3.4.8 \(\int \tan (e+f x) (a+b \tan ^2(e+f x))^{3/2} \, dx\) [308]

Optimal. Leaf size=90 \[ -\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f} \]

[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

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Rubi [A]
time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3751, 455, 52, 65, 214} \begin {gather*} \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 {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]*(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)

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 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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

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

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Mathematica [A]
time = 0.28, size = 80, normalized size = 0.89 \begin {gather*} \frac {-3 (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+\sqrt {a+b \tan ^2(e+f x)} \left (4 a-3 b+b \tan ^2(e+f x)\right )}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(78)=156\).
time = 0.05, size = 197, normalized size = 2.19

method	result	size
derivativedivides	\(\frac {b^{2} \left (\frac {\left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 b}-\frac {2 a \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 b^{2}}-\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{b}+\frac {\arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\right )+2 a b \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{b}-\frac {\arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\right )+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}}{f}\)	\(197\)
default	\(\frac {b^{2} \left (\frac {\left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 b}-\frac {2 a \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 b^{2}}-\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{b}+\frac {\arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\right )+2 a b \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{b}-\frac {\arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\right )+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}}{f}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.94, size = 267, normalized size = 2.97 \begin {gather*} \left [-\frac {3 \, {\left (a - b\right )}^{\frac {3}{2}} \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 (b \tan \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, f}, \frac {3 \, {\left (a - b\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 (b \tan \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(3*(a - b)^(3/2)*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*(b*tan(f*x + e)^2 + 4*a - 3*b)*sqrt(b*tan(f*x + e)^2 + a))/f, 1/6*(3*(a - b)*sqrt(-a + b)*arctan(2*sqrt(b*
tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*(b*tan(f*x + e)^2 + 4*a - 3*b)*sqrt(b*tan(f
*x + e)^2 + a))/f]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)*(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), x)

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Giac [A]
time = 0.52, size = 114, normalized size = 1.27 \begin {gather*} \frac {{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {-a + b}}\right )}{\sqrt {-a + b} f} + \frac {{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} f^{2} + 3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} a f^{2} - 3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b f^{2}}{3 \, f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 14.99, size = 91, normalized size = 1.01 \begin {gather*} \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{3\,f}+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a-b\right )}{f}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,{\left (a-b\right )}^{3/2}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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