∫ (a+b tan(e+fx))2 ____________________________

3.13.55 \(\int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx\) [1255]

Optimal. Leaf size=150 \[ -\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}-\frac {2 (b c-a d)^2}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]

[Out]

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

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

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Rubi [A]
time = 0.22, antiderivative size = 150, 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 = {3623, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

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

h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(b*c - a*d)^2)/(d*(c^2 + d^2)*f*Sqrt[c + d

*Tan[e + f*x]])

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 3623

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[

(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Ta

n[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Tan[e + f*x], x], x], x] /; FreeQ

[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.19, size = 124, normalized size = 0.83 \begin {gather*} \frac {-\frac {2 b^2}{d}-\frac {(a-i b)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{i c+d}+\frac {(a+i b)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{i c-d}}{f \sqrt {c+d \tan (e+f x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

((-2*b^2)/d - ((a - I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(I*c + d) + ((a +

I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(I*c - d))/(f*Sqrt[c + d*Tan[e + f*x]]

)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3310\) vs. \(2(128)=256\).
time = 0.46, size = 3311, normalized size = 22.07


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(3311\)
default	\(\text {Expression too large to display}\)	\(3311\)

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))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/f/d*(d/(c^2+d^2)*(1/4/d^2/(3*c^2-d^2)/(c^2+d^2)^(3/2)*(1/2*(-2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)

*a*b*c^4+2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/

2)*a^2*c^5*d-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*

c)^(1/2)*a^2*c*d^5+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^6-4*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2

)+2*c)^(1/2)*a*b*c^4*d^2-6*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2

+d^2)^(1/2)+2*c)^(1/2)*b^2*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d^3-(c^2+d^2)^(1/2)*(

2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^5+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^6*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)

*a^2*c^4*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d^5+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^7+12*(2*(c^2+d^2)

^(1/2)+2*c)^(1/2)*a*b*c^5*d^2+8*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3*d^4-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*

c*d^6-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^4*d^3+3*(2*(c^2+d^2)^(1/2)

+2*c)^(1/2)*b^2*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^7)*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*(12*a^2*c^5*d^3+8*a^2*c^3*d^5-4*a^2*c*d^7-12*a*b*c^6*d^2+4*a*b*c^4*

d^4+12*a*b*c^2*d^6-4*a*b*d^8-12*b^2*c^5*d^3-8*b^2*c^3*d^5+4*b^2*c*d^7+1/2*(-2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/

2)+2*c)^(1/2)*a*b*c^4+2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(

1/2)+2*c)^(1/2)*a^2*c^5*d-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d^3+(c^2+d^2)^(1/2)*(2*(c^2+

d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^5+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^6-4*(c^2+d^2)^(1/2)*(2*(

c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^4*d^2-6*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^4+3*(c^2+d^2)^

(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d^3-(c^2

+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^5+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^6*d-(2*(c^2+d^2)^(1/

2)+2*c)^(1/2)*a^2*c^4*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d^5+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^7+12

*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^5*d^2+8*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3*d^4-4*(2*(c^2+d^2)^(1/2)+2*

c)^(1/2)*a*b*c*d^6-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^4*d^3+3*(2*(c

^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^7)*(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^2/(3*c^2-d^2)/(c^2+d^2)^(3/2)*(1/2*(2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^4

-2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c

^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)

*a^2*c*d^5-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^6+4*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(

1/2)*a*b*c^4*d^2+6*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^4-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1

/2)+2*c)^(1/2)*b^2*c^5*d-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d

^2)^(1/2)+2*c)^(1/2)*b^2*c*d^5-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^4

*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^7-12*(2*(c^2+d^2)^(1/2)+2

*c)^(1/2)*a*b*c^5*d^2-8*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3*d^4+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d^6+3*

(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^6*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^4*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1

/2)*b^2*c^2*d^5+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^7)*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*(12*a^2*c^5*d^3+8*a^2*c^3*d^5-4*a^2*c*d^7-12*a*b*c^6*d^2+4*a*b*c^4*d^4+12*a

*b*c^2*d^6-4*a*b*d^8-12*b^2*c^5*d^3-8*b^2*c^3*d^5+4*b^2*c*d^7-1/2*(2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(

1/2)*a*b*c^4-2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)

^(1/2)*a^2*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2

)+2*c)^(1/2)*a^2*c*d^5-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^6+4*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^

(1/2)+2*c)^(1/2)*a*b*c^4*d^2+6*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^4-3*(c^2+d^2)^(1/2)*(2*

(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^5*d-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d^3+(c^2+d^2)^(1/

2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^5-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(

1/2)*a^2*c^4*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^7-12*(2*(c^2+

d^2)^(1/2)+2*c)^(1/2)*a*b*c^5*d^2-8*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3*d^4+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*

a*b*c*d^6+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c...

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Maxima [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((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35751 vs. \(2 (123) = 246\).
time = 240.23, size = 35751, normalized size = 238.34 \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))^(3/2),x, algorithm="fricas")

[Out]

1/4*(4*sqrt(2)*((c^10*d + 3*c^8*d^3 + 2*c^6*d^5 - 2*c^4*d^7 - 3*c^2*d^9 - d^11)*f^5*cos(f*x + e)^2 + 2*(c^9*d^

2 + 4*c^7*d^4 + 6*c^5*d^6 + 4*c^3*d^8 + c*d^10)*f^5*cos(f*x + e)*sin(f*x + e) + (c^8*d^3 + 4*c^6*d^5 + 6*c^4*d

^7 + 4*c^2*d^9 + d^11)*f^5)*sqrt(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^6 + 3*(a^8 + 4*a^6*b^2 + 6

*a^4*b^4 + 4*a^2*b^6 + b^8)*c^4*d^2 + 3*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2*d^4 + (a^8 + 4*a^6

*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^6 + ((a^4 - 6*a^2*b^2 + b^4)*c^9 + 12*(a^3*b - a*b^3)*c^8*d + 32*(a^3*b

- a*b^3)*c^6*d^3 - 6*(a^4 - 6*a^2*b^2 + b^4)*c^5*d^4 + 24*(a^3*b - a*b^3)*c^4*d^5 - 8*(a^4 - 6*a^2*b^2 + b^4)*

c^3*d^6 - 3*(a^4 - 6*a^2*b^2 + b^4)*c*d^8 - 4*(a^3*b - a*b^3)*d^9)*f^2*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a

^2*b^6 + b^8)/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^4)))/(16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^6 - 24*(a^7*b

- 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c^5*d + 3*(3*a^8 - 68*a^6*b^2 + 178*a^4*b^4 - 68*a^2*b^6 + 3*b^8)*c^4*d^2 + 8

0*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c^3*d^3 - 6*(a^8 - 36*a^6*b^2 + 86*a^4*b^4 - 36*a^2*b^6 + b^8)*c^2*d

^4 - 24*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d^5 + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^6

))*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^6 - 24*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c^5*d + 3*(3*a^8

- 68*a^6*b^2 + 178*a^4*b^4 - 68*a^2*b^6 + 3*b^8)*c^4*d^2 + 80*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c^3*d^3

- 6*(a^8 - 36*a^6*b^2 + 86*a^4*b^4 - 36*a^2*b^6 + b^8)*c^2*d^4 - 24*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*

d^5 + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 +

15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4))*((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)/((c^6 + 3*c^4*d^2 + 3*c

^2*d^4 + d^6)*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^13 - 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^1

6)*c^12*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^

11*d^2 - 14*(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^10*d^3 - 20*(

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^9*d^4 - 25*(a^1

6 - 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^8*d^5 - 80*(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^7*d^6 - 20*(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^6*d^7 - 100*(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*d^8 - 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^9 - 56*(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^10 + 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^11 - 12*(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^12 + (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^13)*f^4*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^6 - 24*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c^

5*d + 3*(3*a^8 - 68*a^6*b^2 + 178*a^4*b^4 - 68*a^2*b^6 + 3*b^8)*c^4*d^2 + 80*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 -

a*b^7)*c^3*d^3 - 6*(a^8 - 36*a^6*b^2 + 86*a^4*b^4 - 36*a^2*b^6 + b^8)*c^2*d^4 - 24*(a^7*b - 7*a^5*b^3 + 7*a^3*

b^5 - a*b^7)*c*d^5 + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4

+ 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)/((c^

6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*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 - 1

4*a^9*b^11 - 28*a^7*b^13 - 20*a^5*b^15 - 7*a^3*b^17 - a*b^19)*c^10 - 3*(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^9*d

- 8*(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^7*d^3 - 24*(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^6*d^4 - 6*(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^5*d^5 - 32*(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*d^6 - 12*(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^8 + (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^9)*f^2*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^...

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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}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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))**(3/2),x)

[Out]

Integral((a + b*tan(e + f*x))**2/(c + d*tan(e + f*x))**(3/2), 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((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),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 = 12.22, size = 2500, normalized size = 16.67 \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))^(3/2),x)

[Out]

atan((((c + d*tan(e + f*x))^(1/2)*(16*a^4*d^10*f^3 + 16*b^4*d^10*f^3 - 96*a^2*b^2*d^10*f^3 + 32*a^4*c^2*d^8*f^

3 - 32*a^4*c^6*d^4*f^3 - 16*a^4*c^8*d^2*f^3 + 32*b^4*c^2*d^8*f^3 - 32*b^4*c^6*d^4*f^3 - 16*b^4*c^8*d^2*f^3 - 1

92*a^2*b^2*c^2*d^8*f^3 + 192*a^2*b^2*c^6*d^4*f^3 + 96*a^2*b^2*c^8*d^2*f^3 + 128*a*b^3*c*d^9*f^3 - 128*a^3*b*c*

d^9*f^3 + 384*a*b^3*c^3*d^7*f^3 + 384*a*b^3*c^5*d^5*f^3 + 128*a*b^3*c^7*d^3*f^3 - 384*a^3*b*c^3*d^7*f^3 - 384*

a^3*b*c^5*d^5*f^3 - 128*a^3*b*c^7*d^3*f^3) - (-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c^3*f^2*

1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*((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^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*

d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 64*a^2*c*d^11*f^4 + 64*b^2

*c*d^11*f^4 - 256*a^2*c^3*d^9*f^4 - 384*a^2*c^5*d^7*f^4 - 256*a^2*c^7*d^5*f^4 - 64*a^2*c^9*d^3*f^4 + 256*b^2*c

^3*d^9*f^4 + 384*b^2*c^5*d^7*f^4 + 256*b^2*c^7*d^5*f^4 + 64*b^2*c^9*d^3*f^4 - 64*a*b*d^12*f^4 - 192*a*b*c^2*d^

10*f^4 - 128*a*b*c^4*d^8*f^4 + 128*a*b*c^6*d^6*f^4 + 192*a*b*c^8*d^4*f^4 + 64*a*b*c^10*d^2*f^4))*(-(4*a*b^3 -

4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*1i + ((

c + d*tan(e + f*x))^(1/2)*(16*a^4*d^10*f^3 + 16*b^4*d^10*f^3 - 96*a^2*b^2*d^10*f^3 + 32*a^4*c^2*d^8*f^3 - 32*a

^4*c^6*d^4*f^3 - 16*a^4*c^8*d^2*f^3 + 32*b^4*c^2*d^8*f^3 - 32*b^4*c^6*d^4*f^3 - 16*b^4*c^8*d^2*f^3 - 192*a^2*b

^2*c^2*d^8*f^3 + 192*a^2*b^2*c^6*d^4*f^3 + 96*a^2*b^2*c^8*d^2*f^3 + 128*a*b^3*c*d^9*f^3 - 128*a^3*b*c*d^9*f^3

+ 384*a*b^3*c^3*d^7*f^3 + 384*a*b^3*c^5*d^5*f^3 + 128*a*b^3*c^7*d^3*f^3 - 384*a^3*b*c^3*d^7*f^3 - 384*a^3*b*c^

5*d^5*f^3 - 128*a^3*b*c^7*d^3*f^3) - (-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c^3*f^2*1i + d^3

*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*((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^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5

+ 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) + 64*a^2*c*d^11*f^4 - 64*b^2*c*d^11*

f^4 + 256*a^2*c^3*d^9*f^4 + 384*a^2*c^5*d^7*f^4 + 256*a^2*c^7*d^5*f^4 + 64*a^2*c^9*d^3*f^4 - 256*b^2*c^3*d^9*f

^4 - 384*b^2*c^5*d^7*f^4 - 256*b^2*c^7*d^5*f^4 - 64*b^2*c^9*d^3*f^4 + 64*a*b*d^12*f^4 + 192*a*b*c^2*d^10*f^4 +

128*a*b*c^4*d^8*f^4 - 128*a*b*c^6*d^6*f^4 - 192*a*b*c^8*d^4*f^4 - 64*a*b*c^10*d^2*f^4))*(-(4*a*b^3 - 4*a^3*b

+ a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*1i)/(((c + d*ta

n(e + f*x))^(1/2)*(16*a^4*d^10*f^3 + 16*b^4*d^10*f^3 - 96*a^2*b^2*d^10*f^3 + 32*a^4*c^2*d^8*f^3 - 32*a^4*c^6*d

^4*f^3 - 16*a^4*c^8*d^2*f^3 + 32*b^4*c^2*d^8*f^3 - 32*b^4*c^6*d^4*f^3 - 16*b^4*c^8*d^2*f^3 - 192*a^2*b^2*c^2*d

^8*f^3 + 192*a^2*b^2*c^6*d^4*f^3 + 96*a^2*b^2*c^8*d^2*f^3 + 128*a*b^3*c*d^9*f^3 - 128*a^3*b*c*d^9*f^3 + 384*a*

b^3*c^3*d^7*f^3 + 384*a*b^3*c^5*d^5*f^3 + 128*a*b^3*c^7*d^3*f^3 - 384*a^3*b*c^3*d^7*f^3 - 384*a^3*b*c^5*d^5*f^

3 - 128*a^3*b*c^7*d^3*f^3) - (-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c^3*f^2*1i + d^3*f^2 - c

*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*((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^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c

^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 64*a^2*c*d^11*f^4 + 64*b^2*c*d^11*f^4 - 25

6*a^2*c^3*d^9*f^4 - 384*a^2*c^5*d^7*f^4 - 256*a^2*c^7*d^5*f^4 - 64*a^2*c^9*d^3*f^4 + 256*b^2*c^3*d^9*f^4 + 384

*b^2*c^5*d^7*f^4 + 256*b^2*c^7*d^5*f^4 + 64*b^2*c^9*d^3*f^4 - 64*a*b*d^12*f^4 - 192*a*b*c^2*d^10*f^4 - 128*a*b

*c^4*d^8*f^4 + 128*a*b*c^6*d^6*f^4 + 192*a*b*c^8*d^4*f^4 + 64*a*b*c^10*d^2*f^4))*(-(4*a*b^3 - 4*a^3*b + a^4*1i

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

^(1/2)*(16*a^4*d^10*f^3 + 16*b^4*d^10*f^3 - 96*a^2*b^2*d^10*f^3 + 32*a^4*c^2*d^8*f^3 - 32*a^4*c^6*d^4*f^3 - 16

*a^4*c^8*d^2*f^3 + 32*b^4*c^2*d^8*f^3 - 32*b^4*c^6*d^4*f^3 - 16*b^4*c^8*d^2*f^3 - 192*a^2*b^2*c^2*d^8*f^3 + 19

2*a^2*b^2*c^6*d^4*f^3 + 96*a^2*b^2*c^8*d^2*f^3 + 128*a*b^3*c*d^9*f^3 - 128*a^3*b*c*d^9*f^3 + 384*a*b^3*c^3*d^7

*f^3 + 384*a*b^3*c^5*d^5*f^3 + 128*a*b^3*c^7*d^3*f^3 - 384*a^3*b*c^3*d^7*f^3 - 384*a^3*b*c^5*d^5*f^3 - 128*a^3

*b*c^7*d^3*f^3) - (-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i

- 3*c^2*d*f^2)))^(1/2)*((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

^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5

+ 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) + 64*a^2*c*d^11*f^4 - 64*b^2*c*d^11*f^4 + 256*a^2*c^3*d

^9*f^4 + 384*a^2*c^5*d^7*f^4 + 256*a^2*c^7*d^5*f^4 + 64*a^2*c^9*d^3*f^4 - 256*b^2*c^3*d^9*f^4 - 384*b^2*c^5*d^

7*f^4 - 256*b^2*c^7*d^5*f^4 - 64*b^2*c^9*d^3*f^...

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