\(\int (b \tan ^3(e+f x))^{3/2} \, dx\) [8]

Optimal result

Integrand size = 14, antiderivative size = 286 \[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=-\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}-\frac {b \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {b \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f} \]

[Out]

-2/3*b*(b*tan(f*x+e)^3)^(1/2)/f+1/2*b*arctan(-1+2^(1/2)*tan(f*x+e)^(1/2))*(b*tan(f*x+e)^3)^(1/2)/f*2^(1/2)/tan
(f*x+e)^(3/2)+1/2*b*arctan(1+2^(1/2)*tan(f*x+e)^(1/2))*(b*tan(f*x+e)^3)^(1/2)/f*2^(1/2)/tan(f*x+e)^(3/2)+1/4*b
*ln(1-2^(1/2)*tan(f*x+e)^(1/2)+tan(f*x+e))*(b*tan(f*x+e)^3)^(1/2)/f*2^(1/2)/tan(f*x+e)^(3/2)-1/4*b*ln(1+2^(1/2
)*tan(f*x+e)^(1/2)+tan(f*x+e))*(b*tan(f*x+e)^3)^(1/2)/f*2^(1/2)/tan(f*x+e)^(3/2)+2/7*b*(b*tan(f*x+e)^3)^(1/2)*
tan(f*x+e)^2/f

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3554, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=-\frac {b \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b \arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}+\frac {b \sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {b \sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \]

[In]

Int[(b*Tan[e + f*x]^3)^(3/2),x]

[Out]

(-2*b*Sqrt[b*Tan[e + f*x]^3])/(3*f) - (b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[
2]*f*Tan[e + f*x]^(3/2)) + (b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e
+ f*x]^(3/2)) + (b*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan
[e + f*x]^(3/2)) - (b*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*
Tan[e + f*x]^(3/2)) + (2*b*Tan[e + f*x]^2*Sqrt[b*Tan[e + f*x]^3])/(7*f)

Rule 210

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \int \tan ^{\frac {9}{2}}(e+f x) \, dx}{\tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}-\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \int \tan ^{\frac {5}{2}}(e+f x) \, dx}{\tan ^{\frac {3}{2}}(e+f x)} \\ & = -\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \int \sqrt {\tan (e+f x)} \, dx}{\tan ^{\frac {3}{2}}(e+f x)} \\ & = -\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f \tan ^{\frac {3}{2}}(e+f x)} \\ & = -\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}+\frac {\left (2 b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)} \\ & = -\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}-\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)} \\ & = -\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \\ & = -\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}+\frac {b \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {b \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f}+\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\left (b \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \\ & = -\frac {2 b \sqrt {b \tan ^3(e+f x)}}{3 f}-\frac {b \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {b \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {2 b \tan ^2(e+f x) \sqrt {b \tan ^3(e+f x)}}{7 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.40 \[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=\frac {b \sqrt {b \tan ^3(e+f x)} \left (21 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \sqrt [4]{-\tan (e+f x)}-21 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \sqrt [4]{-\tan (e+f x)}+2 \tan ^{\frac {7}{4}}(e+f x) \left (-7+3 \tan ^2(e+f x)\right )\right )}{21 f \tan ^{\frac {7}{4}}(e+f x)} \]

[In]

Integrate[(b*Tan[e + f*x]^3)^(3/2),x]

[Out]

(b*Sqrt[b*Tan[e + f*x]^3]*(21*ArcTan[(-Tan[e + f*x]^2)^(1/4)]*(-Tan[e + f*x])^(1/4) - 21*ArcTanh[(-Tan[e + f*x
]^2)^(1/4)]*(-Tan[e + f*x])^(1/4) + 2*Tan[e + f*x]^(7/4)*(-7 + 3*Tan[e + f*x]^2)))/(21*f*Tan[e + f*x]^(7/4))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\left (b \tan \left (f x +e \right )^{3}\right )^{\frac {3}{2}} \left (24 \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \left (b^{2}\right )^{\frac {1}{4}}-56 b^{2} \left (b \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (b^{2}\right )^{\frac {1}{4}}+21 b^{4} \sqrt {2}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )\right )}{84 f \tan \left (f x +e \right )^{3} \left (b \tan \left (f x +e \right )\right )^{\frac {3}{2}} b^{2} \left (b^{2}\right )^{\frac {1}{4}}}\)	\(236\)
default	\(\frac {\left (b \tan \left (f x +e \right )^{3}\right )^{\frac {3}{2}} \left (24 \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \left (b^{2}\right )^{\frac {1}{4}}-56 b^{2} \left (b \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (b^{2}\right )^{\frac {1}{4}}+21 b^{4} \sqrt {2}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )\right )}{84 f \tan \left (f x +e \right )^{3} \left (b \tan \left (f x +e \right )\right )^{\frac {3}{2}} b^{2} \left (b^{2}\right )^{\frac {1}{4}}}\)	\(236\)

[In]

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

[Out]

1/84/f*(b*tan(f*x+e)^3)^(3/2)*(24*(b*tan(f*x+e))^(7/2)*(b^2)^(1/4)-56*b^2*(b*tan(f*x+e))^(3/2)*(b^2)^(1/4)+21*
b^4*2^(1/2)*ln(-((b^2)^(1/4)*(b*tan(f*x+e))^(1/2)*2^(1/2)-b*tan(f*x+e)-(b^2)^(1/2))/(b*tan(f*x+e)+(b^2)^(1/4)*
(b*tan(f*x+e))^(1/2)*2^(1/2)+(b^2)^(1/2)))+42*b^4*2^(1/2)*arctan((2^(1/2)*(b*tan(f*x+e))^(1/2)+(b^2)^(1/4))/(b
^2)^(1/4))+42*b^4*2^(1/2)*arctan((2^(1/2)*(b*tan(f*x+e))^(1/2)-(b^2)^(1/4))/(b^2)^(1/4)))/tan(f*x+e)^3/(b*tan(
f*x+e))^(3/2)/b^2/(b^2)^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.98 \[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=\frac {21 \, \left (-\frac {b^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (\frac {\left (-\frac {b^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}} b^{4}}{\tan \left (f x + e\right )}\right ) - 21 \, \left (-\frac {b^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (-\frac {\left (-\frac {b^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3} \tan \left (f x + e\right ) - \sqrt {b \tan \left (f x + e\right )^{3}} b^{4}}{\tan \left (f x + e\right )}\right ) - 21 i \, \left (-\frac {b^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (\frac {i \, \left (-\frac {b^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}} b^{4}}{\tan \left (f x + e\right )}\right ) + 21 i \, \left (-\frac {b^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (\frac {-i \, \left (-\frac {b^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}} b^{4}}{\tan \left (f x + e\right )}\right ) + 4 \, \sqrt {b \tan \left (f x + e\right )^{3}} {\left (3 \, b \tan \left (f x + e\right )^{2} - 7 \, b\right )}}{42 \, f} \]

[In]

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

[Out]

1/42*(21*(-b^6/f^4)^(1/4)*f*log(((-b^6/f^4)^(3/4)*f^3*tan(f*x + e) + sqrt(b*tan(f*x + e)^3)*b^4)/tan(f*x + e))
- 21*(-b^6/f^4)^(1/4)*f*log(-((-b^6/f^4)^(3/4)*f^3*tan(f*x + e) - sqrt(b*tan(f*x + e)^3)*b^4)/tan(f*x + e)) -
21*I*(-b^6/f^4)^(1/4)*f*log((I*(-b^6/f^4)^(3/4)*f^3*tan(f*x + e) + sqrt(b*tan(f*x + e)^3)*b^4)/tan(f*x + e))
+ 21*I*(-b^6/f^4)^(1/4)*f*log((-I*(-b^6/f^4)^(3/4)*f^3*tan(f*x + e) + sqrt(b*tan(f*x + e)^3)*b^4)/tan(f*x + e)
) + 4*sqrt(b*tan(f*x + e)^3)*(3*b*tan(f*x + e)^2 - 7*b))/f

Sympy [F]

\[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=\int \left (b \tan ^{3}{\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((b*tan(e + f*x)**3)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.49 \[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=\frac {24 \, b^{\frac {3}{2}} \tan \left (f x + e\right )^{\frac {7}{2}} - 56 \, b^{\frac {3}{2}} \tan \left (f x + e\right )^{\frac {3}{2}} + 21 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )\right )} b^{\frac {3}{2}}}{84 \, f} \]

[In]

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

[Out]

1/84*(24*b^(3/2)*tan(f*x + e)^(7/2) - 56*b^(3/2)*tan(f*x + e)^(3/2) + 21*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2
) + 2*sqrt(tan(f*x + e)))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(f*x + e)))) - sqrt(2)*log(sqr
t(2)*sqrt(tan(f*x + e)) + tan(f*x + e) + 1) + sqrt(2)*log(-sqrt(2)*sqrt(tan(f*x + e)) + tan(f*x + e) + 1))*b^(
3/2))/f

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.88 \[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=\frac {1}{84} \, b {\left (\frac {42 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b f} + \frac {42 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b f} - \frac {21 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b f} + \frac {21 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b f} + \frac {8 \, {\left (3 \, \sqrt {b \tan \left (f x + e\right )} b^{21} f^{6} \tan \left (f x + e\right )^{3} - 7 \, \sqrt {b \tan \left (f x + e\right )} b^{21} f^{6} \tan \left (f x + e\right )\right )}}{b^{21} f^{7}}\right )} \mathrm {sgn}\left (\tan \left (f x + e\right )\right ) \]

[In]

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

[Out]

1/84*b*(42*sqrt(2)*abs(b)^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) + 2*sqrt(b*tan(f*x + e)))/sqrt(abs(b)
))/(b*f) + 42*sqrt(2)*abs(b)^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sqrt(b*tan(f*x + e)))/sqrt(ab
s(b)))/(b*f) - 21*sqrt(2)*abs(b)^(3/2)*log(b*tan(f*x + e) + sqrt(2)*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs(b)
)/(b*f) + 21*sqrt(2)*abs(b)^(3/2)*log(b*tan(f*x + e) - sqrt(2)*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs(b))/(b*
f) + 8*(3*sqrt(b*tan(f*x + e))*b^21*f^6*tan(f*x + e)^3 - 7*sqrt(b*tan(f*x + e))*b^21*f^6*tan(f*x + e))/(b^21*f
^7))*sgn(tan(f*x + e))

Mupad [F(-1)]

Timed out. \[ \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx=\int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^3\right )}^{3/2} \,d x \]

[In]

int((b*tan(e + f*x)^3)^(3/2),x)

[Out]

int((b*tan(e + f*x)^3)^(3/2), x)