3.13.49 \(\int \frac {(a+b \tan (e+f x))^2}{\sqrt {c+d \tan (e+f x)}} \, dx\) [1249]

Optimal. Leaf size=134 \[ -\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 b^2 \sqrt {c+d \tan (e+f x)}}{d f} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3624, 3620, 3618, 65, 214} \begin {gather*} -\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {2 b^2 \sqrt {c+d \tan (e+f x)}}{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*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (I*(a + I*b)^2*ArcTanh[
Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*b^2*Sqrt[c + d*Tan[e + f*x]])/(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 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 \frac {(a+b \tan (e+f x))^2}{\sqrt {c+d \tan (e+f x)}} \, dx &=\frac {2 b^2 \sqrt {c+d \tan (e+f x)}}{d f}+\int \frac {a^2-b^2+2 a b \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 b^2 \sqrt {c+d \tan (e+f x)}}{d f}+\frac {1}{2} (a-i b)^2 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} (a+i b)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 b^2 \sqrt {c+d \tan (e+f x)}}{d f}+\frac {\left (i (a-i b)^2\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)^2\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 b^2 \sqrt {c+d \tan (e+f x)}}{d f}-\frac {(a-i b)^2 \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 {(a+i b)^2 \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 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 b^2 \sqrt {c+d \tan (e+f x)}}{d f}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 129, normalized size = 0.96 \begin {gather*} \frac {-\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}+\frac {2 b^2 \sqrt {c+d \tan (e+f x)}}{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]

(((-I)*(a - I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (I*(a + I*b)^2*ArcTanh[Sqr
t[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*b^2*Sqrt[c + d*Tan[e + f*x]])/d)/f

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2009\) vs. \(2(112)=224\).
time = 0.43, size = 2010, normalized size = 15.00

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2010\)
default	\(\text {Expression too large to display}\)	\(2010\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/d*((c+d*tan(f*x+e))^(1/2)*b^2+d*(1/4/(c^2+d^2)^(3/2)/d^2*(1/2*(-2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a*b*c-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a^2*d^3+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a*b*c*d^2+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b^2*d^3+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^3+2*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*a*b*c^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d-
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^3)*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*(2*a^2*c^2*d^3+2*a^2*d^5-4*a*b*c^3*d^2-4*a*b*c*d^4-2*b^2*c^2*d^3-2*b^2*d^5+1/2*(-2*(c^2+d
^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d-(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^3+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3+2*(c^2+d
^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d^2+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d+(c^2
+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^3+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d+(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*a^2*c*d^3+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4-(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^3)*(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/(c^2+d^2)^(3/2)/d^2*(1/2*(2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c+(c^2+d^2)^(1/2
)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^3-2*(c^2+d^2)^(1
/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d^2-(c^2+d^2)^
(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^3-(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a^2*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^3-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*
d^2-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b^2*c*d^3)*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*(2*a^
2*c^2*d^3+2*a^2*d^5-4*a*b*c^3*d^2-4*a*b*c*d^4-2*b^2*c^2*d^3-2*b^2*d^5-1/2*(2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*b*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a^2*d^3-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*b*c*d^2-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*b^2*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^3-2*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^2-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2
*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^3)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*a
rctan((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((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(1/2),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. 14968 vs. \(2 (107) = 214\).
time = 55.51, size = 14968, normalized size = 111.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*sqrt(2)*(c^2*d + d^3)*f^5*sqrt((((a^4 - 6*a^2*b^2 + b^4)*c^3 + 4*(a^3*b - a*b^3)*c^2*d + (a^4 - 6*a^2*b
^2 + b^4)*c*d^2 + 4*(a^3*b - a*b^3)*d^3)*f^2*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)/((c^2 + 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^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)/((c^4 + 2*c^2*d^2 + d^4)*f
^4))*((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)/((c^2 + d^2)*f^4))^(3/4)*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^5 - (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^4*d + 8*(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*d^2 - 2*(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^3 + 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^4 - (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)*d^5)*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)/((c^4 + 2*c^2*d^2 + d^4)*f^4))*sqrt((a^8 + 4*a^6*b^
2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)/((c^2 + 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 - 30
8*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)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) + sqrt(2)*((2*a*b*c^5 + 4*a*b*c^3*d^2 + 2*a*b*c*d^4 - (a^2 -
b^2)*c^4*d - 2*(a^2 - b^2)*c^2*d^3 - (a^2 - b^2)*d^5)*f^7*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)/((c^4 + 2*
c^2*d^2 + d^4)*f^4))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)/((c^2 + d^2)*f^4)) + 2*((a^5*b + 2*a
^3*b^3 + a*b^5)*c^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*c^2*d^2 + (a^5*b + 2*a^3*b^3 + a*b^5)*d^4)*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)/((c^4 + 2*c^2*d^2 + d^4)*f^4)))*sqrt((((a^4 - 6*a^2*b^2 + b^4)*c^3 + 4*(a^3*b -
a*b^3)*c^2*d + (a^4 - 6*a^2*b^2 + b^4)*c*d^2 + 4*(a^3*b - a*b^3)*d^3)*f^2*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8)/((c^2 + 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 + 1
5*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 + d^2)*f^4))*cos(f*x + e) + sqrt(2)*(2
*(16*(a^7*b^3 - 2*a^5*b^5 + a^3*b^7)*c^4 - 8*(a^8*b^2 - 7*a^6*b^4 + 7*a^4*b^6 - a^2*b^8)*c^3*d + (a^9*b + 4*a^
7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^2*d^2 - 8*(a^8*b^2 - 7*a^6*b^4 + 7*a^4*b^6 - a^2*b^8)*c*d^3 + (a^9*b
- 12*a^7*b^3 + 38*a^5*b^5 - 12*a^3*b^7 + a*b^9)*d^4)*f^3*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)/
((c^2 + d^2)*f^4))*cos(f*x + e) + (32*(a^11*b^3 - 2*a^7*b^7 + a^3*b^11)*c^3 - 32*(a^12*b^2 - 3*a^10*b^4 - 4*a^
8*b^6 + 4*a^6*b^8 + 3*a^4*b^10 - a^2*b^12)*c^2*d + 2*(5*a^13*b - 34*a^11*b^3 + 11*a^9*b^5 + 100*a^7*b^7 + 11*a
^5*b^9 - 34*a^3*b^11 + 5*a*b^13)*c*d^2 - (a^14 - 11*a^12*b^2 + 25*a^10*b^4 + 37*a^8*b^6 - 37*a^6*b^8 - 25*a^4*
b^10 + 11*a^2*b^12 - b^14)*d^3)*f*cos(f*x + e))*sqrt((((a^4 - 6*a^2*b^2 + b^4)*c^3 + 4*(a^3*b - a*b^3)*c^2*d +
(a^4 - 6*a^2*b^2 + b^4)*c*d^2 + 4*(a^3*b - a*b^3)*d^3)*f^2*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^
8)/((c^2 + 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 +...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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((a+b*tan(f*x+e))**2/(c+d*tan(f*x+e))**(1/2),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((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(1/2),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 = 7.23, size = 2287, normalized size = 17.07 \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))^(1/2))/(d*f) - atan(((((16*(2*b^2*d^3*f^2 - 2*a^2*d^3*f^2 + 4*a*b*c*d^2*f^2))/f^3
- 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^
(1/2))*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x)
)^(1/2)*(a^4*d^2 + b^4*d^2 - 6*a^2*b^2*d^2))/f^2)*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 -
d*f^2*1i)))^(1/2)*1i - (((16*(2*b^2*d^3*f^2 - 2*a^2*d^3*f^2 + 4*a*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e +
f*x))^(1/2)*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*(-(a*b^3*4i - a^3*b
*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^2 + b^4*d^2
- 6*a^2*b^2*d^2))/f^2)*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*1i)/((((
16*(2*b^2*d^3*f^2 - 2*a^2*d^3*f^2 + 4*a*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(a*b^3*4i -
a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^
2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^2 + b^4*d^2 - 6*a^2*b^2*d^2))/f^2)*(-
(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (((16*(2*b^2*d^3*f^2 - 2*a^2*d^3
*f^2 + 4*a*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*
b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1
/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^2 + b^4*d^2 - 6*a^2*b^2*d^2))/f^2)*(-(a*b^3*4i - a^3*b*4i + a^4 +
b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (32*(a*b^5*d^2 + a^5*b*d^2 + 2*a^3*b^3*d^2))/f^3))*(-(a*b^3*4
i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*2i - atan(((((16*(2*b^2*d^3*f^2 - 2*a^2*d^
3*f^2 + 4*a*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a
^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2))*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i -
d*f^2)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^2 + b^4*d^2 - 6*a^2*b^2*d^2))/f^2)*(-(4*a*b^3 - 4*a^3*b
+ a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2)*1i - (((16*(2*b^2*d^3*f^2 - 2*a^2*d^3*f^2 + 4*a
*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/
(4*(c*f^2*1i - d*f^2)))^(1/2))*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1
/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^2 + b^4*d^2 - 6*a^2*b^2*d^2))/f^2)*(-(4*a*b^3 - 4*a^3*b + a^4*1i +
b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2)*1i)/((((16*(2*b^2*d^3*f^2 - 2*a^2*d^3*f^2 + 4*a*b*c*d^2*f^
2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1
i - d*f^2)))^(1/2))*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2) - (16*(
c + d*tan(e + f*x))^(1/2)*(a^4*d^2 + b^4*d^2 - 6*a^2*b^2*d^2))/f^2)*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a
^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2) - (32*(a*b^5*d^2 + a^5*b*d^2 + 2*a^3*b^3*d^2))/f^3 + (((16*(2*b^2*d^3
*f^2 - 2*a^2*d^3*f^2 + 4*a*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(4*a*b^3 - 4*a^3*b + a^4*
1i + b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2))*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/
(4*(c*f^2*1i - d*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^2 + b^4*d^2 - 6*a^2*b^2*d^2))/f^2)*(-(4*
a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2)))*(-(4*a*b^3 - 4*a^3*b + a^4*1i
+ b^4*1i - a^2*b^2*6i)/(4*(c*f^2*1i - d*f^2)))^(1/2)*2i

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