3.353 \(\int \frac{(a+a \tan (e+f x))^3}{\sqrt{d \tan (e+f x)}} \, dx\)

Optimal. Leaf size=117 \[ \frac{16 a^3 \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt{d \tan (e+f x)}}{3 d f}-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{\sqrt{d} f} \]

[Out]

(-2*Sqrt[2]*a^3*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[d]*f) + (16*a^
3*Sqrt[d*Tan[e + f*x]])/(3*d*f) + (2*Sqrt[d*Tan[e + f*x]]*(a^3 + a^3*Tan[e + f*x]))/(3*d*f)

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Rubi [A]  time = 0.154886, antiderivative size = 117, 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 = {3566, 3630, 3532, 208} \[ \frac{16 a^3 \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt{d \tan (e+f x)}}{3 d f}-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{\sqrt{d} f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*a^3*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[d]*f) + (16*a^
3*Sqrt[d*Tan[e + f*x]])/(3*d*f) + (2*Sqrt[d*Tan[e + f*x]]*(a^3 + a^3*Tan[e + f*x]))/(3*d*f)

Rule 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

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 208

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

Rubi steps

\begin{align*} \int \frac{(a+a \tan (e+f x))^3}{\sqrt{d \tan (e+f x)}} \, dx &=\frac{2 \sqrt{d \tan (e+f x)} \left (a^3+a^3 \tan (e+f x)\right )}{3 d f}+\frac{2 \int \frac{a^3 d+3 a^3 d \tan (e+f x)+4 a^3 d \tan ^2(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{3 d}\\ &=\frac{16 a^3 \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \sqrt{d \tan (e+f x)} \left (a^3+a^3 \tan (e+f x)\right )}{3 d f}+\frac{2 \int \frac{-3 a^3 d+3 a^3 d \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{3 d}\\ &=\frac{16 a^3 \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \sqrt{d \tan (e+f x)} \left (a^3+a^3 \tan (e+f x)\right )}{3 d f}-\frac{\left (12 a^6 d\right ) \operatorname{Subst}\left (\int \frac{1}{-18 a^6 d^2+d x^2} \, dx,x,\frac{-3 a^3 d-3 a^3 d \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{d}+\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{\sqrt{d} f}+\frac{16 a^3 \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \sqrt{d \tan (e+f x)} \left (a^3+a^3 \tan (e+f x)\right )}{3 d f}\\ \end{align*}

Mathematica [C]  time = 5.65543, size = 292, normalized size = 2.5 \[ \frac{a^3 \cos (e+f x) (\tan (e+f x)+1)^3 \left (8 \sin ^2(e+f x) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right )+4 \sin ^2(e+f x)+18 \sin (2 (e+f x))+6 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \sqrt{\tan (e+f x)}-6 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \sqrt{\tan (e+f x)}+3 \sqrt{2} \cos ^2(e+f x) \sqrt{\tan (e+f x)} \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )-3 \sqrt{2} \cos ^2(e+f x) \sqrt{\tan (e+f x)} \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )\right )}{6 f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Cos[e + f*x]*(4*Sin[e + f*x]^2 + 8*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sin[e + f*x]^2 + 18*Si
n[2*(e + f*x)] + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2*Sqrt[Tan[e + f*x]] - 6*Sqrt[2
]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2*Sqrt[Tan[e + f*x]] + 3*Sqrt[2]*Cos[e + f*x]^2*Log[1 -
Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[Tan[e + f*x]] - 3*Sqrt[2]*Cos[e + f*x]^2*Log[1 + Sqrt[2]*Sqrt[
Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[Tan[e + f*x]])*(1 + Tan[e + f*x])^3)/(6*f*(Cos[e + f*x] + Sin[e + f*x])^3*S
qrt[d*Tan[e + f*x]])

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Maple [B]  time = 0.034, size = 379, normalized size = 3.2 \begin{align*}{\frac{2\,{a}^{3}}{3\,f{d}^{2}} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{3}\sqrt{d\tan \left ( fx+e \right ) }}{df}}-{\frac{{a}^{3}\sqrt{2}}{2\,df}\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}}{df}\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}}{df}\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\,f}\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}}{f}\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}}{f}\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))^(1/2),x)

[Out]

2/3/f*a^3/d^2*(d*tan(f*x+e))^(3/2)+6*a^3*(d*tan(f*x+e))^(1/2)/d/f-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)+1/2/f*a^3/(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)^(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)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)

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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))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74208, size = 525, normalized size = 4.49 \begin{align*} \left [\frac{3 \, \sqrt{2} a^{3} \sqrt{d} \log \left (\frac{\tan \left (f x + e\right )^{2} - \frac{2 \, \sqrt{2} \sqrt{d \tan \left (f x + e\right )}{\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt{d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (a^{3} \tan \left (f x + e\right ) + 9 \, a^{3}\right )} \sqrt{d \tan \left (f x + e\right )}}{3 \, d f}, \frac{2 \,{\left (3 \, \sqrt{2} a^{3} d \sqrt{-\frac{1}{d}} \arctan \left (\frac{\sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{-\frac{1}{d}}{\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) +{\left (a^{3} \tan \left (f x + e\right ) + 9 \, a^{3}\right )} \sqrt{d \tan \left (f x + e\right )}\right )}}{3 \, d f}\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))^(1/2),x, algorithm="fricas")

[Out]

[1/3*(3*sqrt(2)*a^3*sqrt(d)*log((tan(f*x + e)^2 - 2*sqrt(2)*sqrt(d*tan(f*x + e))*(tan(f*x + e) + 1)/sqrt(d) +
4*tan(f*x + e) + 1)/(tan(f*x + e)^2 + 1)) + 2*(a^3*tan(f*x + e) + 9*a^3)*sqrt(d*tan(f*x + e)))/(d*f), 2/3*(3*s
qrt(2)*a^3*d*sqrt(-1/d)*arctan(1/2*sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(-1/d)*(tan(f*x + e) + 1)/tan(f*x + e)) +
(a^3*tan(f*x + e) + 9*a^3)*sqrt(d*tan(f*x + e)))/(d*f)]

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

[Out]

a**3*(Integral(1/sqrt(d*tan(e + f*x)), x) + Integral(3*tan(e + f*x)/sqrt(d*tan(e + f*x)), x) + Integral(3*tan(
e + f*x)**2/sqrt(d*tan(e + f*x)), x) + Integral(tan(e + f*x)**3/sqrt(d*tan(e + f*x)), x))

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Giac [B]  time = 1.35184, size = 423, normalized size = 3.62 \begin{align*} -\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 )}{2 \, 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 )}{2 \, d^{2} f} - \frac{{\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{{\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{d \tan \left (f x + e\right )} a^{3} d^{5} f^{2} \tan \left (f x + e\right ) + 9 \, \sqrt{d \tan \left (f x + e\right )} a^{3} d^{5} f^{2}\right )}}{3 \, d^{6} f^{3}} \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))^(1/2),x, algorithm="giac")

[Out]

-1/2*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(ab
s(d)) + abs(d))/(d^2*f) + 1/2*sqrt(2)*(a^3*d*sqrt(abs(d)) + a^3*abs(d)^(3/2))*log(d*tan(f*x + e) - sqrt(2)*sqr
t(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/(d^2*f) - (sqrt(2)*a^3*d*sqrt(abs(d)) - sqrt(2)*a^3*abs(d)^(3/2))*arc
tan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/(d^2*f) - (sqrt(2)*a^3*d*sqrt(ab
s(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) + 2/3*(sqrt(d*tan(f*x + e))*a^3*d^5*f^2*tan(f*x + e) + 9*sqrt(d*tan(f*x + e))*a^3*d^5*f^2)/(d^6*
f^3)