3.5.1 \(\int \frac {\tan ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\) [401]

Optimal. Leaf size=187 \[ \frac {\sqrt {-1+\sqrt {2}} \text {ArcTan}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}+\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {4 \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2 \tan (e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \]

[Out]

1/2*arctan((3-2*2^(1/2)+(1-2^(1/2))*tan(f*x+e))/(-14+10*2^(1/2))^(1/2)/(1+tan(f*x+e))^(1/2))*(2^(1/2)-1)^(1/2)
/f+1/2*arctanh((3+2*2^(1/2)+(1+2^(1/2))*tan(f*x+e))/(14+10*2^(1/2))^(1/2)/(1+tan(f*x+e))^(1/2))*(1+2^(1/2))^(1
/2)/f-4/3*(1+tan(f*x+e))^(1/2)/f+2/3*(1+tan(f*x+e))^(1/2)*tan(f*x+e)/f

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Rubi [A]
time = 0.16, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3647, 3711, 12, 3617, 3616, 209, 213} \begin {gather*} \frac {\sqrt {\sqrt {2}-1} \text {ArcTan}\left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 f}+\frac {2 \tan (e+f x) \sqrt {\tan (e+f x)+1}}{3 f}-\frac {4 \sqrt {\tan (e+f x)+1}}{3 f}+\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^3/Sqrt[1 + Tan[e + f*x]],x]

[Out]

(Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan
[e + f*x]])])/(2*f) + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*S
qrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) - (4*Sqrt[1 + Tan[e + f*x]])/(3*f) + (2*Tan[e + f*x]*Sqrt[1 + Tan[e +
f*x]])/(3*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 3616

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2*(
d^2/f), Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]
]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[2
*a*c*d - b*(c^2 - d^2), 0]

Rule 3617

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> With[{q =
Rt[a^2 + b^2, 2]}, Dist[1/(2*q), Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*
x]], x], x] - Dist[1/(2*q), Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x
], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2
*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])

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 {\tan ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx &=\frac {2 \tan (e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2}{3} \int \frac {-1-\frac {3}{2} \tan (e+f x)-\tan ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {4 \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2 \tan (e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2}{3} \int -\frac {3 \tan (e+f x)}{2 \sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {4 \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2 \tan (e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\int \frac {\tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {4 \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2 \tan (e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {\int \frac {1+\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}-\frac {\int \frac {1+\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}\\ &=-\frac {4 \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2 \tan (e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {\left (4-3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1+\sqrt {2}\right )-4 \left (-1+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1+\sqrt {2}\right )-\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {\left (4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1-\sqrt {2}\right )-4 \left (-1-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1-\sqrt {2}\right )-\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{2 f}\\ &=\frac {\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}+\frac {\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {4 \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2 \tan (e+f x) \sqrt {1+\tan (e+f x)}}{3 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 90, normalized size = 0.48 \begin {gather*} \frac {\frac {6 \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {6 \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}}+4 (-2+\tan (e+f x)) \sqrt {1+\tan (e+f x)}}{6 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^3/Sqrt[1 + Tan[e + f*x]],x]

[Out]

((6*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (6*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])
/Sqrt[1 + I] + 4*(-2 + Tan[e + f*x])*Sqrt[1 + Tan[e + f*x]])/(6*f)

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Maple [A]
time = 0.15, size = 233, normalized size = 1.25

method	result	size
derivativedivides	\(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\tan \left (f x +e \right )}+\frac {\sqrt {2}\, \left (-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}+\frac {\sqrt {2}\, \left (\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}}{f}\)	\(233\)
default	\(\frac {\frac {2 \left (1+\tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\tan \left (f x +e \right )}+\frac {\sqrt {2}\, \left (-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}-\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\tan \left (f x +e \right )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}+\frac {\sqrt {2}\, \left (\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\sqrt {2}+\sqrt {2 \sqrt {2}+2}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}}{f}\)	\(233\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^3/(1+tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/f*(2/3*(1+tan(f*x+e))^(3/2)-2*(1+tan(f*x+e))^(1/2)+1/4*2^(1/2)*(-1/2*(2*2^(1/2)+2)^(1/2)*ln(1+2^(1/2)-(2*2^(
1/2)+2)^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))+2*(1-2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+tan(f*x+e))^(1/
2)-(2*2^(1/2)+2)^(1/2))/(-2+2*2^(1/2))^(1/2)))+1/4*2^(1/2)*(1/2*(2*2^(1/2)+2)^(1/2)*ln(1+2^(1/2)+(2*2^(1/2)+2)
^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))+2*(1-2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan(((2*2^(1/2)+2)^(1/2)+2*(1+ta
n(f*x+e))^(1/2))/(-2+2*2^(1/2))^(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(tan(f*x+e)^3/(1+tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(tan(f*x + e)^3/sqrt(tan(f*x + e) + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (149) = 298\).
time = 1.26, size = 1028, normalized size = 5.50 \begin {gather*} \frac {12 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {\frac {1}{2}} f^{3}\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (2 \, \sqrt {\frac {1}{2}} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + \cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - 2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {\frac {1}{2}} f^{3}\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - f^{2} \sqrt {\frac {1}{f^{4}}} - 2 \, \sqrt {\frac {1}{2}}\right ) \cos \left (f x + e\right ) + 12 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {\frac {1}{2}} f^{3}\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (2 \, \sqrt {\frac {1}{2}} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + \cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - 2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {\frac {1}{2}} f^{3}\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + f^{2} \sqrt {\frac {1}{f^{4}}} + 2 \, \sqrt {\frac {1}{2}}\right ) \cos \left (f x + e\right ) + 3 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (\sqrt {\frac {1}{2}} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (2 \, \sqrt {\frac {1}{2}} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + \cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) - 3 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (\sqrt {\frac {1}{2}} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (2 \, \sqrt {\frac {1}{2}} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + f \cos \left (f x + e\right )\right )} \sqrt {-4 \, \sqrt {\frac {1}{2}} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + \cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) - 8 \, \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} {\left (2 \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )}}{12 \, f \cos \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^3/(1+tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/12*(12*(1/2)^(3/4)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(2*(1/2)^(3/4)*(f^5*sqrt(f
^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e
) + (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)
) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x
+ e))*(f^(-4))^(3/4) - 2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4
)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - f^2*sqrt(f^(-4)) - 2*sqrt(1/2))*cos(f*x +
e) + 12*(1/2)^(3/4)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(2*(1/2)^(3/4)*(f^5*sqrt(f^
(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e)
- (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4))
+ 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x +
e))*(f^(-4))^(3/4) - 2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)
*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) + f^2*sqrt(f^(-4)) + 2*sqrt(1/2))*cos(f*x + e
) + 3*(1/2)^(1/4)*(sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)
) + 4)*(f^(-4))^(1/4)*log((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e) + (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4
))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/c
os(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) - 3*(1/2)^(1/4)*(sqrt(1/2)*f^3*sqrt(f
^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log((2*sqrt(1/2)*
f^2*sqrt(f^(-4))*cos(f*x + e) - (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(
-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x +
e) + sin(f*x + e))/cos(f*x + e)) - 8*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(2*cos(f*x + e) - sin(f
*x + e)))/(f*cos(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{3}{\left (e + f x \right )}}{\sqrt {\tan {\left (e + f x \right )} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**3/(1+tan(f*x+e))**(1/2),x)

[Out]

Integral(tan(e + f*x)**3/sqrt(tan(e + f*x) + 1), x)

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Giac [A]
time = 0.68, size = 237, normalized size = 1.27 \begin {gather*} -\frac {\sqrt {\sqrt {2} - 1} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f} - \frac {\sqrt {\sqrt {2} - 1} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f} + \frac {\sqrt {\sqrt {2} + 1} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f} - \frac {\sqrt {\sqrt {2} + 1} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f} + \frac {2 \, {\left (f^{2} {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} - 3 \, f^{2} \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{3 \, f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^3/(1+tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(sqrt(2) - 1)*arctan(1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) + 2*sqrt(tan(f*x + e) + 1))/sqrt(-sqrt(2)
+ 2))/f - 1/2*sqrt(sqrt(2) - 1)*arctan(-1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) - 2*sqrt(tan(f*x + e) + 1))/sq
rt(-sqrt(2) + 2))/f + 1/4*sqrt(sqrt(2) + 1)*log(2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e) + 1) + sqrt(2) + t
an(f*x + e) + 1)/f - 1/4*sqrt(sqrt(2) + 1)*log(-2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e) + 1) + sqrt(2) + t
an(f*x + e) + 1)/f + 2/3*(f^2*(tan(f*x + e) + 1)^(3/2) - 3*f^2*sqrt(tan(f*x + e) + 1))/f^3

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Mupad [B]
time = 4.13, size = 103, normalized size = 0.55 \begin {gather*} \frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3\,f}-\frac {2\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{f}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,2{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,2{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^3/(tan(e + f*x) + 1)^(1/2),x)

[Out]

(2*(tan(e + f*x) + 1)^(3/2))/(3*f) - (2*(tan(e + f*x) + 1)^(1/2))/f - atan(f*((1/8 - 1i/8)/f^2)^(1/2)*(tan(e +
f*x) + 1)^(1/2)*2i)*((1/8 - 1i/8)/f^2)^(1/2)*2i - atan(f*((1/8 + 1i/8)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*2i
)*((1/8 + 1i/8)/f^2)^(1/2)*2i

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