\(\int \tan ^3(e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\) [307]

Optimal result

Integrand size = 25, antiderivative size = 116 \[ \int \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}-\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}-\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b f} \]

[Out]

(a-b)^(3/2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/f-(a-b)*(a+b*tan(f*x+e)^2)^(1/2)/f-1/3*(a+b*tan(f*x+
e)^2)^(3/2)/f+1/5*(a+b*tan(f*x+e)^2)^(5/2)/b/f

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3751, 457, 81, 52, 65, 214} \[ \int \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f} \]

[In]

Int[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - ((a - b)*Sqrt[a + b*Tan[e + f*x]^2])/f - (
a + b*Tan[e + f*x]^2)^(3/2)/(3*f) + (a + b*Tan[e + f*x]^2)^(5/2)/(5*b*f)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

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

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {15 (a-b)^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+\sqrt {a+b \tan ^2(e+f x)} \left (3 a^2-20 a b+15 b^2+(6 a-5 b) b \tan ^2(e+f x)+3 b^2 \tan ^4(e+f x)\right )}{15 b f} \]

[In]

Integrate[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

(15*(a - b)^(3/2)*b*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Sqrt[a + b*Tan[e + f*x]^2]*(3*a^2 - 20*a
*b + 15*b^2 + (6*a - 5*b)*b*Tan[e + f*x]^2 + 3*b^2*Tan[e + f*x]^4))/(15*b*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(203\) vs. \(2(100)=200\).

Time = 0.07 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.76

method	result	size
derivativedivides	\(\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{5 b f}-\frac {b \tan \left (f x +e \right )^{2} \sqrt {a +b \tan \left (f x +e \right )^{2}}}{3 f}-\frac {4 a \sqrt {a +b \tan \left (f x +e \right )^{2}}}{3 f}+\frac {b \sqrt {a +b \tan \left (f x +e \right )^{2}}}{f}-\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}+\frac {2 a b \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}\)	\(204\)
default	\(\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{5 b f}-\frac {b \tan \left (f x +e \right )^{2} \sqrt {a +b \tan \left (f x +e \right )^{2}}}{3 f}-\frac {4 a \sqrt {a +b \tan \left (f x +e \right )^{2}}}{3 f}+\frac {b \sqrt {a +b \tan \left (f x +e \right )^{2}}}{f}-\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}+\frac {2 a b \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}\)	\(204\)

[In]

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

[Out]

1/5*(a+b*tan(f*x+e)^2)^(5/2)/b/f-1/3/f*b*tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2)-4/3/f*a*(a+b*tan(f*x+e)^2)^(1/2
)+b*(a+b*tan(f*x+e)^2)^(1/2)/f-1/f*b^2/(-a+b)^(1/2)*arctan((a+b*tan(f*x+e)^2)^(1/2)/(-a+b)^(1/2))+2/f*a*b/(-a+
b)^(1/2)*arctan((a+b*tan(f*x+e)^2)^(1/2)/(-a+b)^(1/2))-1/f*a^2/(-a+b)^(1/2)*arctan((a+b*tan(f*x+e)^2)^(1/2)/(-
a+b)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.88 \[ \int \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (a b - b^{2}\right )} \sqrt {a - b} \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} - 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left (3 \, b^{2} \tan \left (f x + e\right )^{4} + {\left (6 \, a b - 5 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2} - 20 \, a b + 15 \, b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{60 \, b f}, -\frac {15 \, {\left (a b - b^{2}\right )} \sqrt {-a + b} \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) - 2 \, {\left (3 \, b^{2} \tan \left (f x + e\right )^{4} + {\left (6 \, a b - 5 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2} - 20 \, a b + 15 \, b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{30 \, b f}\right ] \]

[In]

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

[Out]

[-1/60*(15*(a*b - b^2)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x
+ e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x +
e)^2 + 1)) - 4*(3*b^2*tan(f*x + e)^4 + (6*a*b - 5*b^2)*tan(f*x + e)^2 + 3*a^2 - 20*a*b + 15*b^2)*sqrt(b*tan(f*
x + e)^2 + a))/(b*f), -1/30*(15*(a*b - b^2)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*t
an(f*x + e)^2 + 2*a - b)) - 2*(3*b^2*tan(f*x + e)^4 + (6*a*b - 5*b^2)*tan(f*x + e)^2 + 3*a^2 - 20*a*b + 15*b^2
)*sqrt(b*tan(f*x + e)^2 + a))/(b*f)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 22.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.34 \[ \int \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{5/2}}{5\,b\,f}-\left (\frac {a}{3\,b\,f}-\frac {a-b}{3\,b\,f}\right )\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}-\left (\frac {a}{b\,f}-\frac {a-b}{b\,f}\right )\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a-b\right )-\frac {\mathrm {atan}\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i}}{f} \]

[In]

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

[Out]

(a + b*tan(e + f*x)^2)^(5/2)/(5*b*f) - (a/(3*b*f) - (a - b)/(3*b*f))*(a + b*tan(e + f*x)^2)^(3/2) - (a/(b*f) -
(a - b)/(b*f))*(a + b*tan(e + f*x)^2)^(1/2)*(a - b) - (atan(((a + b*tan(e + f*x)^2)^(1/2)*(a - b)^(3/2)*1i)/(
a^2 - 2*a*b + b^2))*(a - b)^(3/2)*1i)/f