3.1255 \(\int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=150 \[ -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}} \]
[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 = 0.33, 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 =
{3542, 3539, 3537, 63, 208} \[ -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}} \]
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 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 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 3542
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 \operatorname {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 \operatorname {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 \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 )}{(c-i d) d f}-\frac {(a+i b)^2 \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 )}{(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] time = 0.17, size = 124, normalized size = 0.83 \[ \frac {-\frac {(a-i b)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{d+i c}+\frac {(a+i b)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{-d+i c}-\frac {2 b^2}{d}}{f \sqrt {c+d \tan (e+f x)}} \]
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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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((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),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((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.29, size = 11618, normalized size = 77.45 \[ \text {output too large to display} \]
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)
[Out]
result too large to display
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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((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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mupad [B] time = 12.22, size = 7613, normalized size = 50.75 \[ \text {result too large to display} \]
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))^(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^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) - 16*a^6*d^9*f^2 + 16*b^6*d^9*f^2
+ 16*a^2*b^4*d^9*f^2 - 16*a^4*b^2*d^9*f^2 - 48*a^6*c^2*d^7*f^2 - 48*a^6*c^4*d^5*f^2 - 16*a^6*c^6*d^3*f^2 + 48*
b^6*c^2*d^7*f^2 + 48*b^6*c^4*d^5*f^2 + 16*b^6*c^6*d^3*f^2 + 48*a^2*b^4*c^2*d^7*f^2 + 48*a^2*b^4*c^4*d^5*f^2 +
16*a^2*b^4*c^6*d^3*f^2 + 192*a^3*b^3*c^3*d^6*f^2 + 192*a^3*b^3*c^5*d^4*f^2 + 64*a^3*b^3*c^7*d^2*f^2 - 48*a^4*b
^2*c^2*d^7*f^2 - 48*a^4*b^2*c^4*d^5*f^2 - 16*a^4*b^2*c^6*d^3*f^2 + 32*a*b^5*c*d^8*f^2 + 32*a^5*b*c*d^8*f^2 + 9
6*a*b^5*c^3*d^6*f^2 + 96*a*b^5*c^5*d^4*f^2 + 32*a*b^5*c^7*d^2*f^2 + 64*a^3*b^3*c*d^8*f^2 + 96*a^5*b*c^3*d^6*f^
2 + 96*a^5*b*c^5*d^4*f^2 + 32*a^5*b*c^7*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^3
*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*2i + atan(-(((-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^
2*b^2)/(4*(c^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(1/2)*((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^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(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^1
1*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)
- (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 - 3
2*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))*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c^3*f^2 + d^3*f^2*1
i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(1/2)*1i + ((-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c^3*f^2 + d^
3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(1/2)*((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^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(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) - (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))*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d
*f^2*3i)))^(1/2)*1i)/(((-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2
- c^2*d*f^2*3i)))^(1/2)*((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^3*f^
2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(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 - 1
28*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) - (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))*(-(a*b^
3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(1/2) - ((-(
a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(1/2)*((
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^3*f^2 + d^3*f^2*1i - 3*c*d^2*f
^2 - c^2*d*f^2*3i)))^(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 + 6
4*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) - (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 + 1
28*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))*(-(a*b^3*4i - a^3*b*4i + a^4 + b^
4 - 6*a^2*b^2)/(4*(c^3*f^2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(1/2) - 16*a^6*d^9*f^2 + 16*b^6*d^9*f^
2 + 16*a^2*b^4*d^9*f^2 - 16*a^4*b^2*d^9*f^2 - 48*a^6*c^2*d^7*f^2 - 48*a^6*c^4*d^5*f^2 - 16*a^6*c^6*d^3*f^2 + 4
8*b^6*c^2*d^7*f^2 + 48*b^6*c^4*d^5*f^2 + 16*b^6*c^6*d^3*f^2 + 48*a^2*b^4*c^2*d^7*f^2 + 48*a^2*b^4*c^4*d^5*f^2
+ 16*a^2*b^4*c^6*d^3*f^2 + 192*a^3*b^3*c^3*d^6*f^2 + 192*a^3*b^3*c^5*d^4*f^2 + 64*a^3*b^3*c^7*d^2*f^2 - 48*a^4
*b^2*c^2*d^7*f^2 - 48*a^4*b^2*c^4*d^5*f^2 - 16*a^4*b^2*c^6*d^3*f^2 + 32*a*b^5*c*d^8*f^2 + 32*a^5*b*c*d^8*f^2 +
96*a*b^5*c^3*d^6*f^2 + 96*a*b^5*c^5*d^4*f^2 + 32*a*b^5*c^7*d^2*f^2 + 64*a^3*b^3*c*d^8*f^2 + 96*a^5*b*c^3*d^6*
f^2 + 96*a^5*b*c^5*d^4*f^2 + 32*a^5*b*c^7*d^2*f^2))*(-(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)/(4*(c^3*f^
2 + d^3*f^2*1i - 3*c*d^2*f^2 - c^2*d*f^2*3i)))^(1/2)*2i - (2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(d*f*(c^2 + d^2)
*(c + d*tan(e + f*x))^(1/2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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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