3.13.56 \(\int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx\) [1256]
Optimal. Leaf size=138 \[ -\frac {(i a+b) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {(i a-b) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}+\frac {2 (b c-a d)}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]
[Out]
-(I*a+b)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(3/2)/f+(I*a-b)*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)/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)
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Rubi [A]
time = 0.17, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3610, 3620,
3618, 65, 214} \begin {gather*} \frac {2 (b c-a d)}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {(b+i a) \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) \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])/(c + d*Tan[e + f*x])^(3/2),x]
[Out]
-(((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) + ((I*a - b)*ArcTanh[Sqrt[c
+ d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) + (2*(b*c - a*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 3610
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
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]
Rubi steps
\begin {align*} \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx &=\frac {2 (b c-a d)}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {a c+b d+(b c-a d) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{c^2+d^2}\\ &=\frac {2 (b c-a d)}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {(a-i b) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)}+\frac {(a+i b) \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)}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {(i a+b) \text {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-b) \text {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 (b c-a d)}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {(a-i b) \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) \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+b) \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-b) \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)}{\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 = 0.20, size = 113, normalized size = 0.82 \begin {gather*} \frac {i \left (\frac {(a-i b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{c-i d}-\frac {(a+i b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{c+i d}\right )}{f \sqrt {c+d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(3/2),x]
[Out]
(I*(((a - I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(c - I*d) - ((a + I*b)*Hyperge
ometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*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. \(2267\) vs.
\(2(118)=236\).
time = 0.44, size = 2268, normalized size = 16.43
| | |
| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(2268\) |
| default |
\(\text {Expression too large to display}\) |
\(2268\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/f*(2/(c^2+d^2)*(1/4/d^2/(3*c^2-d^2)/(c^2+d^2)^(3/2)*(1/2*((c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^
4-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d+
2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^
5-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^6+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^
2+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^4-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d+(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*c^4*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^7-6*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^2-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d^4+2*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b*c*d^6)*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*c
^5*d^3+8*a*c^3*d^5-4*a*c*d^7-6*b*c^6*d^2+2*b*c^4*d^4+6*b*c^2*d^6-2*b*d^8-1/2*((c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*b*c^4-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*a*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*a*c*d^5-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^6+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*b*c^4*d^2+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^4-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^5-(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*a*d^7-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^2-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d^4+2*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b*c*d^6)*(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*(-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b
*d^4-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a
*c^3*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^5+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b
*c^6-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*b*c^2*d^4+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^3-3*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*c^2*d^5+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^7+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^2+4*(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d^4-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^6)*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*c^5*d^3+8*a*c^3*d^5-4*a*c*d^7-6*b*c^6*d^2+2*b*c^4*
d^4+6*b*c^2*d^6-2*b*d^8+1/2*(-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b*d^4-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*a*c^3*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^5+(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b*c^6-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*b*c^2*d^4+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^3-
3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^5+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^7+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
b*c^5*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d^4-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^6)*(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))))-2*(a*d-b*c)/(c^2+d^2)/(c+d*tan(f*x+e))^(1/2))
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d-c>0)', see `assume?` for mor
e details)Is
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20798 vs.
\(2 (116) = 232\).
time = 36.06, size = 20798, normalized size = 150.71 \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))/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
-1/4*(4*sqrt(2)*((c^10 + 3*c^8*d^2 + 2*c^6*d^4 - 2*c^4*d^6 - 3*c^2*d^8 - d^10)*f^5*cos(f*x + e)^2 + 2*(c^9*d +
4*c^7*d^3 + 6*c^5*d^5 + 4*c^3*d^7 + c*d^9)*f^5*cos(f*x + e)*sin(f*x + e) + (c^8*d^2 + 4*c^6*d^4 + 6*c^4*d^6 +
4*c^2*d^8 + d^10)*f^5)*sqrt(((a^4 + 2*a^2*b^2 + b^4)*c^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 + 3*(a^4 + 2*a^2
*b^2 + b^4)*c^2*d^4 + (a^4 + 2*a^2*b^2 + b^4)*d^6 + (6*a*b*c^8*d + 16*a*b*c^6*d^3 + 12*a*b*c^4*d^5 - 2*a*b*d^9
+ (a^2 - b^2)*c^9 - 6*(a^2 - b^2)*c^5*d^4 - 8*(a^2 - b^2)*c^3*d^6 - 3*(a^2 - b^2)*c*d^8)*f^2*sqrt((a^4 + 2*a^
2*b^2 + b^4)/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^4)))/(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4
- 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a
*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6))*sqrt((4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2
*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*
d^5 + (a^4 - 2*a^2*b^2 + b^4)*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^4 + 2*a^2*b^2 + b^4)/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^4))^(3/4)*arctan(-((2*(a^7*b + 3*a^
5*b^3 + 3*a^3*b^5 + a*b^7)*c^13 - 3*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^12*d + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*
b^5 + a*b^7)*c^11*d^2 - 14*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^10*d^3 - 10*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 +
a*b^7)*c^9*d^4 - 25*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^8*d^5 - 40*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c
^7*d^6 - 20*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^6*d^7 - 50*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^5*d^8 -
5*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^4*d^9 - 28*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^3*d^10 + 2*(a^8
+ 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^2*d^11 - 6*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c*d^12 + (a^8 + 2*a^6*b^2
- 2*a^2*b^6 - b^8)*d^13)*f^4*sqrt((4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c
^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*
a^2*b^2 + b^4)*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^4 + 2*a^2*b^2 + b^4)/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^4)) + (2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^
3*b^7 + a*b^9)*c^10 - 3*(a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c^9*d - 8*(a^10 + 3*a^8*
b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c^7*d^3 - 12*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^
9)*c^6*d^4 - 6*(a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c^5*d^5 - 16*(a^9*b + 4*a^7*b^3 +
6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^4*d^6 - 6*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^2*d^8 + (a^1
0 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c*d^9)*f^2*sqrt((4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*
c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^
4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*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(2)*((b*c^14 - 2*a*c^13*d + 5*b*c^12*d^2 - 12*a*c^11*d^3 + 9*b*c^10*
d^4 - 30*a*c^9*d^5 + 5*b*c^8*d^6 - 40*a*c^7*d^7 - 5*b*c^6*d^8 - 30*a*c^5*d^9 - 9*b*c^4*d^10 - 12*a*c^3*d^11 -
5*b*c^2*d^12 - 2*a*c*d^13 - b*d^14)*f^7*sqrt((4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2
+ 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5
+ (a^4 - 2*a^2*b^2 + b^4)*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^4 + 2*a^2*b^2 + b^4)/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^4)) + ((a^2*b + b^3)*c^11 - (a^3 +
a*b^2)*c^10*d + 5*(a^2*b + b^3)*c^9*d^2 - 5*(a^3 + a*b^2)*c^8*d^3 + 10*(a^2*b + b^3)*c^7*d^4 - 10*(a^3 + a*b^2
)*c^6*d^5 + 10*(a^2*b + b^3)*c^5*d^6 - 10*(a^3 + a*b^2)*c^4*d^7 + 5*(a^2*b + b^3)*c^3*d^8 - 5*(a^3 + a*b^2)*c^
2*d^9 + (a^2*b + b^3)*c*d^10 - (a^3 + a*b^2)*d^11)*f^5*sqrt((4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a
^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b
- a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*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^4 + 2*a^2*b^2 + b^4)*c^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 + 3*(a^4 + 2*a^2*
b^2 + b^4)*c^2*d^4 + (a^4 + 2*a^2*b^2 + b^4)*d^6 + (6*a*b*c^8*d + 16*a*b*c^6*d^3 + 12*a*b*c^4*d^5 - 2*a*b*d^9
+ (a^2 - b^2)*c^9 - 6*(a^2 - b^2)*c^5*d^4 - 8*(a^2 - b^2)*c^3*d^6 - 3*(a^2 - b^2)*c*d^8)*f^2*sqrt((a^4 + 2*a^2
*b^2 + b^4)/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^4)))/(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 -
14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*
b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6))*sqrt(((4*(a^4*b^2 + a^2*b^4)*c^10 - 12*(a^5*b - a*b^5)*c^9*d + (9*a
^6 - 25*a^4*b^2 - 25*a^2*b^4 + 9*b^6)*c^8*d^2 +...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \tan {\left (e + f x \right )}}{\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))/(c+d*tan(f*x+e))**(3/2),x)
[Out]
Integral((a + b*tan(e + f*x))/(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))/(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 = 11.43, size = 2500, normalized size = 18.12 \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))/(c + d*tan(e + f*x))^(3/2),x)
[Out]
(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)*(32*b*d^12*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)/(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 + 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 - 8*b^3*c*d
^8*f^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)*(((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)*(32*b*d^12*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)/(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 + 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 - 8*b^3*c*d^8*f^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)*(-((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) - 8*b^3*c*d^8*f^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)*(((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 + 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) - 8*b^3*c*d^8*f^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)*(-((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) + (l
og(((((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)*(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
)*(((((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 14...
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