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3.119 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx\)

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

[Out]

-(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(3/2)/f-(B-I*(A-C))*arctanh((c+d*tan(f*x+e)
)^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(3/2)/f-2*(A*d^2-B*c*d+C*c^2)/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)

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Rubi [A]  time = 0.29, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3628, 3539, 3537, 63, 208} \[ -\frac {2 \left (A d^2-B c d+c^2 C\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

-(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) - ((B - I*(A - C))*Ar
cTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(c^2*C - B*c*d + 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 3628

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(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*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
 f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
 + b^2, 0]

Rubi steps

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

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Mathematica [C]  time = 1.01, size = 218, normalized size = 1.39 \[ \frac {\frac {(d (C-A)+B c) \left ((d-i c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )+(d+i c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )}{\left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-i B \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )-\frac {2 C}{\sqrt {c+d \tan (e+f x)}}}{d f} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

((-I)*B*(ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/Sqrt[c - I*d] - ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt
[c + I*d]]/Sqrt[c + I*d]) - (2*C)/Sqrt[c + d*Tan[e + f*x]] + ((B*c + (-A + C)*d)*(((-I)*c + d)*Hypergeometric2
F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] + (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*
x])/(c + I*d)]))/((c^2 + d^2)*Sqrt[c + d*Tan[e + f*x]]))/(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((A+B*tan(f*x+e)+C*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)+C*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 = 0.36, size = 11427, normalized size = 72.78 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e)+C*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)+C*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 = 19.61, size = 8588, normalized size = 54.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(e + f*x) + C*tan(e + f*x)^2)/(c + d*tan(e + f*x))^(3/2),x)

[Out]

(log(((((((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2
)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(64*C*c*d^11*f^4 - ((c + d*tan(e + f*x))^(1/2)*((
(96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(c^6*f^4
 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(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))/4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^7*f^4 + 256*C*c^7*d^5*f^4
 + 64*C*c^9*d^3*f^4))/4 + (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f^3 - 32*C^2*c^6*d^4*f^
3 - 16*C^2*c^8*d^2*f^3))*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 +
 12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 8*C^3*d^9*f^2 - 24*C^3*c^2*
d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4
)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 + (l
og(((((-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)
/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(64*C*c*d^11*f^4 - ((c + d*tan(e + f*x))^(1/2)*(-(
(96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(c^6*f^4
 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(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))/4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^7*f^4 + 256*C*c^7*d^5*f^4
 + 64*C*c^9*d^3*f^4))/4 + (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f^3 - 32*C^2*c^6*d^4*f^
3 - 16*C^2*c^8*d^2*f^3))*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2
- 12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 8*C^3*d^9*f^2 - 24*C^3*c^2
*d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f
^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 -
log(((((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(
16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(((96*C^4*c^2*d^
4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f
^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(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*C*c*d^11*f^4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^7*f^4 + 256*C*c^
7*d^5*f^4 + 64*C*c^9*d^3*f^4) - (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f^3 - 32*C^2*c^6*
d^4*f^3 - 16*C^2*c^8*d^2*f^3))*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3
*f^2 + 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - 8*C^3*d^9*f^2 -
24*C^3*c^2*d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*
c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*
f^4))^(1/2) - log(((-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C
^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(
-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(16*c^
6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(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*C*c*d^11*f^4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^
7*f^4 + 256*C*c^7*d^5*f^4 + 64*C*c^9*d^3*f^4) - (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f
^3 - 32*C^2*c^6*d^4*f^3 - 16*C^2*c^8*d^2*f^3))*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^
(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) -
 8*C^3*d^9*f^2 - 24*C^3*c^2*d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*
d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4
*f^4 + 48*c^4*d^2*f^4))^(1/2) + (log(8*A^3*d^9*f^2 - ((((((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d
^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(
((((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*
f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(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))/4 + 64*A*c*d^11*f^4 + 256*A*c^3
*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(16*A^2*d
^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^
4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*
d^2*f^4))^(1/2))/4 + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*(((96*A^4*c^2*d^4*f^4 - 16*A
^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4
 + 3*c^4*d^2*f^4))^(1/2))/4 + (log(8*A^3*d^9*f^2 - ((((-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^
2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*((
(-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*
f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(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))/4 + 64*A*c*d^11*f^4 + 256*A*c^3
*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(16*A^2*d
^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f
^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4
*d^2*f^4))^(1/2))/4 + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*(-((96*A^4*c^2*d^4*f^4 - 16
*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f
^4 + 3*c^4*d^2*f^4))^(1/2))/4 - log(8*A^3*d^9*f^2 - ((((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*
f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1
/2)*(64*A*c*d^11*f^4 - (((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 1
2*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(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) +
256*A*c^3*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4) + (c + d*tan(e + f*x))^(1/2)*(16
*A^2*d^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(((96*A^4*c^2*d^4*f^4 - 16*A^4*
d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4
*f^4 + 48*c^4*d^2*f^4))^(1/2) + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*(((96*A^4*c^2*d^4
*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^
4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - log(8*A^3*d^9*f^2 - ((-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 1
44*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c
^4*d^2*f^4))^(1/2)*(64*A*c*d^11*f^4 - (-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4
*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan
(e + f*x))^(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) + 256*A*c^3*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4) + (c + d*tan(e +
 f*x))^(1/2)*(16*A^2*d^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(-((96*A^4*c^2*
d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6
*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*
(-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c
^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) + (log(- (((c + d*tan(e + f*x))^(1/2)*(16*B^2*d^
10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d^4*f^3 - 16*B^2*c^8*d^2*f^3) + ((((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f
^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4
*d^2*f^4))^(1/2)*(((((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^
2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(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))/4 + 32*B*d^12
*f^4 + 96*B*c^2*d^10*f^4 + 64*B*c^4*d^8*f^4 - 64*B*c^6*d^6*f^4 - 96*B*c^8*d^4*f^4 - 32*B*c^10*d^2*f^4))/4)*(((
96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(c^6*f^4
+ d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 24*B^3*c^3*d^6*f^2 - 24*B^3*c^5*d^4*f^2 - 8*B^3*c^7*d^2
*f^2 - 8*B^3*c*d^8*f^2)*(((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 -
12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 + (log(- (((c + d*tan(e + f*x)
)^(1/2)*(16*B^2*d^10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d^4*f^3 - 16*B^2*c^8*d^2*f^3) + ((-((96*B^4*c^2*d^4
*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*
c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(((-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4
*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x
))^(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))/4 + 32*B*d^12*f^4 + 96*B*c^2*d^10*f^4 + 64*B*c^4*d^8*f^4 - 64*B*c^6*d^6*f^4 - 96*B*c^8*d^4*f^4 - 32*B*c
^10*d^2*f^4))/4)*(-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2
*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 24*B^3*c^3*d^6*f^2 - 24*B^3*c^5*d^
4*f^2 - 8*B^3*c^7*d^2*f^2 - 8*B^3*c*d^8*f^2)*(-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1
/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - log(((
c + d*tan(e + f*x))^(1/2)*(16*B^2*d^10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d^4*f^3 - 16*B^2*c^8*d^2*f^3) + (
((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6
*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*
c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*
f^4))^(1/2)*(c + d*tan(e + f*x))^(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) - 32*B*d^12*f^4 - 96*B*c^2*d^10*f^4 - 64*B*c^4*d^8*f^4 + 64*B*c^6*d^6*f^4
+ 96*B*c^8*d^4*f^4 + 32*B*c^10*d^2*f^4))*(((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) +
 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - 24*B^3
*c^3*d^6*f^2 - 24*B^3*c^5*d^4*f^2 - 8*B^3*c^7*d^2*f^2 - 8*B^3*c*d^8*f^2)*(((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^
4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 +
 48*c^4*d^2*f^4))^(1/2) - log(((c + d*tan(e + f*x))^(1/2)*(16*B^2*d^10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d
^4*f^3 - 16*B^2*c^8*d^2*f^3) + (-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^
3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((-((96*B^4*c^2*d
^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*
f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 6
40*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) - 32*B*d^12*f^4 - 96*B*c^2*d^10*f^4 - 64
*B*c^4*d^8*f^4 + 64*B*c^6*d^6*f^4 + 96*B*c^8*d^4*f^4 + 32*B*c^10*d^2*f^4))*(-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6
*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^
4 + 48*c^4*d^2*f^4))^(1/2) - 24*B^3*c^3*d^6*f^2 - 24*B^3*c^5*d^4*f^2 - 8*B^3*c^7*d^2*f^2 - 8*B^3*c*d^8*f^2)*(-
((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6
*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - (2*A*d)/(f*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/2
)) + (2*B*c)/(f*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/2)) - (2*C*c^2)/(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 \[ \int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(3/2),x)

[Out]

Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(c + d*tan(e + f*x))**(3/2), x)

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