Integrand size = 23, antiderivative size = 98 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} a \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}-\frac {2 a}{d^2 f \sqrt {d \tan (e+f x)}} \] Output:
2^(1/2)*a*arctan(1/2*(d^(1/2)-d^(1/2)*tan(f*x+e))*2^(1/2)/(d*tan(f*x+e))^( 1/2))/d^(5/2)/f-2/3*a/d/f/(d*tan(f*x+e))^(3/2)-2*a/d^2/f/(d*tan(f*x+e))^(1 /2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=-\frac {\left (\frac {1}{3}+\frac {i}{3}\right ) a \left (\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-i \tan (e+f x)\right )-i \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},i \tan (e+f x)\right )\right )}{d f (d \tan (e+f x))^{3/2}} \] Input:
Integrate[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2),x]
Output:
((-1/3 - I/3)*a*(Hypergeometric2F1[-3/2, 1, -1/2, (-I)*Tan[e + f*x]] - I*H ypergeometric2F1[-3/2, 1, -1/2, I*Tan[e + f*x]]))/(d*f*(d*Tan[e + f*x])^(3 /2))
Time = 0.54 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4012, 3042, 4012, 25, 3042, 4015, 218}
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.
| \(\displaystyle \int \frac {a \tan (e+f x)+a}{(d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \tan (e+f x)+a}{(d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int \frac {a d-a d \tan (e+f x)}{(d \tan (e+f x))^{3/2}}dx}{d^2}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a d-a d \tan (e+f x)}{(d \tan (e+f x))^{3/2}}dx}{d^2}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int -\frac {a d^2+a \tan (e+f x) d^2}{\sqrt {d \tan (e+f x)}}dx}{d^2}-\frac {2 a}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {a d^2+a \tan (e+f x) d^2}{\sqrt {d \tan (e+f x)}}dx}{d^2}-\frac {2 a}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {a d^2+a \tan (e+f x) d^2}{\sqrt {d \tan (e+f x)}}dx}{d^2}-\frac {2 a}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {\frac {2 a^2 d^2 \int \frac {1}{2 a^2 d^4+\cot (e+f x) \left (a d^2-a d^2 \tan (e+f x)\right )^2}d\frac {a d^2-a d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}}{f}-\frac {2 a}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\sqrt {2} a \arctan \left (\frac {a d^2-a d^2 \tan (e+f x)}{\sqrt {2} a d^{3/2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 a}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 d f (d \tan (e+f x))^{3/2}}\) |
Input:
Int[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2),x]
Output:
(-2*a)/(3*d*f*(d*Tan[e + f*x])^(3/2)) + ((Sqrt[2]*a*ArcTan[(a*d^2 - a*d^2* Tan[e + f*x])/(Sqrt[2]*a*d^(3/2)*Sqrt[d*Tan[e + f*x]])])/(Sqrt[d]*f) - (2* a)/(f*Sqrt[d*Tan[e + f*x]]))/d^2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(81)=162\).
Time = 1.54 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.14
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {2}{d^{2} \sqrt {d \tan \left (f x +e \right )}}-\frac {2}{3 d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d^{2}}\right )}{f}\) | \(308\) |
| default | \(\frac {a \left (-\frac {2}{d^{2} \sqrt {d \tan \left (f x +e \right )}}-\frac {2}{3 d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d^{2}}\right )}{f}\) | \(308\) |
| parts | \(\frac {2 a d \left (-\frac {1}{3 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{4}}\right )}{f}+\frac {a \left (-\frac {2}{d^{2} \sqrt {d \tan \left (f x +e \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}\) | \(314\) |
Input:
int((a+a*tan(f*x+e))/(d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/f*a*(-2/d^2/(d*tan(f*x+e))^(1/2)-2/3/d/(d*tan(f*x+e))^(3/2)+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 = 0.09 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.24 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=\left [\frac {3 \, \sqrt {2} a d \sqrt {-\frac {1}{d}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) - 1\right )} - \tan \left (f x + e\right )^{2} + 4 \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, {\left (3 \, a \tan \left (f x + e\right ) + a\right )} \sqrt {d \tan \left (f x + e\right )}}{6 \, d^{3} f \tan \left (f x + e\right )^{2}}, -\frac {3 \, \sqrt {2} a \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{2} + 2 \, {\left (3 \, a \tan \left (f x + e\right ) + a\right )} \sqrt {d \tan \left (f x + e\right )}}{3 \, d^{3} f \tan \left (f x + e\right )^{2}}\right ] \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(5/2),x, algorithm="fricas")
Output:
[1/6*(3*sqrt(2)*a*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 - 4*(3*a*tan(f*x + e) + a)*sqrt(d*tan(f*x + e))) /(d^3*f*tan(f*x + e)^2), -1/3*(3*sqrt(2)*a*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)^2 + 2*(3*a*tan(f*x + e) + a)*sqrt(d*tan(f*x + e)))/(d^3*f*tan(f*x + e)^2)]
\[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=a \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))**(5/2),x)
Output:
a*(Integral((d*tan(e + f*x))**(-5/2), x) + Integral(tan(e + f*x)/(d*tan(e + f*x))**(5/2), x))
Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.19 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=-\frac {\frac {3 \, a {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )}}{d} + \frac {2 \, {\left (3 \, a d \tan \left (f x + e\right ) + a d\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d}}{3 \, d f} \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(5/2),x, algorithm="maxima")
Output:
-1/3*(3*a*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2 *sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d))/d + 2*(3*a*d*tan(f*x + e) + a*d)/ ((d*tan(f*x + e))^(3/2)*d))/(d*f)
Timed out. \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(5/2),x, algorithm="giac")
Output:
Timed out
Time = 2.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=-\frac {2\,a}{d^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}-\frac {2\,a}{3\,d\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (-1+1{}\mathrm {i}\right )}{d^{5/2}\,f}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (1+1{}\mathrm {i}\right )}{d^{5/2}\,f} \] Input:
int((a + a*tan(e + f*x))/(d*tan(e + f*x))^(5/2),x)
Output:
((-1)^(1/4)*a*atanh(((-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*(1 + 1i)) /(d^(5/2)*f) - (2*a)/(3*d*f*(d*tan(e + f*x))^(3/2)) - ((-1)^(1/4)*a*atan(( (-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*(1 - 1i))/(d^(5/2)*f) - (2*a)/ (d^2*f*(d*tan(e + f*x))^(1/2))
\[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {d}\, a \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{3}}d x +\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{2}}d x \right )}{d^{3}} \] Input:
int((a+a*tan(f*x+e))/(d*tan(f*x+e))^(5/2),x)
Output:
(sqrt(d)*a*(int(sqrt(tan(e + f*x))/tan(e + f*x)**3,x) + int(sqrt(tan(e + f *x))/tan(e + f*x)**2,x)))/d**3