Integrand size = 25, antiderivative size = 117 \[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 \sqrt {2} a^3 \text {arctanh}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {d} f}+\frac {16 a^3 \sqrt {d \tan (e+f x)}}{3 d f}+\frac {2 \sqrt {d \tan (e+f x)} \left (a^3+a^3 \tan (e+f x)\right )}{3 d f} \] Output:
-2*2^(1/2)*a^3*arctanh(1/2*(d^(1/2)+d^(1/2)*tan(f*x+e))*2^(1/2)/(d*tan(f*x +e))^(1/2))/d^(1/2)/f+16/3*a^3*(d*tan(f*x+e))^(1/2)/d/f+2/3*(d*tan(f*x+e)) ^(1/2)*(a^3+a^3*tan(f*x+e))/d/f
Leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(117)=234\).
Time = 1.07 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.92 \[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\frac {a^3 \cos (e+f x) (1+\tan (e+f x))^3 \left (4 \sin ^2(e+f x)+18 \sin (2 (e+f x))+6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \sqrt {\tan (e+f x)}-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \sqrt {\tan (e+f x)}+3 \sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {\tan (e+f x)}-3 \sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {\tan (e+f x)}+12 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) \sqrt [4]{-\tan ^2(e+f x)}-12 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) \sqrt [4]{-\tan ^2(e+f x)}\right )}{6 f (\cos (e+f x)+\sin (e+f x))^3 \sqrt {d \tan (e+f x)}} \] Input:
Integrate[(a + a*Tan[e + f*x])^3/Sqrt[d*Tan[e + f*x]],x]
Output:
(a^3*Cos[e + f*x]*(1 + Tan[e + f*x])^3*(4*Sin[e + f*x]^2 + 18*Sin[2*(e + f *x)] + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2*Sqr t[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]] + 12*ArcTan[(-Tan[e + f*x]^2)^(1/4)]*Cos[e + f*x]^2*(-Tan[e + f*x]^2)^(1/4) - 12*ArcTanh[(-Tan[e + f*x]^2)^(1/4)]*Cos[e + f*x]^2*(-Tan[e + f*x]^2)^(1 /4)))/(6*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]])
Time = 0.53 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 4049, 3042, 4113, 3042, 4015, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \tan (e+f x)+a)^3}{\sqrt {d \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \tan (e+f x)+a)^3}{\sqrt {d \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {2 \int \frac {4 d \tan ^2(e+f x) a^3+d a^3+3 d \tan (e+f x) a^3}{\sqrt {d \tan (e+f x)}}dx}{3 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {4 d \tan (e+f x)^2 a^3+d a^3+3 d \tan (e+f x) a^3}{\sqrt {d \tan (e+f x)}}dx}{3 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 \left (\int \frac {3 a^3 d \tan (e+f x)-3 a^3 d}{\sqrt {d \tan (e+f x)}}dx+\frac {8 a^3 \sqrt {d \tan (e+f x)}}{f}\right )}{3 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int \frac {3 a^3 d \tan (e+f x)-3 a^3 d}{\sqrt {d \tan (e+f x)}}dx+\frac {8 a^3 \sqrt {d \tan (e+f x)}}{f}\right )}{3 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {2 \left (\frac {8 a^3 \sqrt {d \tan (e+f x)}}{f}-\frac {18 a^6 d^2 \int \frac {1}{9 \cot (e+f x) \left (d a^3+d \tan (e+f x) a^3\right )^2-18 a^6 d^2}d\left (-\frac {3 \left (d a^3+d \tan (e+f x) a^3\right )}{\sqrt {d \tan (e+f x)}}\right )}{f}\right )}{3 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \left (\frac {8 a^3 \sqrt {d \tan (e+f x)}}{f}-\frac {3 \sqrt {2} a^3 \sqrt {d} \text {arctanh}\left (\frac {a^3 d \tan (e+f x)+a^3 d}{\sqrt {2} a^3 \sqrt {d} \sqrt {d \tan (e+f x)}}\right )}{f}\right )}{3 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) \sqrt {d \tan (e+f x)}}{3 d f}\) |
Input:
Int[(a + a*Tan[e + f*x])^3/Sqrt[d*Tan[e + f*x]],x]
Output:
(2*Sqrt[d*Tan[e + f*x]]*(a^3 + a^3*Tan[e + f*x]))/(3*d*f) + (2*((-3*Sqrt[2 ]*a^3*Sqrt[d]*ArcTanh[(a^3*d + a^3*d*Tan[e + f*x])/(Sqrt[2]*a^3*Sqrt[d]*Sq rt[d*Tan[e + f*x]])])/f + (8*a^3*Sqrt[d*Tan[e + f*x]])/f))/(3*d)
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]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[-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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[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] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
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*Si mp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(308\) vs. \(2(98)=196\).
Time = 1.84 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.64
| method | result | size |
| derivativedivides | \(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+3 d \sqrt {d \tan \left (f x +e \right )}-2 d^{2} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(309\) |
| default | \(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+3 d \sqrt {d \tan \left (f x +e \right )}-2 d^{2} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(309\) |
| parts | \(\frac {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}+\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d^{2} \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 )}{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 \left (d^{2}\right )^{\frac {1}{4}}}+\frac {6 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}\) | \(590\) |
Input:
int((a+a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/f*a^3/d^2*(1/3*(d*tan(f*x+e))^(3/2)+3*d*(d*tan(f*x+e))^(1/2)-2*d^2*(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*arct an(-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))))
Time = 0.10 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.69 \[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\left [\frac {3 \, \sqrt {2} a^{3} \sqrt {d} \log \left (\frac {\tan \left (f x + e\right )^{2} - \frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt {d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (a^{3} \tan \left (f x + e\right ) + 9 \, a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{3 \, d f}, \frac {2 \, {\left (3 \, \sqrt {2} a^{3} d \sqrt {-\frac {1}{d}} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) + {\left (a^{3} \tan \left (f x + e\right ) + 9 \, a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{3 \, d f}\right ] \] Input:
integrate((a+a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
[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*sqr t(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)]
\[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=a^{3} \left (\int \frac {1}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx + \int \frac {3 \tan {\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx + \int \frac {3 \tan ^{2}{\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx\right ) \] Input:
integrate((a+a*tan(f*x+e))**3/(d*tan(f*x+e))**(1/2),x)
Output:
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))
Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {3 \, a^{3} d {\left (\frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {2 \, {\left (\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} + 9 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d\right )}}{d}}{3 \, d f} \] Input:
integrate((a+a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x, algorithm="maxima")
Output:
-1/3*(3*a^3*d*(sqrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*s qrt(d) + d)/sqrt(d) - sqrt(2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d)) - 2*((d*tan(f*x + e))^(3/2)*a^3 + 9*sqrt(d*tan (f*x + e))*a^3*d)/d)/(d*f)
Exception generated. \[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[4,14]%%%}+%%%{6,[4,12]%%%}+%%%{15,[4,10]%%%}+%%%{20 ,[4,8]%%%
Time = 1.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85 \[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\frac {6\,a^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{d\,f}+\frac {2\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{3\,d^2\,f}-\frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,\sqrt {d}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{32\,a^6\,d+32\,a^6\,d\,\mathrm {tan}\left (e+f\,x\right )}\right )}{\sqrt {d}\,f} \] Input:
int((a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(1/2),x)
Output:
(6*a^3*(d*tan(e + f*x))^(1/2))/(d*f) + (2*a^3*(d*tan(e + f*x))^(3/2))/(3*d ^2*f) - (2*2^(1/2)*a^3*atanh((32*2^(1/2)*a^6*d^(1/2)*(d*tan(e + f*x))^(1/2 ))/(32*a^6*d + 32*a^6*d*tan(e + f*x))))/(d^(1/2)*f)
\[ \int \frac {(a+a \tan (e+f x))^3}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\sqrt {d}\, a^{3} \left (6 \sqrt {\tan \left (f x +e \right )}-2 \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right ) f +3 \left (\int \sqrt {\tan \left (f x +e \right )}d x \right ) f +\left (\int \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}d x \right ) f \right )}{d f} \] Input:
int((a+a*tan(f*x+e))^3/(d*tan(f*x+e))^(1/2),x)
Output:
(sqrt(d)*a**3*(6*sqrt(tan(e + f*x)) - 2*int(sqrt(tan(e + f*x))/tan(e + f*x ),x)*f + 3*int(sqrt(tan(e + f*x)),x)*f + int(sqrt(tan(e + f*x))*tan(e + f* x)**2,x)*f))/(d*f)