\(\int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [332]

Optimal result

Integrand size = 33, antiderivative size = 252 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {(a-i b)^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 (A b+a B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d} \]

[Out]

(a-I*b)^(5/2)*(I*A+B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-(a+I*b)^(5/2)*(I*A-B)*arctanh((a+b*tan(d
*x+c))^(1/2)/(a+I*b)^(1/2))/d-2*(2*A*a*b+B*a^2-B*b^2)*(a+b*tan(d*x+c))^(1/2)/d-2/3*(A*b+B*a)*(a+b*tan(d*x+c))^
(3/2)/d-2/5*B*(a+b*tan(d*x+c))^(5/2)/d+2/63*(9*A*b-2*B*a)*(a+b*tan(d*x+c))^(7/2)/b^2/d+2/9*B*tan(d*x+c)*(a+b*t
an(d*x+c))^(7/2)/b/d

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3688, 3711, 3609, 3620, 3618, 65, 214} \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {2 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {(a-i b)^{5/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}-\frac {2 (a B+A b) (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d} \]

[In]

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

[Out]

((a - I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(I*A - B)*Arc
Tanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[a + b*Tan[c + d*x]])/d - (
2*(A*b + a*B)*(a + b*Tan[c + d*x])^(3/2))/(3*d) - (2*B*(a + b*Tan[c + d*x])^(5/2))/(5*d) + (2*(9*A*b - 2*a*B)*
(a + b*Tan[c + d*x])^(7/2))/(63*b^2*d) + (2*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(7/2))/(9*b*d)

Rule 65

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

Rule 214

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

Rule 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 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3688

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 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 \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}+\frac {2 \int (a+b \tan (c+d x))^{5/2} \left (-a B-\frac {9}{2} b B \tan (c+d x)+\frac {1}{2} (9 A b-2 a B) \tan ^2(c+d x)\right ) \, dx}{9 b} \\ & = \frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}+\frac {2 \int (a+b \tan (c+d x))^{5/2} \left (-\frac {9 A b}{2}-\frac {9}{2} b B \tan (c+d x)\right ) \, dx}{9 b} \\ & = -\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}+\frac {2 \int (a+b \tan (c+d x))^{3/2} \left (-\frac {9}{2} b (a A-b B)-\frac {9}{2} b (A b+a B) \tan (c+d x)\right ) \, dx}{9 b} \\ & = -\frac {2 (A b+a B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}+\frac {2 \int \sqrt {a+b \tan (c+d x)} \left (-\frac {9}{2} b \left (a^2 A-A b^2-2 a b B\right )-\frac {9}{2} b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx}{9 b} \\ & = -\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 (A b+a B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}+\frac {2 \int \frac {-\frac {9}{2} b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )-\frac {9}{2} b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{9 b} \\ & = -\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 (A b+a B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}-\frac {1}{2} \left ((a-i 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 ((a+i b)^3 (A+i B)\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 (A b+a B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}+\frac {\left (i (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 (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 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 (A b+a B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}+\frac {\left ((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 (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)^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 (A b+a B) (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {2 B (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac {2 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.97 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.17 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\frac {2 (9 A b-2 a B) (a+b \tan (c+d x))^{7/2}}{b}+14 B \tan (c+d x) (a+b \tan (c+d x))^{7/2}-\frac {63}{2} i b (A-i B) \left (\frac {2}{5} (a+b \tan (c+d x))^{5/2}+\frac {2}{3} (a-i b) \left (-3 (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+\sqrt {a+b \tan (c+d x)} (4 a-3 i b+b \tan (c+d x))\right )\right )+\frac {63}{2} i b (A+i B) \left (\frac {2}{5} (a+b \tan (c+d x))^{5/2}+\frac {2}{3} (a+i b) \left (-3 (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\sqrt {a+b \tan (c+d x)} (4 a+3 i b+b \tan (c+d x))\right )\right )}{63 b d} \]

[In]

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

[Out]

((2*(9*A*b - 2*a*B)*(a + b*Tan[c + d*x])^(7/2))/b + 14*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(7/2) - ((63*I)/2)*
b*(A - I*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) + ((63*I)/2)*b*(A + I*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))/(63*b*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2437\) vs. \(2(216)=432\).

Time = 0.18 (sec) , antiderivative size = 2438, normalized size of antiderivative = 9.67

method	result	size
parts	\(\text {Expression too large to display}\)	\(2438\)
derivativedivides	\(\text {Expression too large to display}\)	\(2469\)
default	\(\text {Expression too large to display}\)	\(2469\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4963 vs. \(2 (210) = 420\).

Time = 0.86 (sec) , antiderivative size = 4963, normalized size of antiderivative = 19.69 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/630*(315*b^2*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) + ((A*a^2 - 2*B*a*b - A*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*B^2*a^7 + (9*A^2*B - 5*B^3)*a^6*b + 2*(5*A^3 - 16*A*B^2)*a^5*b^2 -
5*(11*A^2*B - 3*B^3)*a^4*b^3 - 10*(2*A^3 - 5*A*B^2)*a^3*b^4 + (31*A^2*B - 11*B^3)*a^2*b^5 + 2*(A^3 - 6*A*B^2)
*a*b^6 - (A^2*B - B^3)*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)) - 315*b^2*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) - ((A*a^2 - 2*B*a*b - A*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*B^2*a^7 + (9*A^2*B - 5*B^3)*a^6*b + 2*(5*A^3 - 16*A*B^2)*a^5*b^2 - 5*(11*A^2*B - 3*B^3)*a^4*b^3 -
10*(2*A^3 - 5*A*B^2)*a^3*b^4 + (31*A^2*B - 11*B^3)*a^2*b^5 + 2*(A^3 - 6*A*B^2)*a*b^6 - (A^2*B - B^3)*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)) - 315*b^2*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) + ((A*a^2 - 2*B*a*b - A*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*B^2*a^7 + (9*A^2*B
- 5*B^3)*a^6*b + 2*(5*A^3 - 16*A*B^2)*a^5*b^2 - 5*(11*A^2*B - 3*B^3)*a^4*b^3 - 10*(2*A^3 - 5*A*B^2)*a^3*b^4 +
(31*A^2*B - 11*B^3)*a^2*b^5 + 2*(A^3 - 6*A*B^2)*a*b^6 - (A^2*B - B^3)*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)) + 315*b^2*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) - ((A*a^2 - 2*B*
a*b - A*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 - 2
40*(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*B^2*a^7 + (9*A^2*B - 5*B^3)*a^6*b + 2*(5*A^3 - 16
*A*B^2)*a^5*b^2 - 5*(11*A^2*B - 3*B^3)*a^4*b^3 - 10*(2*A^3 - 5*A*B^2)*a^3*b^4 + (31*A^2*B - 11*B^3)*a^2*b^5 +
2*(A^3 - 6*A*B^2)*a*b^6 - (A^2*B - B^3)*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*(35*B
*b^4*tan(d*x + c)^4 - 10*B*a^4 + 45*A*a^3*b - 483*B*a^2*b^2 - 735*A*a*b^3 + 315*B*b^4 + 5*(19*B*a*b^3 + 9*A*b^
4)*tan(d*x + c)^3 + 3*(25*B*a^2*b^2 + 45*A*a*b^3 - 21*B*b^4)*tan(d*x + c)^2 + (5*B*a^3*b + 135*A*a^2*b^2 - 231
*B*a*b^3 - 105*A*b^4)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(b^2*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}