3.4.0 ∫ tan(c + dx)(a + b tan(c + dx))5∕2(A + B tan(c + dx)) dx [333]

Integrand size = 31, antiderivative size = 213 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {(a-i b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a A-b B) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 A (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 B (a+b \tan (c+d x))^{7/2}}{7 b d} \]

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

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

^(3/2)/d+2/5*A*(a+b*tan(d*x+c))^(5/2)/d+2/7*B*(a+b*tan(d*x+c))^(7/2)/b/d

Rubi [A] (veriﬁed)

Time = 0.61 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3673, 3609, 3620, 3618, 65, 214} \[ \int \tan (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {2 \left (a^2 A-2 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}-\frac {(a-i b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 (a A-b B) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 A (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 B (a+b \tan (c+d x))^{7/2}}{7 b d} \]

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

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

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

+ (2*(a*A - b*B)*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*A*(a + b*Tan[c + d*x])^(5/2))/(5*d) + (2*B*(a + b*Tan[

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]

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(

e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B*d*((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*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b

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

Mathematica [A] (veriﬁed)

Time = 1.74 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.21 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\frac {4 B (a+b \tan (c+d x))^{7/2}}{b}-7 i (i A+B) \left (\frac {2}{5} (a+b \tan (c+d x))^{5/2}+\frac {2}{3} (a-i b) \left (-3 (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+\sqrt {a+b \tan (c+d x)} (4 a-3 i b+b \tan (c+d x))\right )\right )-7 i (i A-B) \left (\frac {2}{5} (a+b \tan (c+d x))^{5/2}+\frac {2}{3} (a+i b) \left (-3 (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\sqrt {a+b \tan (c+d x)} (4 a+3 i b+b \tan (c+d x))\right )\right )}{14 d} \]

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

((4*B*(a + b*Tan[c + d*x])^(7/2))/b - (7*I)*(I*A + B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a - I*b)*(-3*(a

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

[c + d*x])))/3) - (7*I)*(I*A - B)*((2*(a + b*Tan[c + d*x])^(5/2))/5 + (2*(a + I*b)*(-3*(a + I*b)^(3/2)*ArcTanh

[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a + (3*I)*b + b*Tan[c + d*x])))/3))/(14

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(2406\) vs. \(2(181)=362\).

Time = 0.12 (sec) , antiderivative size = 2407, normalized size of antiderivative = 11.30


method	result	size

parts	\(\text {Expression too large to display}\)	\(2407\)
derivativedivides	\(\text {Expression too large to display}\)	\(2426\)
default	\(\text {Expression too large to display}\)	\(2426\)

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

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

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

^2+b^2)^(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*(a^2+b^2)^(1/2)*a^2+1/d*b^2/(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*(a^2+b^2)^(1/2)-1/2*A/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-A/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+A/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)*b^2+B*(2/7/b/d

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

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

an(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)+1/4/b/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^3-3/4*b/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/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+3*b/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^3/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*

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4916 vs. \(2 (175) = 350\).

Time = 0.84 (sec) , antiderivative size = 4916, normalized size of antiderivative = 23.08 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

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

1/210*(105*b*d*sqrt(-(10*A*B*a^4*b - 20*A*B*a^2*b^3 + 2*A*B*b^5 - (A^2 - B^2)*a^5 + 10*(A^2 - B^2)*a^3*b^2 - 5

*(A^2 - B^2)*a*b^4 + d^2*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8

*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 1

0*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 +

20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4))/d^2)*log(-(2*(A^3*B + A*B^3)*a^9 + 5*(A^4 - B^4

)*a^8*b - 16*(A^3*B + A*B^3)*a^7*b^2 - 28*(A^3*B + A*B^3)*a^5*b^4 - 14*(A^4 - B^4)*a^4*b^5 - 8*(A^4 - B^4)*a^2

*b^7 + 10*(A^3*B + A*B^3)*a*b^8 + (A^4 - B^4)*b^9)*sqrt(b*tan(d*x + c) + a) + ((B*a^2 + 2*A*a*b - B*b^2)*d^3*s

qrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)

*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 1

1*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 +

(A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4) + (2*A^2*B*a^7 + (5*A^3 - 9*A*B^2)*a^6*b - 2*(16*A^2*B - 5*B^3)*a^5*b^2 -

5*(3*A^3 - 11*A*B^2)*a^4*b^3 + 10*(5*A^2*B - 2*B^3)*a^3*b^4 + (11*A^3 - 31*A*B^2)*a^2*b^5 - 2*(6*A^2*B - B^3)*

a*b^6 - (A^3 - A*B^2)*b^7)*d)*sqrt(-(10*A*B*a^4*b - 20*A*B*a^2*b^3 + 2*A*B*b^5 - (A^2 - B^2)*a^5 + 10*(A^2 - B

^2)*a^3*b^2 - 5*(A^2 - B^2)*a*b^4 + d^2*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B

^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B

^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 +

B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4))/d^2)) - 105*b*d*sqrt(-(10*A*B*a^

4*b - 20*A*B*a^2*b^3 + 2*A*B*b^5 - (A^2 - B^2)*a^5 + 10*(A^2 - B^2)*a^3*b^2 - 5*(A^2 - B^2)*a*b^4 + d^2*sqrt(-

(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*

b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4

)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4

- 2*A^2*B^2 + B^4)*b^10)/d^4))/d^2)*log(-(2*(A^3*B + A*B^3)*a^9 + 5*(A^4 - B^4)*a^8*b - 16*(A^3*B + A*B^3)*a^

7*b^2 - 28*(A^3*B + A*B^3)*a^5*b^4 - 14*(A^4 - B^4)*a^4*b^5 - 8*(A^4 - B^4)*a^2*b^7 + 10*(A^3*B + A*B^3)*a*b^8

+ (A^4 - B^4)*b^9)*sqrt(b*tan(d*x + c) + a) - ((B*a^2 + 2*A*a*b - B*b^2)*d^3*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*

B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B

^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A

*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/

d^4) + (2*A^2*B*a^7 + (5*A^3 - 9*A*B^2)*a^6*b - 2*(16*A^2*B - 5*B^3)*a^5*b^2 - 5*(3*A^3 - 11*A*B^2)*a^4*b^3 +

10*(5*A^2*B - 2*B^3)*a^3*b^4 + (11*A^3 - 31*A*B^2)*a^2*b^5 - 2*(6*A^2*B - B^3)*a*b^6 - (A^3 - A*B^2)*b^7)*d)*s

qrt(-(10*A*B*a^4*b - 20*A*B*a^2*b^3 + 2*A*B*b^5 - (A^2 - B^2)*a^5 + 10*(A^2 - B^2)*a^3*b^2 - 5*(A^2 - B^2)*a*b

^4 + d^2*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*

B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A

^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^

3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4))/d^2)) - 105*b*d*sqrt(-(10*A*B*a^4*b - 20*A*B*a^2*b^3 + 2*A*B*b^

5 - (A^2 - B^2)*a^5 + 10*(A^2 - B^2)*a^3*b^2 - 5*(A^2 - B^2)*a*b^4 - d^2*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A

*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 +

5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)

*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4))

/d^2)*log(-(2*(A^3*B + A*B^3)*a^9 + 5*(A^4 - B^4)*a^8*b - 16*(A^3*B + A*B^3)*a^7*b^2 - 28*(A^3*B + A*B^3)*a^5*

b^4 - 14*(A^4 - B^4)*a^4*b^5 - 8*(A^4 - B^4)*a^2*b^7 + 10*(A^3*B + A*B^3)*a*b^8 + (A^4 - B^4)*b^9)*sqrt(b*tan(

d*x + c) + a) + ((B*a^2 + 2*A*a*b - B*b^2)*d^3*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 2

6*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*

B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2

*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4) - (2*A^2*B*a^7 + (5*A^3 -

9*A*B^2)*a^6*b - 2*(16*A^2*B - 5*B^3)*a^5*b^2 - 5*(3*A^3 - 11*A*B^2)*a^4*b^3 + 10*(5*A^2*B - 2*B^3)*a^3*b^4 +

(11*A^3 - 31*A*B^2)*a^2*b^5 - 2*(6*A^2*B - B^3)*a*b^6 - (A^3 - A*B^2)*b^7)*d)*sqrt(-(10*A*B*a^4*b - 20*A*B*a^2

*b^3 + 2*A*B*b^5 - (A^2 - B^2)*a^5 + 10*(A^2 - B^2)*a^3*b^2 - 5*(A^2 - B^2)*a*b^4 - d^2*sqrt(-(4*A^2*B^2*a^10

+ 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4

- 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*

(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B

^4)*b^10)/d^4))/d^2)) + 105*b*d*sqrt(-(10*A*B*a^4*b - 20*A*B*a^2*b^3 + 2*A*B*b^5 - (A^2 - B^2)*a^5 + 10*(A^2 -

B^2)*a^3*b^2 - 5*(A^2 - B^2)*a*b^4 - d^2*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2

*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A

*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2

+ B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4))/d^2)*log(-(2*(A^3*B + A*B^3)*a

^9 + 5*(A^4 - B^4)*a^8*b - 16*(A^3*B + A*B^3)*a^7*b^2 - 28*(A^3*B + A*B^3)*a^5*b^4 - 14*(A^4 - B^4)*a^4*b^5 -

8*(A^4 - B^4)*a^2*b^7 + 10*(A^3*B + A*B^3)*a*b^8 + (A^4 - B^4)*b^9)*sqrt(b*tan(d*x + c) + a) - ((B*a^2 + 2*A*a

*b - B*b^2)*d^3*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 24

0*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 + 504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4

- 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B

- A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4) - (2*A^2*B*a^7 + (5*A^3 - 9*A*B^2)*a^6*b - 2*(16*A^2*B -

5*B^3)*a^5*b^2 - 5*(3*A^3 - 11*A*B^2)*a^4*b^3 + 10*(5*A^2*B - 2*B^3)*a^3*b^4 + (11*A^3 - 31*A*B^2)*a^2*b^5 - 2

*(6*A^2*B - B^3)*a*b^6 - (A^3 - A*B^2)*b^7)*d)*sqrt(-(10*A*B*a^4*b - 20*A*B*a^2*b^3 + 2*A*B*b^5 - (A^2 - B^2)*

a^5 + 10*(A^2 - B^2)*a^3*b^2 - 5*(A^2 - B^2)*a*b^4 - d^2*sqrt(-(4*A^2*B^2*a^10 + 20*(A^3*B - A*B^3)*a^9*b + 5*

(5*A^4 - 26*A^2*B^2 + 5*B^4)*a^8*b^2 - 240*(A^3*B - A*B^3)*a^7*b^3 - 20*(5*A^4 - 32*A^2*B^2 + 5*B^4)*a^6*b^4 +

504*(A^3*B - A*B^3)*a^5*b^5 + 10*(11*A^4 - 62*A^2*B^2 + 11*B^4)*a^4*b^6 - 240*(A^3*B - A*B^3)*a^3*b^7 - 20*(A

^4 - 7*A^2*B^2 + B^4)*a^2*b^8 + 20*(A^3*B - A*B^3)*a*b^9 + (A^4 - 2*A^2*B^2 + B^4)*b^10)/d^4))/d^2)) + 4*(15*B

*b^3*tan(d*x + c)^3 + 15*B*a^3 + 161*A*a^2*b - 245*B*a*b^2 - 105*A*b^3 + 3*(15*B*a*b^2 + 7*A*b^3)*tan(d*x + c)

^2 + (45*B*a^2*b + 77*A*a*b^2 - 35*B*b^3)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(b*d)

Sympy [F]

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

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

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

Maxima [F]

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

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

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

Giac [F(-1)]

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

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

Mupad [B] (veriﬁcation not implemented)

Time = 94.32 (sec) , antiderivative size = 3932, normalized size of antiderivative = 18.46 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

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

log(- ((((((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + A^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 5*A^2*a*b^4*

d^2)/d^4)^(1/2)*(32*A*b^6 - 32*A*a^4*b^2 + 32*a*b^2*d*(((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + A^

2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 5*A^2*a*b^4*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) - (16*A^2*b^2*

(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b

^2)^2)^(1/2) + A^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 5*A^2*a*b^4*d^2)/d^4)^(1/2))/2 - (8*A^3*a*b^2*(a^2 - 3*b^2)*

(a^2 + b^2)^3)/d^3)*((20*A^4*a^2*b^8*d^4 - A^4*b^10*d^4 - 110*A^4*a^4*b^6*d^4 + 100*A^4*a^6*b^4*d^4 - 25*A^4*a

^8*b^2*d^4)^(1/2)/(4*d^4) + (A^2*a^5)/(4*d^2) - (5*A^2*a^3*b^2)/(2*d^2) + (5*A^2*a*b^4)/(4*d^2))^(1/2) - log((

(((((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + A^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 5*A^2*a*b^4*d^2)/d^

4)^(1/2)*(32*A*a^4*b^2 - 32*A*b^6 + 32*a*b^2*d*(((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + A^2*a^5*d

^2 - 10*A^2*a^3*b^2*d^2 + 5*A^2*a*b^4*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) - (16*A^2*b^2*(a + b*

tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^

(1/2) + A^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 5*A^2*a*b^4*d^2)/d^4)^(1/2))/2 - (8*A^3*a*b^2*(a^2 - 3*b^2)*(a^2 +

b^2)^3)/d^3)*(((20*A^4*a^2*b^8*d^4 - A^4*b^10*d^4 - 110*A^4*a^4*b^6*d^4 + 100*A^4*a^6*b^4*d^4 - 25*A^4*a^8*b^2

*d^4)^(1/2) + A^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 5*A^2*a*b^4*d^2)/(4*d^4))^(1/2) - ((2*B*(a^2 + b^2))/(3*b*d)

- (2*B*a^2)/(3*b*d))*(a + b*tan(c + d*x))^(3/2) - log(((((-((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2)

- A^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 5*A^2*a*b^4*d^2)/d^4)^(1/2)*(32*A*a^4*b^2 - 32*A*b^6 + 32*a*b^2*d*(-((-A^

4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - A^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 5*A^2*a*b^4*d^2)/d^4)^(1/2)

*(a + b*tan(c + d*x))^(1/2)))/(2*d) - (16*A^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*

b^2))/d^2)*(-((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - A^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 5*A^2*a*b

^4*d^2)/d^4)^(1/2))/2 - (8*A^3*a*b^2*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3)*(-((20*A^4*a^2*b^8*d^4 - A^4*b^10*d^4 -

110*A^4*a^4*b^6*d^4 + 100*A^4*a^6*b^4*d^4 - 25*A^4*a^8*b^2*d^4)^(1/2) - A^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 5*

A^2*a*b^4*d^2)/(4*d^4))^(1/2) + log(- ((((-((-A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - A^2*a^5*d^2 +

10*A^2*a^3*b^2*d^2 - 5*A^2*a*b^4*d^2)/d^4)^(1/2)*(32*A*b^6 - 32*A*a^4*b^2 + 32*a*b^2*d*(-((-A^4*b^2*d^4*(5*a^4

+ b^4 - 10*a^2*b^2)^2)^(1/2) - A^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 5*A^2*a*b^4*d^2)/d^4)^(1/2)*(a + b*tan(c +

d*x))^(1/2)))/(2*d) - (16*A^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-((-

A^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - A^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 5*A^2*a*b^4*d^2)/d^4)^(1/

2))/2 - (8*A^3*a*b^2*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3)*((A^2*a^5)/(4*d^2) - (20*A^4*a^2*b^8*d^4 - A^4*b^10*d^4

- 110*A^4*a^4*b^6*d^4 + 100*A^4*a^6*b^4*d^4 - 25*A^4*a^8*b^2*d^4)^(1/2)/(4*d^4) - (5*A^2*a^3*b^2)/(2*d^2) + (

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

(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3 - ((-((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + B^2*a^5*d^2 - 10*B^

2*a^3*b^2*d^2 + 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(((-((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + B^2*a^5*d

^2 - 10*B^2*a^3*b^2*d^2 + 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(64*B*a^3*b^3 + 64*B*a*b^5 - 32*a*b^2*d*(-((-B^4*b^2*d^4

*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + B^2*a^5*d^2 - 10*B^2*a^3*b^2*d^2 + 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(a + b*t

an(c + d*x))^(1/2)))/(2*d) - (16*B^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2

))/2)*(-((20*B^4*a^2*b^8*d^4 - B^4*b^10*d^4 - 110*B^4*a^4*b^6*d^4 + 100*B^4*a^6*b^4*d^4 - 25*B^4*a^8*b^2*d^4)^

(1/2) + B^2*a^5*d^2 - 10*B^2*a^3*b^2*d^2 + 5*B^2*a*b^4*d^2)/(4*d^4))^(1/2) - log((8*B^3*b^3*(3*a^2 - b^2)*(a^2

+ b^2)^3)/d^3 - ((((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - B^2*a^5*d^2 + 10*B^2*a^3*b^2*d^2 - 5*B

^2*a*b^4*d^2)/d^4)^(1/2)*(((((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - B^2*a^5*d^2 + 10*B^2*a^3*b^2*

d^2 - 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(64*B*a^3*b^3 + 64*B*a*b^5 - 32*a*b^2*d*(((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^

2*b^2)^2)^(1/2) - B^2*a^5*d^2 + 10*B^2*a^3*b^2*d^2 - 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/

(2*d) - (16*B^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2))/2)*(((20*B^4*a^2*b

^8*d^4 - B^4*b^10*d^4 - 110*B^4*a^4*b^6*d^4 + 100*B^4*a^6*b^4*d^4 - 25*B^4*a^8*b^2*d^4)^(1/2) - B^2*a^5*d^2 +

10*B^2*a^3*b^2*d^2 - 5*B^2*a*b^4*d^2)/(4*d^4))^(1/2) + log((8*B^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3 - ((((-

B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - B^2*a^5*d^2 + 10*B^2*a^3*b^2*d^2 - 5*B^2*a*b^4*d^2)/d^4)^(1/

2)*(((((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - B^2*a^5*d^2 + 10*B^2*a^3*b^2*d^2 - 5*B^2*a*b^4*d^2)

/d^4)^(1/2)*(64*B*a^3*b^3 + 64*B*a*b^5 + 32*a*b^2*d*(((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - B^2*

a^5*d^2 + 10*B^2*a^3*b^2*d^2 - 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) + (16*B^2*b^2*(a

+ b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2))/2)*((20*B^4*a^2*b^8*d^4 - B^4*b^10*d^4 -

110*B^4*a^4*b^6*d^4 + 100*B^4*a^6*b^4*d^4 - 25*B^4*a^8*b^2*d^4)^(1/2)/(4*d^4) - (B^2*a^5)/(4*d^2) + (5*B^2*a^

3*b^2)/(2*d^2) - (5*B^2*a*b^4)/(4*d^2))^(1/2) + log((8*B^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3 - ((-((-B^4*b^

2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + B^2*a^5*d^2 - 10*B^2*a^3*b^2*d^2 + 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(((

-((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + B^2*a^5*d^2 - 10*B^2*a^3*b^2*d^2 + 5*B^2*a*b^4*d^2)/d^4)

^(1/2)*(64*B*a^3*b^3 + 64*B*a*b^5 + 32*a*b^2*d*(-((-B^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + B^2*a^5*

d^2 - 10*B^2*a^3*b^2*d^2 + 5*B^2*a*b^4*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) + (16*B^2*b^2*(a + b

*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2))/2)*((5*B^2*a^3*b^2)/(2*d^2) - (B^2*a^5)/(4*d

^2) - (20*B^4*a^2*b^8*d^4 - B^4*b^10*d^4 - 110*B^4*a^4*b^6*d^4 + 100*B^4*a^6*b^4*d^4 - 25*B^4*a^8*b^2*d^4)^(1/

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

))^(1/2) + (2*A*(a + b*tan(c + d*x))^(5/2))/(5*d) + (2*A*a*(a + b*tan(c + d*x))^(3/2))/(3*d) + (2*B*(a + b*tan

\(\int \tan (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [333]

Optimal result