\(\int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx\) [354]

Optimal result

Integrand size = 25, antiderivative size = 114 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}+\frac {4 a^3 \sqrt {d \tan (e+f x)}}{d^2 f}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt {d \tan (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3646, 3711, 3613, 211} \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}+\frac {4 a^3 \sqrt {d \tan (e+f x)}}{d^2 f}-\frac {2 \left (a^3 \tan (e+f x)+a^3\right )}{d f \sqrt {d \tan (e+f x)}} \]

[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 211

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

Rule 3613

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 3646

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 3711

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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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 \arctan \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 [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(357\) vs. \(2(114)=228\).

Time = 3.09 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.13 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {a^3 \cos (e+f x) (1+\tan (e+f x))^3 \left (-4 \sin ^2(e+f x)+2 \sin (2 (e+f x))+4 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) (-\tan (e+f x))^{5/4} \sqrt [4]{\tan (e+f x)}+4 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) \sqrt [4]{-\tan (e+f x)} \tan ^{\frac {5}{4}}(e+f x)+2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)+\sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)-\sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)\right )}{2 f (\cos (e+f x)+\sin (e+f x))^3 (d \tan (e+f x))^{3/2}} \]

[In]

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

[Out]

-1/2*(a^3*Cos[e + f*x]*(1 + Tan[e + f*x])^3*(-4*Sin[e + f*x]^2 + 2*Sin[2*(e + f*x)] + 4*ArcTan[(-Tan[e + f*x]^
2)^(1/4)]*Cos[e + f*x]^2*(-Tan[e + f*x])^(5/4)*Tan[e + f*x]^(1/4) + 4*ArcTanh[(-Tan[e + f*x]^2)^(1/4)]*Cos[e +
f*x]^2*(-Tan[e + f*x])^(1/4)*Tan[e + f*x]^(5/4) + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*
x]^2*Tan[e + f*x]^(3/2) - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2*Tan[e + f*x]^(3/2) +
Sqrt[2]*Cos[e + f*x]^2*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]^2*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2)))/(f*(Cos[e + f*x] + Sin[e + f*
x])^3*(d*Tan[e + f*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(99)=198\).

Time = 0.93 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.68

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {d}{\sqrt {d \tan \left (f x +e \right )}}+2 d \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\)	\(305\)
default	\(\frac {2 a^{3} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {d}{\sqrt {d \tan \left (f x +e \right )}}+2 d \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\)	\(305\)
parts	\(\frac {2 a^{3} d \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}-\frac {1}{d^{2} \sqrt {d \tan \left (f x +e \right )}}\right )}{f}+\frac {2 a^{3} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{f \,d^{2}}+\frac {3 a^{3} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f \,d^{2}}+\frac {3 a^{3} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f d \left (d^{2}\right )^{\frac {1}{4}}}\)	\(594\)

[In]

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

[Out]

2/f*a^3/d^2*((d*tan(f*x+e))^(1/2)-d/(d*tan(f*x+e))^(1/2)+2*d*(1/8/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)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/
2)+1))+1/8/(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)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2
)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.96 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=\left [\frac {\sqrt {2} a^{3} d \sqrt {-\frac {1}{d}} \log \left (\frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) - 1\right )} + \tan \left (f x + e\right )^{2} - 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right ) + 2 \, {\left (a^{3} \tan \left (f x + e\right ) - a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{d^{2} f \tan \left (f x + e\right )}, \frac {2 \, {\left (\sqrt {2} a^{3} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) + {\left (a^{3} \tan \left (f x + e\right ) - a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{d^{2} f \tan \left (f x + e\right )}\right ] \]

[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]

\[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=a^{3} \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {3 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=\frac {2 \, {\left (a^{3} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )} - \frac {a^{3}}{\sqrt {d \tan \left (f x + e\right )}} + \frac {\sqrt {d \tan \left (f x + e\right )} a^{3}}{d}\right )}}{d f} \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 5.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=\frac {2\,a^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{d^2\,f}-\frac {2\,a^3}{d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}+\frac {\sqrt {2}\,a^3\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{d^{3/2}\,f} \]

[In]

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

[Out]

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