3.1230 \(\int (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)} \, dx\)

Optimal. Leaf size=157 \[ \frac {4 a b \sqrt {c+d \tan (e+f x)}}{f}-\frac {i (a-i b)^2 \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 b^2 (c+d \tan (e+f x))^{3/2}}{3 d f} \]

[Out]

-I*(a-I*b)^2*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I*d)^(1/2)/f+I*(a+I*b)^2*arctanh((c+d*tan(f*x+e)
)^(1/2)/(c+I*d)^(1/2))*(c+I*d)^(1/2)/f+4*a*b*(c+d*tan(f*x+e))^(1/2)/f+2/3*b^2*(c+d*tan(f*x+e))^(3/2)/d/f

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Rubi [A]  time = 0.35, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3543, 3528, 3539, 3537, 63, 208} \[ \frac {4 a b \sqrt {c+d \tan (e+f x)}}{f}-\frac {i (a-i b)^2 \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 b^2 (c+d \tan (e+f x))^{3/2}}{3 d f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]],x]

[Out]

((-I)*(a - I*b)^2*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (I*(a + I*b)^2*Sqrt[c + I
*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (4*a*b*Sqrt[c + d*Tan[e + f*x]])/f + (2*b^2*(c + d*Ta
n[e + f*x])^(3/2))/(3*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 3543

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

Rubi steps

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

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Mathematica [A]  time = 0.56, size = 149, normalized size = 0.95 \[ \frac {2 b \sqrt {c+d \tan (e+f x)} (6 a d+b c+b d \tan (e+f x))-3 i d (a-i b)^2 \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+3 i d (a+i b)^2 \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{3 d f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]],x]

[Out]

((-3*I)*(a - I*b)^2*Sqrt[c - I*d]*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (3*I)*(a + I*b)^2*Sqrt[c
+ I*d]*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + 2*b*Sqrt[c + d*Tan[e + f*x]]*(b*c + 6*a*d + b*d*Ta
n[e + f*x]))/(3*d*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))^(1/2)*(a+b*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))^(1/2)*(a+b*tan(f*x+e))^2,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.24, size = 1507, normalized size = 9.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*(c+d*tan(f*x+e))^(3/2)/d/f+4*a*b*(c+d*tan(f*x+e))^(1/2)/f-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))*(c^2+d^2)^(1/2)*a*b+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))*a*b*c+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^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))*b^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^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))*b^
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))*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c-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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a*b-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))*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*b^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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^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))*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a^2*c+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))*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*a*b+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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c+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))*(c^2+d^2)^(1/2)*a*b-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))*a*
b*c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right ) + a\right )}^{2} \sqrt {d \tan \left (f x + e\right ) + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e) + a)^2*sqrt(d*tan(f*x + e) + c), x)

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mupad [B]  time = 9.66, size = 3722, normalized size = 23.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^2*(c + d*tan(e + f*x))^(3/2))/(3*d*f) - atan(((((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c
+ d*tan(e + f*x))^(1/2)*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^
3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2))*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b
^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^
4*d^4 + b^4*d^4 - 6*a^2*b^2*d^4 - a^4*c^2*d^2 - b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^
3))/f^2)*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b
^3*d - 4*a^3*b*d)/(4*f^2))^(1/2)*1i - (((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e +
f*x))^(1/2)*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*
a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2))*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b
^3*c*4i - a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^4 + b^4*d
^4 - 6*a^2*b^2*d^4 - a^4*c^2*d^2 - b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^3))/f^2)*(-(a
^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b^3*d - 4*a^3*
b*d)/(4*f^2))^(1/2)*1i)/((((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(
-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b^3*d - 4*a
^3*b*d)/(4*f^2))^(1/2))*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3
*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^4 + b^4*d^4 - 6*a^2*b^
2*d^4 - a^4*c^2*d^2 - b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^3))/f^2)*(-(a^4*c - a^4*d*
1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))
^(1/2) - (16*(a^6*d^5 - b^6*d^5 - a^2*b^4*d^5 + a^4*b^2*d^5 + a^6*c^2*d^3 - b^6*c^2*d^3 + 2*a*b^5*c^3*d^2 + 4*
a^3*b^3*c*d^4 + 2*a^5*b*c^3*d^2 - a^2*b^4*c^2*d^3 + 4*a^3*b^3*c^3*d^2 + a^4*b^2*c^2*d^3 + 2*a*b^5*c*d^4 + 2*a^
5*b*c*d^4))/f^3 + (((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(a^4*c
- a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)
/(4*f^2))^(1/2))*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i
+ 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^4 + b^4*d^4 - 6*a^2*b^2*d^4 -
a^4*c^2*d^2 - b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^3))/f^2)*(-(a^4*c - a^4*d*1i + b^
4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2))
)*(-(a^4*c - a^4*d*1i + b^4*c - b^4*d*1i - 6*a^2*b^2*c + a^2*b^2*d*6i + a*b^3*c*4i - a^3*b*c*4i + 4*a*b^3*d -
4*a^3*b*d)/(4*f^2))^(1/2)*2i - atan(((((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f
*x))^(1/2)*(-(a^4*c + a^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a
*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2))*(-(a^4*c + a^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^
3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^4 + b^4*d^
4 - 6*a^2*b^2*d^4 - a^4*c^2*d^2 - b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^3))/f^2)*(-(a^
4*c + a^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b
*d)/(4*f^2))^(1/2)*1i - (((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-
(a^4*c + a^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^
3*b*d)/(4*f^2))^(1/2))*(-(a^4*c + a^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*
b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^4 + b^4*d^4 - 6*a^2*b^2
*d^4 - a^4*c^2*d^2 - b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^3))/f^2)*(-(a^4*c + a^4*d*1
i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^
(1/2)*1i)/((((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(a^4*c + a^4*
d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2
))^(1/2))*(-(a^4*c + a^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*
b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^4 + b^4*d^4 - 6*a^2*b^2*d^4 - a^4*c^
2*d^2 - b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^3))/f^2)*(-(a^4*c + a^4*d*1i + b^4*c + b
^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2) - (16*(
a^6*d^5 - b^6*d^5 - a^2*b^4*d^5 + a^4*b^2*d^5 + a^6*c^2*d^3 - b^6*c^2*d^3 + 2*a*b^5*c^3*d^2 + 4*a^3*b^3*c*d^4
+ 2*a^5*b*c^3*d^2 - a^2*b^4*c^2*d^3 + 4*a^3*b^3*c^3*d^2 + a^4*b^2*c^2*d^3 + 2*a*b^5*c*d^4 + 2*a^5*b*c*d^4))/f^
3 + (((8*(8*a*b*d^4*f^2 + 8*a*b*c^2*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(a^4*c + a^4*d*1i +
b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2
))*(-(a^4*c + a^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d -
4*a^3*b*d)/(4*f^2))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^4 + b^4*d^4 - 6*a^2*b^2*d^4 - a^4*c^2*d^2 -
b^4*c^2*d^2 + 6*a^2*b^2*c^2*d^2 - 8*a*b^3*c*d^3 + 8*a^3*b*c*d^3))/f^2)*(-(a^4*c + a^4*d*1i + b^4*c + b^4*d*1i
- 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*f^2))^(1/2)))*(-(a^4*c + a
^4*d*1i + b^4*c + b^4*d*1i - 6*a^2*b^2*c - a^2*b^2*d*6i - a*b^3*c*4i + a^3*b*c*4i + 4*a*b^3*d - 4*a^3*b*d)/(4*
f^2))^(1/2)*2i - ((4*b^2*c - 4*a*b*d)/(d*f) - (4*b^2*c)/(d*f))*(c + d*tan(e + f*x))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \sqrt {c + d \tan {\left (e + f x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tan(e + f*x))**2*sqrt(c + d*tan(e + f*x)), x)

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