3.1.99 \(\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [99]

Optimal. Leaf size=273 \[ -\frac {(i a+b) (A-i B-C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(i a-b) (A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f} \]

[Out]

-(I*a+b)*(A-I*B-C)*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f+(I*a-b)*(A+I*B-C)*(c+I*d)^(3/
2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/f+2*(A*a*d+A*b*c+B*a*c-B*b*d-C*a*d-C*b*c)*(c+d*tan(f*x+e))^(1
/2)/f+2/3*(A*b+B*a-C*b)*(c+d*tan(f*x+e))^(3/2)/f-2/35*(-7*B*b*d-7*C*a*d+2*C*b*c)*(c+d*tan(f*x+e))^(5/2)/d^2/f+
2/7*b*C*tan(f*x+e)*(c+d*tan(f*x+e))^(5/2)/d/f

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Rubi [A]
time = 0.60, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3718, 3711, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (a B+A b-b C) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \sqrt {c+d \tan (e+f x)} (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac {(b+i a) (c-i d)^{3/2} (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(-b+i a) (c+i d)^{3/2} (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}-\frac {2 (-7 a C d-7 b B d+2 b c C) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((I*a + b)*(A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(A
+ I*B - C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(A*b*c + a*B*c - b*c*C + a
*A*d - b*B*d - a*C*d)*Sqrt[c + d*Tan[e + f*x]])/f + (2*(A*b + a*B - b*C)*(c + d*Tan[e + f*x])^(3/2))/(3*f) - (
2*(2*b*c*C - 7*b*B*d - 7*a*C*d)*(c + d*Tan[e + f*x])^(5/2))/(35*d^2*f) + (2*b*C*Tan[e + f*x]*(c + d*Tan[e + f*
x])^(5/2))/(7*d*f)

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 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]

Rule 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rubi steps

\begin {align*} \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} (2 b c C-7 a A d)-\frac {7}{2} (A b+a B-b C) d \tan (e+f x)+\frac {1}{2} (2 b c C-7 b B d-7 a C d) \tan ^2(e+f x)\right ) \, dx}{7 d}\\ &=-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {7}{2} (b B-a (A-C)) d-\frac {7}{2} (A b+a B-b C) d \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (\frac {7}{2} d (b B c+b (A-C) d-a (A c-c C-B d))-\frac {7}{2} d (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int \frac {\frac {7}{2} d \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-\frac {7}{2} d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{7 d}\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {1}{2} \left ((a-i b) (A-i B-C) (c-i d)^2\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b) (A+i B-C) (c+i d)^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {\left ((i a+b) (A-i B-C) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {\left ((i a-b) (A+i B-C) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {\left ((a-i b) (A-i B-C) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b) (A+i B-C) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac {(a-i b) (i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(i a-b) (A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}\\ \end {align*}

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Mathematica [A]
time = 3.03, size = 260, normalized size = 0.95 \begin {gather*} \frac {\frac {2 (-2 b c C+7 b B d+7 a C d) (c+d \tan (e+f x))^{5/2}}{d}+10 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}+\frac {35}{3} (i a+b) (A-i B-C) d \left (-3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c-3 i d+d \tan (e+f x))\right )+\frac {35}{3} (-i a+b) (A+i B-C) d \left (-3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c+3 i d+d \tan (e+f x))\right )}{35 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(-2*b*c*C + 7*b*B*d + 7*a*C*d)*(c + d*Tan[e + f*x])^(5/2))/d + 10*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5
/2) + (35*(I*a + b)*(A - I*B - C)*d*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt
[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])))/3 + (35*((-I)*a + b)*(A + I*B - C)*d*(-3*(c + I*d)^(3/
2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f*x])
))/3)/(35*d*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2314\) vs. \(2(239)=478\).
time = 0.49, size = 2315, normalized size = 8.48

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2315\)
default	\(\text {Expression too large to display}\)	\(2315\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/d^2*(1/7*C*b*(c+d*tan(f*x+e))^(7/2)+1/5*B*b*d*(c+d*tan(f*x+e))^(5/2)+1/5*C*a*d*(c+d*tan(f*x+e))^(5/2)-1/5*
C*b*c*(c+d*tan(f*x+e))^(5/2)+1/3*A*b*d^2*(c+d*tan(f*x+e))^(3/2)+1/3*B*a*d^2*(c+d*tan(f*x+e))^(3/2)-1/3*C*b*d^2
*(c+d*tan(f*x+e))^(3/2)+A*a*d^3*(c+d*tan(f*x+e))^(1/2)+A*b*c*d^2*(c+d*tan(f*x+e))^(1/2)+B*a*c*d^2*(c+d*tan(f*x
+e))^(1/2)-B*b*d^3*(c+d*tan(f*x+e))^(1/2)-C*a*d^3*(c+d*tan(f*x+e))^(1/2)-C*b*c*d^2*(c+d*tan(f*x+e))^(1/2)-d^2*
(1/4/d*(1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b*d+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2-2*A*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b*c*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b*c-2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2+B*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b*d^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2+2*C*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b*c*d)*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(2*A*
(c^2+d^2)^(1/2)*a*d^2+2*A*(c^2+d^2)^(1/2)*b*c*d+2*B*(c^2+d^2)^(1/2)*a*c*d-2*B*(c^2+d^2)^(1/2)*b*d^2-2*C*(c^2+d
^2)^(1/2)*a*d^2-2*C*(c^2+d^2)^(1/2)*b*c*d+1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+A*(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*a*d^2-2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d+B*(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c-2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b*c^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-C*(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*a*d^2+2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*
arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/4/d*(1/2*(A*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d-A*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*a*c^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d-B*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c+2*B*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^2-C*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d+C*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d)*l
n(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(2*A*(c^2+d^2)^(1/2)*
a*d^2+2*A*(c^2+d^2)^(1/2)*b*c*d+2*B*(c^2+d^2)^(1/2)*a*c*d-2*B*(c^2+d^2)^(1/2)*b*d^2-2*C*(c^2+d^2)^(1/2)*a*d^2-
2*C*(c^2+d^2)^(1/2)*b*c*d-1/2*(A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-A*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*b*d-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2+2*A*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d-B*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*b*c+2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2-B*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b*d^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+C*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*b*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2-2*C*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*ta
n(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))))

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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