3.2.6 \(\int (c+d \tan (e+f x))^{5/2} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [106]

Optimal. Leaf size=229 \[ -\frac {(i A+B-i C) (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (B c+(A-C) d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 B (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 C (c+d \tan (e+f x))^{7/2}}{7 d f} \]

[Out]

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

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Rubi [A]
time = 0.42, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3711, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (d (A-C)+B c) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 B (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 C (c+d \tan (e+f x))^{7/2}}{7 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.39, size = 262, normalized size = 1.14 \begin {gather*} \frac {\frac {4 C (c+d \tan (e+f x))^{7/2}}{d}+7 i (A-i B-C) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c-i 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 )\right )-7 i (A+i B-C) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c+i 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 )\right )}{14 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*C*(c + d*Tan[e + f*x])^(7/2))/d + (7*I)*(A - I*B - C)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c - I*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) - (7*I)*(A + I*B - C)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c + I*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))/(14*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1786\) vs. \(2(196)=392\).
time = 0.48, size = 1787, normalized size = 7.80

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1787\)
default	\(\text {Expression too large to display}\)	\(1787\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/d*(-d*(1/4/d*(1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*d^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3-3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2+2*B*(c^2+d^2)^(1
/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d-3*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
d^3+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^2-C*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2)*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*(4*A*(c^2+d^2)^(1/2)*c*d^2+2*B*(c^2+d^2)^(1/2)*c^2*d-2*
B*(c^2+d^2)^(1/2)*d^3-4*C*(c^2+d^2)^(1/2)*c*d^2+1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+A*(c
^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3-3*A*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*c*d^2+2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d-3*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d+B*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2-C*(c^2+d^2)^(1/2)*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*d^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2)*(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)*c^2-A*(c
^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^2-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+3*A*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*c*d^2-2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d+3*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d-B*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+C*(c^2+d^2)^(1/2)*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*d^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3-3*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2)*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*(4*A*(c^2+d^2)^(1/2)*c*d^
2+2*B*(c^2+d^2)^(1/2)*c^2*d-2*B*(c^2+d^2)^(1/2)*d^3-4*C*(c^2+d^2)^(1/2)*c*d^2-1/2*(A*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*c^2-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^2-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c
^3+3*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2-2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d+3*B*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*c^2*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*c^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3-3*C*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*c*d^2)*(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))))+1/7*C*(c+d*tan(f*x+e))^(7/2)+1/5*B*d*(
c+d*tan(f*x+e))^(5/2)+1/3*A*d^2*(c+d*tan(f*x+e))^(3/2)+1/3*B*c*d*(c+d*tan(f*x+e))^(3/2)-1/3*C*d^2*(c+d*tan(f*x
+e))^(3/2)+2*A*c*d^2*(c+d*tan(f*x+e))^(1/2)+B*c^2*d*(c+d*tan(f*x+e))^(1/2)-B*d^3*(c+d*tan(f*x+e))^(1/2)-2*C*c*
d^2*(c+d*tan(f*x+e))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral((c + d*tan(e + f*x))**(5/2)*(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((c+d*tan(f*x+e))^(5/2)*(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 [B]
time = 117.31, size = 2500, normalized size = 10.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*C*c^2)/(3*d*f) - (2*C*(d^3*f + c^2*d*f))/(3*d^2*f^2))*(c + d*tan(e + f*x))^(3/2) - log(((((-B^4*d^2*f^4*(5
*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + B^2*c^5*f^2 - 10*B^2*c^3*d^2*f^2 + 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(((((-B^4*d
^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + B^2*c^5*f^2 - 10*B^2*c^3*d^2*f^2 + 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(3
2*B*c^4*d^2 - 32*B*d^6 + 32*c*d^2*f*(((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + B^2*c^5*f^2 - 10*B^2
*c^3*d^2*f^2 + 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) - (16*B^2*d^2*(c + d*tan(e + f*x
))^(1/2)*(c^6 - d^6 + 15*c^2*d^4 - 15*c^4*d^2))/f^2))/2 - (8*B^3*c*d^2*(c^2 - 3*d^2)*(c^2 + d^2)^3)/f^3)*(((20
*B^4*c^2*d^8*f^4 - B^4*d^10*f^4 - 110*B^4*c^4*d^6*f^4 + 100*B^4*c^6*d^4*f^4 - 25*B^4*c^8*d^2*f^4)^(1/2) + B^2*
c^5*f^2 - 10*B^2*c^3*d^2*f^2 + 5*B^2*c*d^4*f^2)/(4*f^4))^(1/2) + log(- ((((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*
d^2)^2)^(1/2) + B^2*c^5*f^2 - 10*B^2*c^3*d^2*f^2 + 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(((((-B^4*d^2*f^4*(5*c^4 + d^4
- 10*c^2*d^2)^2)^(1/2) + B^2*c^5*f^2 - 10*B^2*c^3*d^2*f^2 + 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(32*B*d^6 - 32*B*c^4*d
^2 + 32*c*d^2*f*(((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + B^2*c^5*f^2 - 10*B^2*c^3*d^2*f^2 + 5*B^2
*c*d^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) - (16*B^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^6 - d^6
+ 15*c^2*d^4 - 15*c^4*d^2))/f^2))/2 - (8*B^3*c*d^2*(c^2 - 3*d^2)*(c^2 + d^2)^3)/f^3)*((20*B^4*c^2*d^8*f^4 - B^
4*d^10*f^4 - 110*B^4*c^4*d^6*f^4 + 100*B^4*c^6*d^4*f^4 - 25*B^4*c^8*d^2*f^4)^(1/2)/(4*f^4) + (B^2*c^5)/(4*f^2)
- (5*B^2*c^3*d^2)/(2*f^2) + (5*B^2*c*d^4)/(4*f^2))^(1/2) - log(((-((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2
)^(1/2) - B^2*c^5*f^2 + 10*B^2*c^3*d^2*f^2 - 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(((-((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*
c^2*d^2)^2)^(1/2) - B^2*c^5*f^2 + 10*B^2*c^3*d^2*f^2 - 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(32*B*c^4*d^2 - 32*B*d^6 +
32*c*d^2*f*(-((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - B^2*c^5*f^2 + 10*B^2*c^3*d^2*f^2 - 5*B^2*c*d
^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) - (16*B^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^6 - d^6 + 15
*c^2*d^4 - 15*c^4*d^2))/f^2))/2 - (8*B^3*c*d^2*(c^2 - 3*d^2)*(c^2 + d^2)^3)/f^3)*(-((20*B^4*c^2*d^8*f^4 - B^4*
d^10*f^4 - 110*B^4*c^4*d^6*f^4 + 100*B^4*c^6*d^4*f^4 - 25*B^4*c^8*d^2*f^4)^(1/2) - B^2*c^5*f^2 + 10*B^2*c^3*d^
2*f^2 - 5*B^2*c*d^4*f^2)/(4*f^4))^(1/2) + log(- ((-((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - B^2*c^
5*f^2 + 10*B^2*c^3*d^2*f^2 - 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(((-((-B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2
) - B^2*c^5*f^2 + 10*B^2*c^3*d^2*f^2 - 5*B^2*c*d^4*f^2)/f^4)^(1/2)*(32*B*d^6 - 32*B*c^4*d^2 + 32*c*d^2*f*(-((-
B^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - B^2*c^5*f^2 + 10*B^2*c^3*d^2*f^2 - 5*B^2*c*d^4*f^2)/f^4)^(1/
2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) - (16*B^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^6 - d^6 + 15*c^2*d^4 - 15*c^
4*d^2))/f^2))/2 - (8*B^3*c*d^2*(c^2 - 3*d^2)*(c^2 + d^2)^3)/f^3)*((B^2*c^5)/(4*f^2) - (20*B^4*c^2*d^8*f^4 - B^
4*d^10*f^4 - 110*B^4*c^4*d^6*f^4 + 100*B^4*c^6*d^4*f^4 - 25*B^4*c^8*d^2*f^4)^(1/2)/(4*f^4) - (5*B^2*c^3*d^2)/(
2*f^2) + (5*B^2*c*d^4)/(4*f^2))^(1/2) + ((4*B*c^2)/f - (2*B*(c^2*f + d^2*f))/f^2)*(c + d*tan(e + f*x))^(1/2) -
log(((((-((-A^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + A^2*c^5*f^2 - 10*A^2*c^3*d^2*f^2 + 5*A^2*c*d^4*
f^2)/f^4)^(1/2)*(64*A*c^3*d^3 + 64*A*c*d^5 + 32*c*d^2*f*(-((-A^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) +
A^2*c^5*f^2 - 10*A^2*c^3*d^2*f^2 + 5*A^2*c*d^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) + (16*A^2*d
^2*(c + d*tan(e + f*x))^(1/2)*(c^6 - d^6 + 15*c^2*d^4 - 15*c^4*d^2))/f^2)*(-((-A^4*d^2*f^4*(5*c^4 + d^4 - 10*c
^2*d^2)^2)^(1/2) + A^2*c^5*f^2 - 10*A^2*c^3*d^2*f^2 + 5*A^2*c*d^4*f^2)/f^4)^(1/2))/2 - (8*A^3*d^3*(3*c^2 - d^2
)*(c^2 + d^2)^3)/f^3)*(-((20*A^4*c^2*d^8*f^4 - A^4*d^10*f^4 - 110*A^4*c^4*d^6*f^4 + 100*A^4*c^6*d^4*f^4 - 25*A
^4*c^8*d^2*f^4)^(1/2) + A^2*c^5*f^2 - 10*A^2*c^3*d^2*f^2 + 5*A^2*c*d^4*f^2)/(4*f^4))^(1/2) - log(((((((-A^4*d^
2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - A^2*c^5*f^2 + 10*A^2*c^3*d^2*f^2 - 5*A^2*c*d^4*f^2)/f^4)^(1/2)*(64
*A*c^3*d^3 + 64*A*c*d^5 + 32*c*d^2*f*(((-A^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - A^2*c^5*f^2 + 10*A^
2*c^3*d^2*f^2 - 5*A^2*c*d^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) + (16*A^2*d^2*(c + d*tan(e + f*
x))^(1/2)*(c^6 - d^6 + 15*c^2*d^4 - 15*c^4*d^2))/f^2)*(((-A^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - A^
2*c^5*f^2 + 10*A^2*c^3*d^2*f^2 - 5*A^2*c*d^4*f^2)/f^4)^(1/2))/2 - (8*A^3*d^3*(3*c^2 - d^2)*(c^2 + d^2)^3)/f^3)
*(((20*A^4*c^2*d^8*f^4 - A^4*d^10*f^4 - 110*A^4*c^4*d^6*f^4 + 100*A^4*c^6*d^4*f^4 - 25*A^4*c^8*d^2*f^4)^(1/2)
- A^2*c^5*f^2 + 10*A^2*c^3*d^2*f^2 - 5*A^2*c*d^4*f^2)/(4*f^4))^(1/2) + log(((((((-A^4*d^2*f^4*(5*c^4 + d^4 - 1
0*c^2*d^2)^2)^(1/2) - A^2*c^5*f^2 + 10*A^2*c^3*d^2*f^2 - 5*A^2*c*d^4*f^2)/f^4)^(1/2)*(64*A*c^3*d^3 + 64*A*c*d^
5 - 32*c*d^2*f*(((-A^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - A^2*c^5*f^2 + 10*A^2*c^3*d^2*f^2 - 5*A^2*
c*d^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2...

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