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

Optimal. Leaf size=286 \[ \frac{2 b \tan ^2(e+f x) \sqrt{b \tan ^3(e+f x)}}{7 f}-\frac{2 b \sqrt{b \tan ^3(e+f x)}}{3 f}-\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \sqrt{b \tan ^3(e+f x)} \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b \sqrt{b \tan ^3(e+f x)} \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.126522, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {3658, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{2 b \tan ^2(e+f x) \sqrt{b \tan ^3(e+f x)}}{7 f}-\frac{2 b \sqrt{b \tan ^3(e+f x)}}{3 f}-\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \sqrt{b \tan ^3(e+f x)} \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b \sqrt{b \tan ^3(e+f x)} \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[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 3658

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
]])

Rule 3473

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 3476

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 329

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 297

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 1162

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 617

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 204

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

Rule 1165

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 628

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]

Rubi steps

\begin{align*} \int \left (b \tan ^3(e+f x)\right )^{3/2} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\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 \tan ^{-1}\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 [C]  time = 0.0783766, size = 54, normalized size = 0.19 \[ \frac{2 b \sqrt{b \tan ^3(e+f x)} \left (7 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\tan ^2(e+f x)\right )+3 \tan ^2(e+f x)-7\right )}{21 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[e + f*x]^3)^(3/2),x]

[Out]

(2*b*Sqrt[b*Tan[e + f*x]^3]*(-7 + 7*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2] + 3*Tan[e + f*x]^2))/(21*f
)

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 236, normalized size = 0.8 \begin{align*}{\frac{1}{84\,f \left ( \tan \left ( fx+e \right ) \right ) ^{3}{b}^{2}} \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 24\, \left ( b\tan \left ( fx+e \right ) \right ) ^{7/2}\sqrt [4]{{b}^{2}}-56\,{b}^{2} \left ( b\tan \left ( fx+e \right ) \right ) ^{3/2}\sqrt [4]{{b}^{2}}+21\,{b}^{4}\sqrt{2}\ln \left ( -{\frac{\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( fx+e \right ) }\sqrt{2}-b\tan \left ( fx+e \right ) -\sqrt{{b}^{2}}}{b\tan \left ( fx+e \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{b}^{2}}}} \right ) +42\,{b}^{4}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) +42\,{b}^{4}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }-\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) \right ) \left ( b\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

________________________________________________________________________________________

Maxima [A]  time = 1.7221, size = 189, normalized size = 0.66 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan ^{3}{\left (e + f x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.42653, size = 356, normalized size = 1.24 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))