Integrand size = 23, antiderivative size = 121 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=-\frac {\sqrt {2} a \text {arctanh}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}-\frac {2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}+\frac {2 a}{d^3 f \sqrt {d \tan (e+f x)}} \] Output:
-2^(1/2)*a*arctanh(1/2*(d^(1/2)+d^(1/2)*tan(f*x+e))*2^(1/2)/(d*tan(f*x+e)) ^(1/2))/d^(7/2)/f-2/5*a/d/f/(d*tan(f*x+e))^(5/2)-2/3*a/d^2/f/(d*tan(f*x+e) )^(3/2)+2*a/d^3/f/(d*tan(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.56 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=-\frac {\left (\frac {1}{5}+\frac {i}{5}\right ) a \left (\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-i \tan (e+f x)\right )-i \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},i \tan (e+f x)\right )\right )}{d f (d \tan (e+f x))^{5/2}} \] Input:
Integrate[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2),x]
Output:
((-1/5 - I/5)*a*(Hypergeometric2F1[-5/2, 1, -3/2, (-I)*Tan[e + f*x]] - I*H ypergeometric2F1[-5/2, 1, -3/2, I*Tan[e + f*x]]))/(d*f*(d*Tan[e + f*x])^(5 /2))
Time = 0.68 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4012, 3042, 4012, 25, 3042, 4012, 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.
| \(\displaystyle \int \frac {a \tan (e+f x)+a}{(d \tan (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \tan (e+f x)+a}{(d \tan (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int \frac {a d-a d \tan (e+f x)}{(d \tan (e+f x))^{5/2}}dx}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a d-a d \tan (e+f x)}{(d \tan (e+f x))^{5/2}}dx}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int -\frac {a d^2+a \tan (e+f x) d^2}{(d \tan (e+f x))^{3/2}}dx}{d^2}-\frac {2 a}{3 f (d \tan (e+f x))^{3/2}}}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {a d^2+a \tan (e+f x) d^2}{(d \tan (e+f x))^{3/2}}dx}{d^2}-\frac {2 a}{3 f (d \tan (e+f x))^{3/2}}}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {a d^2+a \tan (e+f x) d^2}{(d \tan (e+f x))^{3/2}}dx}{d^2}-\frac {2 a}{3 f (d \tan (e+f x))^{3/2}}}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {a d^3-a d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{d^2}-\frac {2 a d}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 f (d \tan (e+f x))^{3/2}}}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {a d^3-a d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{d^2}-\frac {2 a d}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 f (d \tan (e+f x))^{3/2}}}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {-\frac {-\frac {2 a^2 d^4 \int \frac {1}{\cot (e+f x) \left (a d^3+a \tan (e+f x) d^3\right )^2-2 a^2 d^6}d\frac {a d^3+a \tan (e+f x) d^3}{\sqrt {d \tan (e+f x)}}}{f}-\frac {2 a d}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 f (d \tan (e+f x))^{3/2}}}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {\sqrt {2} a \sqrt {d} \text {arctanh}\left (\frac {a d^3 \tan (e+f x)+a d^3}{\sqrt {2} a d^{5/2} \sqrt {d \tan (e+f x)}}\right )}{f}-\frac {2 a d}{f \sqrt {d \tan (e+f x)}}}{d^2}-\frac {2 a}{3 f (d \tan (e+f x))^{3/2}}}{d^2}-\frac {2 a}{5 d f (d \tan (e+f x))^{5/2}}\) |
Input:
Int[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2),x]
Output:
(-2*a)/(5*d*f*(d*Tan[e + f*x])^(5/2)) + ((-2*a)/(3*f*(d*Tan[e + f*x])^(3/2 )) - ((Sqrt[2]*a*Sqrt[d]*ArcTanh[(a*d^3 + a*d^3*Tan[e + f*x])/(Sqrt[2]*a*d ^(5/2)*Sqrt[d*Tan[e + f*x]])])/f - (2*a*d)/(f*Sqrt[d*Tan[e + f*x]]))/d^2)/ d^2
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[((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. \(322\) vs. \(2(100)=200\).
Time = 1.49 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.67
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {2}{3 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{5 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {2}{d^{3} \sqrt {d \tan \left (f x +e \right )}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d^{3}}\right )}{f}\) | \(323\) |
| default | \(\frac {a \left (-\frac {2}{3 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{5 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {2}{d^{3} \sqrt {d \tan \left (f x +e \right )}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d^{3}}\right )}{f}\) | \(323\) |
| parts | \(\frac {2 a d \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{4} \left (d^{2}\right )^{\frac {1}{4}}}-\frac {1}{5 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {1}{d^{4} \sqrt {d \tan \left (f x +e \right )}}\right )}{f}+\frac {a \left (-\frac {2}{3 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d^{4}}\right )}{f}\) | \(328\) |
Input:
int((a+a*tan(f*x+e))/(d*tan(f*x+e))^(7/2),x,method=_RETURNVERBOSE)
Output:
1/f*a*(-2/3/d^2/(d*tan(f*x+e))^(3/2)-2/5/d/(d*tan(f*x+e))^(5/2)+2/d^3/(d*t an(f*x+e))^(1/2)+2/d^3*(-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))))
Time = 0.09 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.01 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=\left [\frac {15 \, \sqrt {2} a \sqrt {d} \log \left (\frac {\tan \left (f x + e\right )^{2} - \frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt {d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{3} + 4 \, {\left (15 \, a \tan \left (f x + e\right )^{2} - 5 \, a \tan \left (f x + e\right ) - 3 \, a\right )} \sqrt {d \tan \left (f x + e\right )}}{30 \, d^{4} f \tan \left (f x + e\right )^{3}}, \frac {15 \, \sqrt {2} a d \sqrt {-\frac {1}{d}} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{3} + 2 \, {\left (15 \, a \tan \left (f x + e\right )^{2} - 5 \, a \tan \left (f x + e\right ) - 3 \, a\right )} \sqrt {d \tan \left (f x + e\right )}}{15 \, d^{4} f \tan \left (f x + e\right )^{3}}\right ] \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(7/2),x, algorithm="fricas")
Output:
[1/30*(15*sqrt(2)*a*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))*tan(f*x + e)^3 + 4*(15*a*tan(f*x + e)^2 - 5*a*tan(f*x + e) - 3*a)*sqrt (d*tan(f*x + e)))/(d^4*f*tan(f*x + e)^3), 1/15*(15*sqrt(2)*a*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))*tan(f*x + e)^3 + 2*(15*a*tan(f*x + e)^2 - 5*a*tan(f*x + e) - 3*a )*sqrt(d*tan(f*x + e)))/(d^4*f*tan(f*x + e)^3)]
\[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=a \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {\tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx\right ) \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))**(7/2),x)
Output:
a*(Integral((d*tan(e + f*x))**(-7/2), x) + Integral(tan(e + f*x)/(d*tan(e + f*x))**(7/2), x))
Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.12 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=-\frac {\frac {15 \, a {\left (\frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )}}{d^{2}} - \frac {4 \, {\left (15 \, a d^{2} \tan \left (f x + e\right )^{2} - 5 \, a d^{2} \tan \left (f x + e\right ) - 3 \, a d^{2}\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} d^{2}}}{30 \, d f} \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(7/2),x, algorithm="maxima")
Output:
-1/30*(15*a*(sqrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqr t(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))/d^2 - 4*(15*a*d^2*tan(f*x + e)^2 - 5*a*d^2*tan(f *x + e) - 3*a*d^2)/((d*tan(f*x + e))^(5/2)*d^2))/(d*f)
Timed out. \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(7/2),x, algorithm="giac")
Output:
Timed out
Time = 3.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=-\frac {\frac {2\,a}{5\,d}-\frac {2\,a\,{\mathrm {tan}\left (e+f\,x\right )}^2}{d}}{f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}-\frac {2\,a}{3\,d^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (1+1{}\mathrm {i}\right )}{d^{7/2}\,f}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (-1+1{}\mathrm {i}\right )}{d^{7/2}\,f} \] Input:
int((a + a*tan(e + f*x))/(d*tan(e + f*x))^(7/2),x)
Output:
((-1)^(1/4)*a*atan(((-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*(1 + 1i))/ (d^(7/2)*f) - (2*a)/(3*d^2*f*(d*tan(e + f*x))^(3/2)) - ((2*a)/(5*d) - (2*a *tan(e + f*x)^2)/d)/(f*(d*tan(e + f*x))^(5/2)) - ((-1)^(1/4)*a*atanh(((-1) ^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*(1 - 1i))/(d^(7/2)*f)
\[ \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx=\frac {\sqrt {d}\, a \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{4}}d x +\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{3}}d x \right )}{d^{4}} \] Input:
int((a+a*tan(f*x+e))/(d*tan(f*x+e))^(7/2),x)
Output:
(sqrt(d)*a*(int(sqrt(tan(e + f*x))/tan(e + f*x)**4,x) + int(sqrt(tan(e + f *x))/tan(e + f*x)**3,x)))/d**4