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

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

[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 = 0.63, 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 = {3630, 3528, 3539, 3537, 63, 208} \[ \frac {2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (d (A-C)+B c) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 B (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 C (c+d \tan (e+f x))^{7/2}}{7 d f} \]

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 63

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 3528

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 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 3539

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 3630

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 2.07, size = 262, normalized size = 1.14 \[ \frac {7 i (A-i B-C) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c-i d) \left (\sqrt {c+d \tan (e+f x)} (4 c+d \tan (e+f x)-3 i d)-3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )\right )\right )-7 i (A+i B-C) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+\frac {2}{3} (c+i d) \left (\sqrt {c+d \tan (e+f x)} (4 c+d \tan (e+f x)+3 i d)-3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )\right )\right )+\frac {4 C (c+d \tan (e+f x))^{7/2}}{d}}{14 f} \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [B]  time = 0.41, size = 3614, normalized size = 15.78 \[ \text {output too large to display} \]

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)

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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mupad [B]  time = 117.31, size = 5863, normalized size = 25.60 \[ \text {result too large to display} \]

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)))/(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)/(4*f^4) - (A^2*c^5)/(4*
f^2) + (5*A^2*c^3*d^2)/(2*f^2) - (5*A^2*c*d^4)/(4*f^2))^(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)*((5*A^2*c^3*d^2)/(2*f^2)
- (A^2*c^5)/(4*f^2) - (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)/(4*f^4) - (5*A^2*c*d^4)/(4*f^2))^(1/2) - log((8*C^3*d^3*(3*c^2 - d^2)*(c^2 + d^2)^3)/f^3 -
((((-((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + C^2*c^5*f^2 - 10*C^2*c^3*d^2*f^2 + 5*C^2*c*d^4*f^2)/
f^4)^(1/2)*(64*C*c^3*d^3 + 64*C*c*d^5 - 32*c*d^2*f*(-((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + C^2*
c^5*f^2 - 10*C^2*c^3*d^2*f^2 + 5*C^2*c*d^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) - (16*C^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)*(-((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^
2)^2)^(1/2) + C^2*c^5*f^2 - 10*C^2*c^3*d^2*f^2 + 5*C^2*c*d^4*f^2)/f^4)^(1/2))/2)*(-((20*C^4*c^2*d^8*f^4 - C^4*
d^10*f^4 - 110*C^4*c^4*d^6*f^4 + 100*C^4*c^6*d^4*f^4 - 25*C^4*c^8*d^2*f^4)^(1/2) + C^2*c^5*f^2 - 10*C^2*c^3*d^
2*f^2 + 5*C^2*c*d^4*f^2)/(4*f^4))^(1/2) - log((8*C^3*d^3*(3*c^2 - d^2)*(c^2 + d^2)^3)/f^3 - ((((((-C^4*d^2*f^4
*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - C^2*c^5*f^2 + 10*C^2*c^3*d^2*f^2 - 5*C^2*c*d^4*f^2)/f^4)^(1/2)*(64*C*c^
3*d^3 + 64*C*c*d^5 - 32*c*d^2*f*(((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - C^2*c^5*f^2 + 10*C^2*c^3
*d^2*f^2 - 5*C^2*c*d^4*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) - (16*C^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)*(((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - C^2*c^5
*f^2 + 10*C^2*c^3*d^2*f^2 - 5*C^2*c*d^4*f^2)/f^4)^(1/2))/2)*(((20*C^4*c^2*d^8*f^4 - C^4*d^10*f^4 - 110*C^4*c^4
*d^6*f^4 + 100*C^4*c^6*d^4*f^4 - 25*C^4*c^8*d^2*f^4)^(1/2) - C^2*c^5*f^2 + 10*C^2*c^3*d^2*f^2 - 5*C^2*c*d^4*f^
2)/(4*f^4))^(1/2) + log((8*C^3*d^3*(3*c^2 - d^2)*(c^2 + d^2)^3)/f^3 - ((((((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2
*d^2)^2)^(1/2) - C^2*c^5*f^2 + 10*C^2*c^3*d^2*f^2 - 5*C^2*c*d^4*f^2)/f^4)^(1/2)*(64*C*c^3*d^3 + 64*C*c*d^5 + 3
2*c*d^2*f*(((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - C^2*c^5*f^2 + 10*C^2*c^3*d^2*f^2 - 5*C^2*c*d^4
*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) + (16*C^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)*(((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - C^2*c^5*f^2 + 10*C^2*c^3*d^2*
f^2 - 5*C^2*c*d^4*f^2)/f^4)^(1/2))/2)*((20*C^4*c^2*d^8*f^4 - C^4*d^10*f^4 - 110*C^4*c^4*d^6*f^4 + 100*C^4*c^6*
d^4*f^4 - 25*C^4*c^8*d^2*f^4)^(1/2)/(4*f^4) - (C^2*c^5)/(4*f^2) + (5*C^2*c^3*d^2)/(2*f^2) - (5*C^2*c*d^4)/(4*f
^2))^(1/2) + log((8*C^3*d^3*(3*c^2 - d^2)*(c^2 + d^2)^3)/f^3 - ((((-((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^
2)^(1/2) + C^2*c^5*f^2 - 10*C^2*c^3*d^2*f^2 + 5*C^2*c*d^4*f^2)/f^4)^(1/2)*(64*C*c^3*d^3 + 64*C*c*d^5 + 32*c*d^
2*f*(-((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + C^2*c^5*f^2 - 10*C^2*c^3*d^2*f^2 + 5*C^2*c*d^4*f^2)
/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) + (16*C^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)*(-((-C^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) + C^2*c^5*f^2 - 10*C^2*c^3*d^2*f^2
+ 5*C^2*c*d^4*f^2)/f^4)^(1/2))/2)*((5*C^2*c^3*d^2)/(2*f^2) - (C^2*c^5)/(4*f^2) - (20*C^4*c^2*d^8*f^4 - C^4*d^1
0*f^4 - 110*C^4*c^4*d^6*f^4 + 100*C^4*c^6*d^4*f^4 - 25*C^4*c^8*d^2*f^4)^(1/2)/(4*f^4) - (5*C^2*c*d^4)/(4*f^2))
^(1/2) + 2*c*((2*C*c^2)/(d*f) - (2*C*(d^3*f + c^2*d*f))/(d^2*f^2))*(c + d*tan(e + f*x))^(1/2) + (2*B*(c + d*ta
n(e + f*x))^(5/2))/(5*f) + (2*A*d*(c + d*tan(e + f*x))^(3/2))/(3*f) + (2*B*c*(c + d*tan(e + f*x))^(3/2))/(3*f)
+ (2*C*(c + d*tan(e + f*x))^(7/2))/(7*d*f) + (4*A*c*d*(c + d*tan(e + f*x))^(1/2))/f

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]

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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