3.2.23 \(\int \frac {(a+b \tan (e+f x))^2 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^{5/2}} \, dx\) [123]

Optimal. Leaf size=358 \[ -\frac {(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f} \]

[Out]

-(a-I*b)^2*(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f-(a+I*b)^2*(B-I*(A-C))*arc
tanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(5/2)/f+2/3*(-a*d+b*c)*(b*(4*c^4*C-B*c^3*d-2*c^2*(A-5*C)*d^
2-7*B*c*d^3+4*A*d^4)+3*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/d^3/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^(1/2)+2/3*b^2*(4*c^
2*C-B*c*d+(A+3*C)*d^2)*(c+d*tan(f*x+e))^(1/2)/d^3/(c^2+d^2)/f-2/3*(A*d^2-B*c*d+C*c^2)*(a+b*tan(f*x+e))^2/d/(c^
2+d^2)/f/(c+d*tan(f*x+e))^(3/2)

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Rubi [A]
time = 1.04, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3726, 3716, 3711, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{3 d^3 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}-\frac {(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}+\frac {2 b^2 \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 f \left (c^2+d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a - I*b)^2*(I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) - ((a + I
*b)^2*(B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(c^2*C - B*c*d
+ A*d^2)*(a + b*Tan[e + f*x])^2)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(b*c - a*d)*(b*(4*c^4*C
- B*c^3*d - 2*c^2*(A - 5*C)*d^2 - 7*B*c*d^3 + 4*A*d^4) + 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2))))/(3*d^3*(c^2
+ d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b^2*(4*c^2*C - B*c*d + (A + 3*C)*d^2)*Sqrt[c + d*Tan[e + f*x]])/(3*
d^3*(c^2 + d^2)*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 3716

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 - a*d))*(c^2*C - B*c*d + A*d^2)
*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f
*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d
+ a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &
& NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]

Rule 3726

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

Rubi steps

\begin {align*} \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {(a+b \tan (e+f x)) \left (\frac {1}{2} \left (2 A d \left (\frac {3 a c}{2}+2 b d\right )+2 \left (2 b c-\frac {3 a d}{2}\right ) (c C-B d)\right )+\frac {3}{2} d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+\frac {1}{2} b \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{3 d \left (c^2+d^2\right )}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 \int \frac {\frac {1}{2} \left (b^2 \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )-3 a^2 d^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+6 a b d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )-\frac {3}{2} d^2 \left (2 a b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)+\frac {1}{2} b^2 \left (c^2+d^2\right ) \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d^2 \left (c^2+d^2\right )^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {2 \int \frac {-\frac {3}{2} d^2 \left (a^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-2 a b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )-\frac {3}{2} d^2 \left (2 a b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d^2 \left (c^2+d^2\right )^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {\left ((a-i b)^2 (A-i B-C)\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)^2}+\frac {\left ((a+i b)^2 (A+i B-C)\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c+i d)^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {\left ((a-i b)^2 (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 (c-i d)^2 f}-\frac {\left (i (a+i b)^2 (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 (c+i d)^2 f}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}-\frac {\left ((a-i b)^2 (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 )}{(c-i d)^2 d f}-\frac {\left ((a+i b)^2 (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 )}{(c+i d)^2 d f}\\ &=-\frac {(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 4.68, size = 414, normalized size = 1.16 \begin {gather*} -\frac {-2 (c-i d) (c+i d) \left (8 a^2 C d^2+a b d (-16 c C+B d)+b^2 \left (8 c^2 C-2 B c d+(-A+C) d^2\right )\right )-d^2 \left (-2 a b (A c-c C+B d)-a^2 (B c+(-A+C) d)+b^2 (B c+(-A+C) d)\right ) \left (i (c+i d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )-6 (c-i d) (c+i d) d (4 b c C-b B d-4 a C d) (a+b \tan (e+f x))-6 C (c-i d) (c+i d) d^2 (a+b \tan (e+f x))^2-3 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \left (i (c+i d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right ) (c+d \tan (e+f x))}{3 d^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(-2*(c - I*d)*(c + I*d)*(8*a^2*C*d^2 + a*b*d*(-16*c*C + B*d) + b^2*(8*c^2*C - 2*B*c*d + (-A + C)*d^2)) -
d^2*(-2*a*b*(A*c - c*C + B*d) - a^2*(B*c + (-A + C)*d) + b^2*(B*c + (-A + C)*d))*(I*(c + I*d)*Hypergeometric2F
1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f
*x])/(c + I*d)]) - 6*(c - I*d)*(c + I*d)*d*(4*b*c*C - b*B*d - 4*a*C*d)*(a + b*Tan[e + f*x]) - 6*C*(c - I*d)*(c
+ I*d)*d^2*(a + b*Tan[e + f*x])^2 - 3*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2*(I*(c + I*d)*Hypergeometric2F1[-1/2
, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c
+ I*d)])*(c + d*Tan[e + f*x]))/(d^3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(15608\) vs. \(2(325)=650\).
time = 0.70, size = 15609, normalized size = 43.60

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(15609\)
default	\(\text {Expression too large to display}\)	\(15609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),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))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),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))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),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 )^{2} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tan(e + f*x))**2*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(c + d*tan(e + f*x))**(5/2), 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))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),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 = 116.90, size = 2500, normalized size = 6.98 \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))^2*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(5/2),x)

[Out]

atan((((c + d*tan(e + f*x))^(1/2)*(96*A^2*a^2*b^2*d^18*f^3 - 16*A^2*b^4*d^18*f^3 - 16*A^2*a^4*d^18*f^3 + 320*A
^2*a^4*c^4*d^14*f^3 + 1024*A^2*a^4*c^6*d^12*f^3 + 1440*A^2*a^4*c^8*d^10*f^3 + 1024*A^2*a^4*c^10*d^8*f^3 + 320*
A^2*a^4*c^12*d^6*f^3 - 16*A^2*a^4*c^16*d^2*f^3 + 320*A^2*b^4*c^4*d^14*f^3 + 1024*A^2*b^4*c^6*d^12*f^3 + 1440*A
^2*b^4*c^8*d^10*f^3 + 1024*A^2*b^4*c^10*d^8*f^3 + 320*A^2*b^4*c^12*d^6*f^3 - 16*A^2*b^4*c^16*d^2*f^3 - 256*A^2
*a*b^3*c*d^17*f^3 + 256*A^2*a^3*b*c*d^17*f^3 - 1280*A^2*a*b^3*c^3*d^15*f^3 - 2304*A^2*a*b^3*c^5*d^13*f^3 - 128
0*A^2*a*b^3*c^7*d^11*f^3 + 1280*A^2*a*b^3*c^9*d^9*f^3 + 2304*A^2*a*b^3*c^11*d^7*f^3 + 1280*A^2*a*b^3*c^13*d^5*
f^3 + 256*A^2*a*b^3*c^15*d^3*f^3 + 1280*A^2*a^3*b*c^3*d^15*f^3 + 2304*A^2*a^3*b*c^5*d^13*f^3 + 1280*A^2*a^3*b*
c^7*d^11*f^3 - 1280*A^2*a^3*b*c^9*d^9*f^3 - 2304*A^2*a^3*b*c^11*d^7*f^3 - 1280*A^2*a^3*b*c^13*d^5*f^3 - 256*A^
2*a^3*b*c^15*d^3*f^3 - 1920*A^2*a^2*b^2*c^4*d^14*f^3 - 6144*A^2*a^2*b^2*c^6*d^12*f^3 - 8640*A^2*a^2*b^2*c^8*d^
10*f^3 - 6144*A^2*a^2*b^2*c^10*d^8*f^3 - 1920*A^2*a^2*b^2*c^12*d^6*f^3 + 96*A^2*a^2*b^2*c^16*d^2*f^3) + ((((8*
A^2*a^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2
- 32*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c
^4*d*f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2
*d^3*f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b
^2)*(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) -
4*A^2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*
f^2 + 16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3
*c^4*d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^
2*d^3*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^
4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(32*A*b^2*d^21*f^4 - 32*A*a^2*d^21*f^4 - (c + d*tan(e + f*x))^(1/2)*((((8*A^2*a
^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2 - 32
*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*
f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2*d^3*
f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b^2)*(
16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - 4*A^
2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*f^2 +
16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3*c^4*
d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^2*d^3
*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4
+ 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^
9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*
f^5 + 64*c^21*d^2*f^5) - 160*A*a^2*c^2*d^19*f^4 - 128*A*a^2*c^4*d^17*f^4 + 896*A*a^2*c^6*d^15*f^4 + 3136*A*a^2
*c^8*d^13*f^4 + 4928*A*a^2*c^10*d^11*f^4 + 4480*A*a^2*c^12*d^9*f^4 + 2432*A*a^2*c^14*d^7*f^4 + 736*A*a^2*c^16*
d^5*f^4 + 96*A*a^2*c^18*d^3*f^4 + 160*A*b^2*c^2*d^19*f^4 + 128*A*b^2*c^4*d^17*f^4 - 896*A*b^2*c^6*d^15*f^4 - 3
136*A*b^2*c^8*d^13*f^4 - 4928*A*b^2*c^10*d^11*f^4 - 4480*A*b^2*c^12*d^9*f^4 - 2432*A*b^2*c^14*d^7*f^4 - 736*A*
b^2*c^16*d^5*f^4 - 96*A*b^2*c^18*d^3*f^4 + 192*A*a*b*c*d^20*f^4 + 1472*A*a*b*c^3*d^18*f^4 + 4864*A*a*b*c^5*d^1
6*f^4 + 8960*A*a*b*c^7*d^14*f^4 + 9856*A*a*b*c^9*d^12*f^4 + 6272*A*a*b*c^11*d^10*f^4 + 1792*A*a*b*c^13*d^8*f^4
- 256*A*a*b*c^15*d^6*f^4 - 320*A*a*b*c^17*d^4*f^4 - 64*A*a*b*c^19*d^2*f^4))*((((8*A^2*a^4*c^5*f^2 + 8*A^2*b^4
*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2 - 32*A^2*a*b^3*d^5*f^2 + 3
2*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*f^2 + 160*A^2*a^3*b*c^
4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2*d^3*f^2 + 480*A^2*a^2*b^2*
c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b^2)*(16*c^10*f^4 + 16*d^10*
f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - 4*A^2*a^4*c^5*f^2 - 4*A^2*
b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*f^2 + 16*A^2*a*b^3*d^5*f^2
- 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3*c^4*d*f^2 - 80*A^2*a^3*b*c
^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*...

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