3.13.30 \(\int (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)} \, dx\) [1230]

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

[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 = 0.24, 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 = {3624, 3609, 3620, 3618, 65, 214} \begin {gather*} \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} \end {gather*}

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 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 3624

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 ) \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 ((a-i b)^2 (i c+d)\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 {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 ) \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)^2 (c+i d)\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-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 = 0.59, size = 149, normalized size = 0.95 \begin {gather*} \frac {-3 i (a-i b)^2 \sqrt {c-i d} d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+3 i (a+i b)^2 \sqrt {c+i d} d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+2 b \sqrt {c+d \tan (e+f x)} (b c+6 a d+b d \tan (e+f x))}{3 d f} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(889\) vs. \(2(131)=262\).
time = 0.43, size = 890, normalized size = 5.67

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 a b d \sqrt {c +d \tan \left (f x +e \right )}+2 d \left (\frac {\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-4 \sqrt {c^{2}+d^{2}}\, a b d -\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}+\frac {\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-4 \sqrt {c^{2}+d^{2}}\, a b d +\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}\right )}{d f}\)	\(890\)
default	\(\frac {\frac {2 b^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 a b d \sqrt {c +d \tan \left (f x +e \right )}+2 d \left (\frac {\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-4 \sqrt {c^{2}+d^{2}}\, a b d -\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}+\frac {\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-4 \sqrt {c^{2}+d^{2}}\, a b d +\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}\right )}{d f}\)	\(890\)

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,method=_RETURNVERBOSE)

[Out]

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

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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))^(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 16199 vs. \(2 (127) = 254\).
time = 99.74, size = 16199, normalized size = 103.18 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/12*(12*sqrt(2)*d*f^5*sqrt((((a^4 - 6*a^2*b^2 + b^4)*c - 4*(a^3*b - a*b^3)*d)*f^2*sqrt(((a^8 + 4*a^6*b^2 + 6
*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/f^4) + (a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/(16*(a^6*b^2 - 2*a^
4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2
*b^6 + b^8)*d^2))*(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2
*b^6 + b^8)*d^2)/f^4)^(3/4)*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 -
a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/f^4)*arctan(-((4*(a^15*b + 5*a^13*b^3 + 9
*a^11*b^5 + 5*a^9*b^7 - 5*a^7*b^9 - 9*a^5*b^11 - 5*a^3*b^13 - a*b^15)*c^3 + (a^16 - 20*a^12*b^4 - 64*a^10*b^6
- 90*a^8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*c^2*d + 4*(a^15*b + 5*a^13*b^3 + 9*a^11*b^5 + 5*a^9*b^7 - 5*a
^7*b^9 - 9*a^5*b^11 - 5*a^3*b^13 - a*b^15)*c*d^2 + (a^16 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^1
0 - 20*a^4*b^12 + b^16)*d^3)*f^4*sqrt(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/f^4)*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 +
7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/f^4) + (4*(a^19*b + 7*a^17*b
^3 + 20*a^15*b^5 + 28*a^13*b^7 + 14*a^11*b^9 - 14*a^9*b^11 - 28*a^7*b^13 - 20*a^5*b^15 - 7*a^3*b^17 - a*b^19)*
c^4 + (a^20 + 2*a^18*b^2 - 19*a^16*b^4 - 104*a^14*b^6 - 238*a^12*b^8 - 308*a^10*b^10 - 238*a^8*b^12 - 104*a^6*
b^14 - 19*a^4*b^16 + 2*a^2*b^18 + b^20)*c^3*d + 4*(a^19*b + 7*a^17*b^3 + 20*a^15*b^5 + 28*a^13*b^7 + 14*a^11*b
^9 - 14*a^9*b^11 - 28*a^7*b^13 - 20*a^5*b^15 - 7*a^3*b^17 - a*b^19)*c^2*d^2 + (a^20 + 2*a^18*b^2 - 19*a^16*b^4
- 104*a^14*b^6 - 238*a^12*b^8 - 308*a^10*b^10 - 238*a^8*b^12 - 104*a^6*b^14 - 19*a^4*b^16 + 2*a^2*b^18 + b^20
)*c*d^3)*f^2*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a
^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/f^4) + sqrt(2)*(2*(4*(a^8*b^2 + a^6*b^4 - a^4*b^6 - a^2*
b^8)*c + (a^9*b - 4*a^7*b^3 - 10*a^5*b^5 - 4*a^3*b^7 + a*b^9)*d)*f^7*sqrt(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^
2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/f^4)*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a
^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^
8)*d^2)/f^4) + (8*(a^12*b^2 + 3*a^10*b^4 + 2*a^8*b^6 - 2*a^6*b^8 - 3*a^4*b^10 - a^2*b^12)*c^2 + 2*(3*a^13*b +
2*a^11*b^3 - 19*a^9*b^5 - 36*a^7*b^7 - 19*a^5*b^9 + 2*a^3*b^11 + 3*a*b^13)*c*d + (a^14 - 3*a^12*b^2 - 15*a^10*
b^4 - 11*a^8*b^6 + 11*a^6*b^8 + 15*a^4*b^10 + 3*a^2*b^12 - b^14)*d^2)*f^5*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*
b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*
d^2)/f^4))*sqrt((((a^4 - 6*a^2*b^2 + b^4)*c - 4*(a^3*b - a*b^3)*d)*f^2*sqrt(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*
a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/f^4) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4
+ 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/(16*(a^6*b^2 - 2*a^4*b^4 + a^2*b
^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d
^2))*sqrt((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))*(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^
2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/f^4)^(3/4) - sqrt(2)*(2*a*b*f^7*sqrt(((a^8 + 4*a^6*b^
2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/f^4)*sqrt((16*(a^6
*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b
^4 - 12*a^2*b^6 + b^8)*d^2)/f^4) + (2*(a^5*b + 2*a^3*b^3 + a*b^5)*c + (a^6 + a^4*b^2 - a^2*b^4 - b^6)*d)*f^5*s
qrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^
2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/f^4))*sqrt((((a^4 - 6*a^2*b^2 + b^4)*c - 4*(a^3*b - a*b^3)*d)*f^2*sqrt
(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/f
^4) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^
2)/(16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2
+ 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2))*sqrt(((16*(a^10*b^2 - 2*a^6*b^6 + a^2*b^10)*c^4 + 8*(a^11*b - 5*a^9*b^
3 - 6*a^7*b^5 + 6*a^5*b^7 + 5*a^3*b^9 - a*b^11)*c^3*d + (a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*
b^8 + 6*a^2*b^10 + b^12)*c^2*d^2 + 8*(a^11*b - 5*a^9*b^3 - 6*a^7*b^5 + 6*a^5*b^7 + 5*a^3*b^9 - a*b^11)*c*d^3 +
(a^12 - 10*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4*b^8 - 10*a^2*b^10 + b^12)*d^4)*f^2*sqrt(((a^8 + 4*a^6*
b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 +...

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

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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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))^(1/2)*(a+b*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 = 9.66, size = 2500, normalized size = 15.92 \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*(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 +...

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