3.354 \(\int \frac{(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{2 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{3/2} f}+\frac{4 a^3 \sqrt{d \tan (e+f x)}}{d^2 f}-\frac{2 \left (a^3 \tan (e+f x)+a^3\right )}{d f \sqrt{d \tan (e+f x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.154426, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3565, 3630, 3532, 205} \[ -\frac{2 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{3/2} f}+\frac{4 a^3 \sqrt{d \tan (e+f x)}}{d^2 f}-\frac{2 \left (a^3 \tan (e+f x)+a^3\right )}{d f \sqrt{d \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, 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] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3630

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]

Rule 3532

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx &=-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt{d \tan (e+f x)}}+\frac{2 \int \frac{2 a^3 d^2+a^3 d^2 \tan (e+f x)+a^3 d^2 \tan ^2(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{d^3}\\ &=\frac{4 a^3 \sqrt{d \tan (e+f x)}}{d^2 f}-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt{d \tan (e+f x)}}+\frac{2 \int \frac{a^3 d^2+a^3 d^2 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{d^3}\\ &=\frac{4 a^3 \sqrt{d \tan (e+f x)}}{d^2 f}-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt{d \tan (e+f x)}}-\frac{\left (4 a^6 d\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^6 d^4+d x^2} \, dx,x,\frac{a^3 d^2-a^3 d^2 \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=-\frac{2 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{3/2} f}+\frac{4 a^3 \sqrt{d \tan (e+f x)}}{d^2 f}-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt{d \tan (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 1.70994, size = 314, normalized size = 2.75 \[ \frac{(a \tan (e+f x)+a)^3 \left (4 \sin ^3(e+f x) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right )-4 \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right )-2 \sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \tan ^{\frac{3}{2}}(e+f x)+2 \sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \tan ^{\frac{3}{2}}(e+f x)+4 \sin ^2(e+f x) \cos (e+f x)-\sqrt{2} \cos ^3(e+f x) \tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )+\sqrt{2} \cos ^3(e+f x) \tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )\right )}{2 f (d \tan (e+f x))^{3/2} (\sin (e+f x)+\cos (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + a*Tan[e + f*x])^3*(-4*Cos[e + f*x]^2*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Sin[e + f*x] + 4*C
os[e + f*x]*Sin[e + f*x]^2 + 4*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sin[e + f*x]^3 - 2*Sqrt[2]*ArcT
an[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^3*Tan[e + f*x]^(3/2) + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e
+ f*x]]]*Cos[e + f*x]^3*Tan[e + f*x]^(3/2) - Sqrt[2]*Cos[e + f*x]^3*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[
e + f*x]]*Tan[e + f*x]^(3/2) + Sqrt[2]*Cos[e + f*x]^3*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e
+ f*x]^(3/2)))/(2*f*(Cos[e + f*x] + Sin[e + f*x])^3*(d*Tan[e + f*x])^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.022, size = 388, normalized size = 3.4 \begin{align*} 2\,{\frac{{a}^{3}\sqrt{d\tan \left ( fx+e \right ) }}{{d}^{2}f}}-2\,{\frac{{a}^{3}}{fd\sqrt{d\tan \left ( fx+e \right ) }}}+{\frac{{a}^{3}\sqrt{2}}{2\,{d}^{2}f}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) }+{\frac{{a}^{3}\sqrt{2}}{{d}^{2}f}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }-{\frac{{a}^{3}\sqrt{2}}{{d}^{2}f}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{{a}^{3}\sqrt{2}}{2\,fd}\ln \left ({ \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}+{\frac{{a}^{3}\sqrt{2}}{fd}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{{a}^{3}\sqrt{2}}{fd}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.74311, size = 585, normalized size = 5.13 \begin{align*} \left [\frac{\sqrt{2} a^{3} d \sqrt{-\frac{1}{d}} \log \left (\frac{2 \, \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{-\frac{1}{d}}{\left (\tan \left (f x + e\right ) - 1\right )} + \tan \left (f x + e\right )^{2} - 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right ) + 2 \,{\left (a^{3} \tan \left (f x + e\right ) - a^{3}\right )} \sqrt{d \tan \left (f x + e\right )}}{d^{2} f \tan \left (f x + e\right )}, \frac{2 \,{\left (\sqrt{2} a^{3} \sqrt{d} \arctan \left (\frac{\sqrt{2} \sqrt{d \tan \left (f x + e\right )}{\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt{d} \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) +{\left (a^{3} \tan \left (f x + e\right ) - a^{3}\right )} \sqrt{d \tan \left (f x + e\right )}\right )}}{d^{2} f \tan \left (f x + e\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \frac{1}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{3 \tan{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{\tan ^{3}{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.36073, size = 404, normalized size = 3.54 \begin{align*} -\frac{\frac{4 \, a^{3}}{\sqrt{d \tan \left (f x + e\right )} f} - \frac{4 \, \sqrt{d \tan \left (f x + e\right )} a^{3}}{d f} - \frac{\sqrt{2}{\left (a^{3} d \sqrt{{\left | d \right |}} - a^{3}{\left | d \right |}^{\frac{3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{d^{2} f} + \frac{\sqrt{2}{\left (a^{3} d \sqrt{{\left | d \right |}} - a^{3}{\left | d \right |}^{\frac{3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{d^{2} f} - \frac{2 \,{\left (\sqrt{2} a^{3} d \sqrt{{\left | d \right |}} + \sqrt{2} a^{3}{\left | d \right |}^{\frac{3}{2}}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} + 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{d^{2} f} - \frac{2 \,{\left (\sqrt{2} a^{3} d \sqrt{{\left | d \right |}} + \sqrt{2} a^{3}{\left | d \right |}^{\frac{3}{2}}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} - 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{d^{2} f}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*a^3/(sqrt(d*tan(f*x + e))*f) - 4*sqrt(d*tan(f*x + e))*a^3/(d*f) - sqrt(2)*(a^3*d*sqrt(abs(d)) - a^3*ab
s(d)^(3/2))*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/(d^2*f) + sqrt(2)*(a^3*d*
sqrt(abs(d)) - a^3*abs(d)^(3/2))*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/(d^2
*f) - 2*(sqrt(2)*a^3*d*sqrt(abs(d)) + sqrt(2)*a^3*abs(d)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*s
qrt(d*tan(f*x + e)))/sqrt(abs(d)))/(d^2*f) - 2*(sqrt(2)*a^3*d*sqrt(abs(d)) + sqrt(2)*a^3*abs(d)^(3/2))*arctan(
-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/(d^2*f))/d