3.2.25 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx\) [125]

Optimal. Leaf size=209 \[ -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right )}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{\left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3709, 3610, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 \left (A d^2-B c d+c^2 C\right )}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )}{f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {(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 {(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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3610

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

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 3709

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.65, size = 223, normalized size = 1.07 \begin {gather*} -\frac {2 C \left (c^2+d^2\right )+(B c+(-A+C) d) \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 )-3 B \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 \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] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(5/2),x]

[Out]

-1/3*(2*C*(c^2 + d^2) + (B*c + (-A + C)*d)*(I*(c + I*d)*Hypergeometric2F1[-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)]) - 3*B*(I*(c + I*d)*Hy
pergeometric2F1[-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*(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. \(4917\) vs. \(2(184)=368\).
time = 0.46, size = 4918, normalized size = 23.53

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4918\)
default	\(\text {Expression too large to display}\)	\(4918\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/f/d*(-1/3*(A*d^2-B*c*d+C*c^2)/(c^2+d^2)/(c+d*tan(f*x+e))^(3/2)-d*(2*A*c*d-B*c^2+B*d^2-2*C*c*d)/(c^2+d^2)^2/(
c+d*tan(f*x+e))^(1/2)+d/(c^2+d^2)^2*(1/4/d/(5*c^4-10*c^2*d^2+d^4)/(c^2+d^2)^(3/2)*(1/2*(-10*C*(c^2+d^2)^(1/2)*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^4+10*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^6-5*A*(c^2+d^2)
^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^2-A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^4+18*A*(c^2
+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6*d^2+10*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^4-1
0*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^6+C*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*
d^4-18*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6*d^2-10*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*c^7*d+10*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^5*d^3+18*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*c^3*d^5-2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^7+5*C*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*c^4*d^2+12*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^7+3*C*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(
1/2)*c^6-C*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^6+15*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^8*d+5*A*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*c^9-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^9-5*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^9-22*B*(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^5-20*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6*d^3+2*C*(c^2+d^2)^(1/2)*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*c^8+20*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^7*d^2-6*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^5*d^4-3
*A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6-28*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d^6+3*C*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*c*d^8+A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^6-2*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*c^8-20*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^7*d^2+6*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^5*d^4+28*
A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d^6-3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^8)*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*(30*A*c^8*d^2-40*A*c^6*d^4-44*A*c^4*d^6+24*A*c^2*
d^8-2*A*d^10-10*B*c^9*d+40*B*c^7*d^3-12*B*c^5*d^5-56*B*c^3*d^7+6*B*c*d^9-30*C*c^8*d^2+40*C*c^6*d^4+44*C*c^4*d^
6-24*C*c^2*d^8+2*C*d^10+1/2*(-10*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^4+10*C*(c^2+d^2)^(1/2)*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^6-5*A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^2-A*(c^2+d^2)^(3
/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^4+18*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6*d^2+10*A*(c^2
+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^4-10*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^6+C
*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^4-18*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6*
d^2-10*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^7*d+10*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*c^5*d^3+18*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d^5-2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*c*d^7+5*C*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^2+12*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^
2*d^7+3*C*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6-C*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^
6+15*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^8*d+5*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^9-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*d^9-5*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^9-22*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^5-20*B*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*c^6*d^3+2*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^8+20*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
c^7*d^2-6*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^5*d^4-3*A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6-28*C*(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d^6+3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^8+A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*d^6-2*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^8-20*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^
7*d^2+6*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^5*d^4+28*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d^6-3*A*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*c*d^8)*(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/(5*c^4-10*c^2*d^2+d^4)/(c^2+d^2)^(3
/2)*(1/2*(10*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^4-10*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*c^2*d^6+5*A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4*d^2+A*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*c^2*d^4-18*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6*d^2-10*A*(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*c^4*d^4+10*A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^6-C*(c^2+d^2)^(3/2)*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^4+18*C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^6*d^2+10*B*(c^2+d^2)^(
1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^7*d-10*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^5*d^3-18*B*(c^2+
d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d^5+2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^7-5*C*(
c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^...

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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*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)+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 {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \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)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(5/2),x)

[Out]

Integral((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)+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 = 37.59, size = 2500, normalized size = 11.96 \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) + C*tan(e + f*x)^2)/(c + d*tan(e + f*x))^(5/2),x)

[Out]

(log(96*A^3*c^3*d^13*f^2 - ((((((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d
^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(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)*(((((320*A^4*c^2*d^8*f^4 - 16*A^
4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2
*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(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)*(c + d*tan(e + f*x))^(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))/4 - 32*A*d^21*f^4 - 160*A*c^2*d^19*f^4 - 128*A*c^4*d^17*f^4 + 896*A
*c^6*d^15*f^4 + 3136*A*c^8*d^13*f^4 + 4928*A*c^10*d^11*f^4 + 4480*A*c^12*d^9*f^4 + 2432*A*c^14*d^7*f^4 + 736*A
*c^16*d^5*f^4 + 96*A*c^18*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(320*A^2*c^4*d^14*f^3 - 16*A^2*d^18*f^3 + 1
024*A^2*c^6*d^12*f^3 + 1440*A^2*c^8*d^10*f^3 + 1024*A^2*c^10*d^8*f^3 + 320*A^2*c^12*d^6*f^3 - 16*A^2*c^16*d^2*
f^3))*(((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2
*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(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))/4 + 240*A^3*c^5*d^11*f^2 + 320*A^3*c^7*d^9*f^2 + 240*A^
3*c^9*d^7*f^2 + 96*A^3*c^11*d^5*f^2 + 16*A^3*c^13*d^3*f^2 + 16*A^3*c*d^15*f^2)*(((320*A^4*c^2*d^8*f^4 - 16*A^4
*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*
c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(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))/4 + (log(96*A^3*c^3*d^13*f^2 - ((((-((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d
^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) + 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4
*f^2)/(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)*(((-((320
*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2)
+ 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(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)*(c + d*tan(e + f*x))^(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))/4 - 32*A*d^21*f^4 - 160*A*c^2*d^19*f^4 - 12
8*A*c^4*d^17*f^4 + 896*A*c^6*d^15*f^4 + 3136*A*c^8*d^13*f^4 + 4928*A*c^10*d^11*f^4 + 4480*A*c^12*d^9*f^4 + 243
2*A*c^14*d^7*f^4 + 736*A*c^16*d^5*f^4 + 96*A*c^18*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(320*A^2*c^4*d^14*f
^3 - 16*A^2*d^18*f^3 + 1024*A^2*c^6*d^12*f^3 + 1440*A^2*c^8*d^10*f^3 + 1024*A^2*c^10*d^8*f^3 + 320*A^2*c^12*d^
6*f^3 - 16*A^2*c^16*d^2*f^3))*(-((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*
d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) + 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(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))/4 + 240*A^3*c^5*d^11*f^2 + 320
*A^3*c^7*d^9*f^2 + 240*A^3*c^9*d^7*f^2 + 96*A^3*c^11*d^5*f^2 + 16*A^3*c^13*d^3*f^2 + 16*A^3*c*d^15*f^2)*(-((32
0*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2)
+ 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(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))/4 - log(96*A^3*c^3*d^13*f^2 - ((((320*A^4*c^2*d^8*f^4 - 16*A^4*d^1
0*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*
d^2*f^2 - 20*A^2*c*d^4*f^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)*(896*A*c^6*d^15*f^4 - (((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 +
1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(1
6*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)*(c + d*
tan(e + f*x))^(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*c^2*d^19*f^4 - 128*A*c^4*d^17*f^4 - 32*A*d^21*f^4 + 3136*A*c^8*d^13*f^4 + 4928*A*c^10
*d^11*f^4 + 4480*A*c^12*d^9*f^4 + 2432*A*c^14*d^7*f^4 + 736*A*c^16*d^5*f^4 + 96*A*c^18*d^3*f^4) + (c + d*tan(e
+ f*x))^(1/2)*(320*A^2*c^4*d^14*f^3 - 16*A^2*d^18*f^3 + 1024*A^2*c^6*d^12*f^3 + 1440*A^2*c^8*d^10*f^3 + 1024*
A^2*c^10*d^8*f^3 + 320*A^2*c^12*d^6*f^3 - 16*A^...

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