3.13.48 \(\int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx\) [1248]
Optimal. Leaf size=178 \[ \frac {(i a+b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f} \]
[Out]
(I*a+b)^3*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f/(c-I*d)^(1/2)-(I*a-b)^3*arctanh((c+d*tan(f*x+e))^(1/
2)/(c+I*d)^(1/2))/f/(c+I*d)^(1/2)-4/3*b^2*(-4*a*d+b*c)*(c+d*tan(f*x+e))^(1/2)/d^2/f+2/3*b^2*(c+d*tan(f*x+e))^(
1/2)*(a+b*tan(f*x+e))/d/f
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Rubi [A]
time = 0.29, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3647, 3711,
3620, 3618, 65, 214} \begin {gather*} -\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {(-b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {(b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*Tan[e + f*x])^3/Sqrt[c + d*Tan[e + f*x]],x]
[Out]
((I*a + b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) - ((I*a - b)^3*ArcTanh[Sqrt[c
+ d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) - (4*b^2*(b*c - 4*a*d)*Sqrt[c + d*Tan[e + f*x]])/(3*d^2*f)
+ (2*b^2*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*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 3647
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))
Rule 3711
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&& !LeQ[m, -1]
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx &=\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {2 \int \frac {\frac {1}{2} \left (3 a^3 d-b^2 (2 b c+a d)\right )+\frac {3}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)-b^2 (b c-4 a d) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d}\\ &=-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {2 \int \frac {\frac {3}{2} a \left (a^2-3 b^2\right ) d+\frac {3}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d}\\ &=-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {1}{2} (a-i b)^3 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} (a+i b)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {(i a-b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}-\frac {(i a+b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}\\ &=-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f}-\frac {(a-i b)^3 \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)^3 \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+b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f}\\ \end {align*}
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Mathematica [A]
time = 1.11, size = 178, normalized size = 1.00 \begin {gather*} \frac {2 \left (-\frac {3 i (a-i b)^3 d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 \sqrt {c-i d}}+\frac {3 i (a+i b)^3 d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 \sqrt {c+i d}}+\frac {2 b^2 (-b c+4 a d) \sqrt {c+d \tan (e+f x)}}{d}+b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}\right )}{3 d f} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Tan[e + f*x])^3/Sqrt[c + d*Tan[e + f*x]],x]
[Out]
(2*((((-3*I)/2)*(a - I*b)^3*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (((3*I)/2)*(a +
I*b)^3*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*b^2*(-(b*c) + 4*a*d)*Sqrt[c + d*
Tan[e + f*x]])/d + b^2*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(3*d*f)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2771\) vs.
\(2(152)=304\).
time = 0.47, size = 2772, normalized size = 15.57
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| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(2772\) |
| default |
\(\text {Expression too large to display}\) |
\(2772\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
[Out]
2/f/d^2*(1/3*b^3*(c+d*tan(f*x+e))^(3/2)+3*a*b^2*d*(c+d*tan(f*x+e))^(1/2)-b^3*c*(c+d*tan(f*x+e))^(1/2)+d^2*(1/4
/(c^2+d^2)^(3/2)/d^2*(1/2*(3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c-(c^2+d^2)^(3/2)*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*b^3*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*a^3*d^3-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^3-3*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2*d-3*(c^2+d^2)^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^3+(c^2+d^2)^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a^3*c*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^4+3*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^3+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3
*c^2*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^4)*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^3*c^2*d^3+2*a^3*d^5-6*a^2*b*c^3*d^2-6*a^2*b*c*d^4-6*a*b^2*c^2*d^3-6*a*b^2*d^5
+2*b^3*c^3*d^2+2*b^3*c*d^4-1/2*(3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c-(c^2+d^2)^(3/2)*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*b^3*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*a^3*d^3-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^3-3*(c^2+d^2)^(1/2)*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2*d-3*(c^2+d^2)^
(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^3+(c^2+d^2)^
(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a^3*c*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^4+3*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^3+(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*b^3*c^2*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^4)*(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*(-3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*b^3*c-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a^3*d^3+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^3+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*a^2*b*c*d^2+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2*d+3*(c^2+d^2)^(1/2)*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^3-(c^2+d^2)^(1/2)*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c
*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d^2+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^4-3*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*a*b^2*c^3*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2*d
^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^4)*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^3*c^2*d^3+2*a^3*d^5-6*a^2*b*c^3*d^2-6*a^2*b*c*d^4-6*a*b^2*c^2*d^3-6*a*b^2*d^5+2*b^3
*c^3*d^2+2*b^3*c*d^4+1/2*(-3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c+(c^2+d^2)^(3/2)*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*b^3*c-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*a^3*d^3+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^3+3*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d^2+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2*d+3*(c^2+d^2)^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^3-(c^2+d^2)^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a^3*c*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d^2+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^4-3*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3
*c^2*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^4)*(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))^3/(c+d*tan(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate((b*tan(f*x + e) + a)^3/sqrt(d*tan(f*x + e) + c), x)
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Fricas [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))^3/(c+d*tan(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
Timed out
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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 )^{3}}{\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))**3/(c+d*tan(f*x+e))**(1/2),x)
[Out]
Integral((a + b*tan(e + f*x))**3/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))^3/(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 = 9.57, size = 3017, normalized size = 16.95 \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))^3/(c + d*tan(e + f*x))^(1/2),x)
[Out]
atan(((((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*ta
n(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i
- d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1
i - d*f^2)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2
)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/2
)*1i - (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*t
an(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1
i - d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*
1i - d*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^
2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/
2)*1i)/((((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*
tan(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*
1i - d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2
*1i - d*f^2)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f
^2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1
/2) + (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*ta
n(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i
- d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1
i - d*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2
)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/2
) + (16*(3*a^8*b*d^2 - b^9*d^2 + 6*a^4*b^5*d^2 + 8*a^6*b^3*d^2))/f^3))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i -
a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/2)*2i + atan(((((8*(4*a^3*d^3*f^2 - 12*a*b
^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5
*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6
i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x)
)^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2
*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*1i - (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4
*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5*b*6i - a^6 +
b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6i - a^6 + b^6
- 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d
^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3
*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*1i)/((((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2
- 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b
^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 -
a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 +
15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*
b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2
*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i
+ 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15
*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 -
15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 -
d*f^2*1i)))^(1/2) + (16*(3*a^8*b*d^2 - b^9*d^2 + 6*a^4*b^5*d^2 + 8*a^6*b^3*d^2))/f^3))*((a*b^5*6i + a^5*b*6i -
a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*2i - ((6*b^3*c - 6*a*b^2*d)/
(d^2*f) - (4*b^3*c)/(d^2*f))*(c + d*tan(e + f*x))^(1/2) + (2*b^3*(c + d*tan(e + f*x))^(3/2))/(3*d^2*f)
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