3.2.10 \(\int \frac {(a+b \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{\sqrt {c+d \tan (e+f x)}} \, dx\) [110]

Optimal. Leaf size=407 \[ \frac {(i a+b)^3 (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f} \]

[Out]

(I*a+b)^3*(A-I*B-C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f/(c-I*d)^(1/2)-(I*a-b)^3*(A+I*B-C)*arctanh(
(c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/f/(c+I*d)^(1/2)+2/105*(72*a^3*C*d^3-6*a^2*b*d^2*(-49*B*d+32*C*c)+21*a*b^
2*d*(8*c^2*C-10*B*c*d+15*(A-C)*d^2)-b^3*(48*c^3*C-56*B*c^2*d+70*c*(A-C)*d^2+105*B*d^3))*(c+d*tan(f*x+e))^(1/2)
/d^4/f+2/105*b*(35*b*(A*b+B*a-C*b)*d^2+4*(-a*d+b*c)*(-7*B*b*d-6*C*a*d+6*C*b*c))*(c+d*tan(f*x+e))^(1/2)*tan(f*x
+e)/d^3/f-2/35*(-7*B*b*d-6*C*a*d+6*C*b*c)*(c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^2/d^2/f+2/7*C*(c+d*tan(f*x+e
))^(1/2)*(a+b*tan(f*x+e))^3/d/f

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Rubi [A]
time = 1.11, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3728, 3718, 3711, 3620, 3618, 65, 214} \begin {gather*} \frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{105 d^4 f}+\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{105 d^3 f}-\frac {(-b+i a)^3 (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {(b+i a)^3 (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I*a + b)^3*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) - ((I*a - b)^3*(
A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*(72*a^3*C*d^3 - 6*a^2*b*d
^2*(32*c*C - 49*B*d) + 21*a*b^2*d*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2) - b^3*(48*c^3*C - 56*B*c^2*d + 70*c*(A
- C)*d^2 + 105*B*d^3))*Sqrt[c + d*Tan[e + f*x]])/(105*d^4*f) + (2*b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*
d)*(6*b*c*C - 7*b*B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(105*d^3*f) - (2*(6*b*c*C - 7*b*B*d -
6*a*C*d)*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(35*d^2*f) + (2*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d
*Tan[e + f*x]])/(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 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]

Rule 3728

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

Rubi steps

\begin {align*} \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx &=\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}+\frac {2 \int \frac {(a+b \tan (e+f x))^2 \left (\frac {1}{2} (-6 b c C+a (7 A-C) d)+\frac {7}{2} (A b+a B-b C) d \tan (e+f x)-\frac {1}{2} (6 b c C-7 b B d-6 a C d) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{7 d}\\ &=-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}+\frac {4 \int \frac {(a+b \tan (e+f x)) \left (\frac {1}{4} (-5 a d (6 b c C-a (7 A-C) d)+(4 b c+a d) (6 b c C-7 b B d-6 a C d))+\frac {35}{4} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac {1}{4} \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{35 d^2}\\ &=\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {8 \int \frac {\frac {1}{8} \left (-3 a^3 (35 A-11 C) d^3-42 a b^2 c d (4 c C-5 B d)+3 a^2 b d^2 (64 c C+7 B d)+2 b^3 c \left (24 c^2 C-28 B c d+35 (A-C) d^2\right )\right )-\frac {105}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)-\frac {1}{8} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{105 d^3}\\ &=\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {8 \int \frac {\frac {105}{8} \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-\frac {105}{8} \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{105 d^3}\\ &=\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}+\frac {1}{2} \left ((a-i b)^3 (A-i B-C)\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (A+i B-C)\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\left (i (a+i b)^3 (A+i B-C)\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 ((a-i b)^3 (i A+B-i C)\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 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\left ((a-i b)^3 (A-i B-C)\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)^3 (A+i B-C)\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)^3 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}\\ \end {align*}

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Mathematica [A]
time = 5.77, size = 392, normalized size = 0.96 \begin {gather*} \frac {-\frac {105 (a-i b)^3 (i A+B-i C) d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {105 i (a+i b)^3 (A+i B-C) d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}-\frac {2 \left (-72 a^3 C d^3+6 a^2 b d^2 (32 c C-49 B d)-21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )+b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{d^3}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{d^2}+\frac {6 (-6 b c C+7 b B d+6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{d}+30 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{105 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-105*(a - I*b)^3*(I*A + B - I*C)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((105*I)
*(a + I*b)^3*(A + I*B - C)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] - (2*(-72*a^3*C*d^
3 + 6*a^2*b*d^2*(32*c*C - 49*B*d) - 21*a*b^2*d*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2) + b^3*(48*c^3*C - 56*B*c^
2*d + 70*c*(A - C)*d^2 + 105*B*d^3))*Sqrt[c + d*Tan[e + f*x]])/d^3 + (2*b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c
- a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/d^2 + (6*(-6*b*c*C + 7*b*B*d + 6
*a*C*d)*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/d + 30*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*
x]])/(105*d*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8227\) vs. \(2(371)=742\).
time = 0.46, size = 8228, normalized size = 20.22

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(8228\)
default	\(\text {Expression too large to display}\)	\(8228\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),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))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(1/2),x)

[Out]

Integral((a + b*tan(e + f*x))**3*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/sqrt(c + d*tan(e + f*x)), 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))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),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 [B]
time = 122.08, size = 2500, normalized size = 6.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*tan(e + f*x))^3*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(1/2),x)

[Out]

atan(((((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(
c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 12
0*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12
+ A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A
^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b
^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A
^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2
/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^
4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f
^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16
*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 - 15*A^2*a^4*b^2*d^2))/f^2)*((((8*
A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^
4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 1
5*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 -
24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)
/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i - (((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12*A*a^
2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5
*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16
*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^
4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*
A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*A^2*a^
6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*
c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*
a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^
2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(
c^2*f^4 + d^2*f^4)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 -
15*A^2*a^4*b^2*d^2))/f^2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 12
0*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12
+ A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A
^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b
^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*1i)/((16*(6*A^3*a^4*b^5*d^2 - A^3*b^9*d^2 + 8
*A^3*a^6*b^3*d^2 + 3*A^3*a^8*b*d^2))/f^3 + (((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12
*A*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*
a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4
- (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a
^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2
- 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2))*((((8*A
^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4
*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15
*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 -
24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/
(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d
^2 - 15*A^2*a^4*b^2*d^2))/f^2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2
+ 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4
*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2)
- 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*...

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