Mupad [B] (veriﬁcation not implemented)

Time = 20.84 (sec) , antiderivative size = 2823, normalized size of antiderivative = 18.82 \[ \int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

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

log((((16*b^2*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/d^4)^(1/2)*(A*b^3 + A*a^

2*b - a*d*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*

x))^(1/2)))/d - (16*A^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2)*(((-A^4*b^2*d^4*(3*a^2 -

b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2 - (16*A^3*a*b^3*(a^2 + b^2)^2)/d^3)*((6*A^4*a^2*b

^4*d^4 - A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a^3)/(4*d^2) + (3*A^2*a*b^2)/(4*d^2))^(1/2) - l

og((((16*b^2*(((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/d^4)^(1/2)*(A*b^3 + A*a^2

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

))^(1/2)))/d + (16*A^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2)*(((-A^4*b^2*d^4*(3*a^2 - b

^2)^2)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2 - (16*A^3*a*b^3*(a^2 + b^2)^2)/d^3)*(((6*A^4*a^2*b

^4*d^4 - A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2) - A^2*a^3*d^2 + 3*A^2*a*b^2*d^2)/(4*d^4))^(1/2) - log((((16*b^

2*(-((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1/2)*(A*b^3 + A*a^2*b + a*d*(

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

)/d + (16*A^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2)*(-((-A^4*b^2*d^4*(3*a^2 - b^2)^2)^(

1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2 - (16*A^3*a*b^3*(a^2 + b^2)^2)/d^3)*(-((6*A^4*a^2*b^4*d^4

- A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/(4*d^4))^(1/2) + log((((16*b^2*(-((-

A^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1/2)*(A*b^3 + A*a^2*b - a*d*(-((-A^4

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

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

A^2*a^3*d^2 - 3*A^2*a*b^2*d^2)/d^4)^(1/2))/2 - (16*A^3*a*b^3*(a^2 + b^2)^2)/d^3)*((3*A^2*a*b^2)/(4*d^2) - (A^2

*a^3)/(4*d^2) - (6*A^4*a^2*b^4*d^4 - A^4*b^6*d^4 - 9*A^4*a^4*b^2*d^4)^(1/2)/(4*d^4))^(1/2) - log(- ((((-B^4*b^

2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/d^4)^(1/2)*((16*B^2*b^2*(a + b*tan(c + d*x))^(1/

2)*(a^4 + b^4 - 6*a^2*b^2))/d^2 - (16*a*b^2*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2

*d^2)/d^4)^(1/2)*(B*a^2 + B*b^2 + d*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/d^

4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/d))/2 - (8*B^3*b^2*(a^2 - b^2)*(a^2 + b^2)^2)/d^3)*(((6*B^4*a^2*b^4*d^4

- B^4*b^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/(4*d^4))^(1/2) - log(- ((-((-B^4*b^2

*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2)*((16*B^2*b^2*(a + b*tan(c + d*x))^(1/2

)*(a^4 + b^4 - 6*a^2*b^2))/d^2 - (16*a*b^2*(-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2

*d^2)/d^4)^(1/2)*(B*a^2 + B*b^2 + d*(-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d

^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/d))/2 - (8*B^3*b^2*(a^2 - b^2)*(a^2 + b^2)^2)/d^3)*(-((6*B^4*a^2*b^4*d^

4 - B^4*b^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/(4*d^4))^(1/2) + log(((((-B^4*b^2*

d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/d^4)^(1/2)*((16*B^2*b^2*(a + b*tan(c + d*x))^(1/2)

*(a^4 + b^4 - 6*a^2*b^2))/d^2 + (16*a*b^2*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d

^2)/d^4)^(1/2)*(B*a^2 + B*b^2 - d*(((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) + B^2*a^3*d^2 - 3*B^2*a*b^2*d^2)/d^4)

^(1/2)*(a + b*tan(c + d*x))^(1/2)))/d))/2 - (8*B^3*b^2*(a^2 - b^2)*(a^2 + b^2)^2)/d^3)*((6*B^4*a^2*b^4*d^4 - B

^4*b^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2)/(4*d^4) + (B^2*a^3)/(4*d^2) - (3*B^2*a*b^2)/(4*d^2))^(1/2) + log(((-((-B

^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*d^2)/d^4)^(1/2)*((16*B^2*b^2*(a + b*tan(c + d*x)

)^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2 + (16*a*b^2*(-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2

*a*b^2*d^2)/d^4)^(1/2)*(B*a^2 + B*b^2 - d*(-((-B^4*b^2*d^4*(3*a^2 - b^2)^2)^(1/2) - B^2*a^3*d^2 + 3*B^2*a*b^2*

d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/d))/2 - (8*B^3*b^2*(a^2 - b^2)*(a^2 + b^2)^2)/d^3)*((B^2*a^3)/(4*

d^2) - (6*B^4*a^2*b^4*d^4 - B^4*b^6*d^4 - 9*B^4*a^4*b^2*d^4)^(1/2)/(4*d^4) - (3*B^2*a*b^2)/(4*d^2))^(1/2) + (2

*B*(a + b*tan(c + d*x))^(3/2))/(3*d) + (2*A*b*(a + b*tan(c + d*x))^(1/2))/d + (2*B*a*(a + b*tan(c + d*x))^(1/2

\(\int (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [327]

Optimal result