\(\int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx\) [348]

Optimal result

Integrand size = 25, antiderivative size = 193 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{5/2} f}-\frac {\sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{5/2} f}+\frac {\sqrt {2} a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}+\sqrt {d} \tan (e+f x)}\right )}{d^{5/2} f}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 f \sqrt {d \tan (e+f x)}} \] Output:

2^(1/2)*a^2*arctan(1-2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/d^(5/2)/f-2^(1/ 
2)*a^2*arctan(1+2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/d^(5/2)/f+2^(1/2)*a^ 
2*arctanh(2^(1/2)*(d*tan(f*x+e))^(1/2)/(d^(1/2)+d^(1/2)*tan(f*x+e)))/d^(5/ 
2)/f-2/3*a^2/d/f/(d*tan(f*x+e))^(3/2)-4*a^2/d^2/f/(d*tan(f*x+e))^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(399\) vs. \(2(193)=386\).

Time = 3.39 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.07 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {a^2 (1+\cot (e+f x))^2 \left (-48 \sin ^2(e+f x)-4 \sin (2 (e+f x))-6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {5}{2}}(e+f x)+6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {5}{2}}(e+f x)-3 \sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {5}{2}}(e+f x)+3 \sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {5}{2}}(e+f x)+12 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \left (\cos ^2(e+f x) (-\tan (e+f x))^{3/4} \tan ^{\frac {7}{4}}(e+f x)+2 \sin ^2(e+f x) \sqrt [4]{-\tan ^2(e+f x)}\right )+12 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) \left ((-\tan (e+f x))^{3/4} \tan ^{\frac {7}{4}}(e+f x)+2 \left (-\tan ^2(e+f x)\right )^{5/4}\right )\right )}{12 d^2 f (\cos (e+f x)+\sin (e+f x))^2 \sqrt {d \tan (e+f x)}} \] Input:

Integrate[(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(5/2),x]
 

Output:

(a^2*(1 + Cot[e + f*x])^2*(-48*Sin[e + f*x]^2 - 4*Sin[2*(e + f*x)] - 6*Sqr 
t[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2*Tan[e + f*x]^(5 
/2) + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2*Tan[ 
e + f*x]^(5/2) - 3*Sqrt[2]*Cos[e + f*x]^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x 
]] + Tan[e + f*x]]*Tan[e + f*x]^(5/2) + 3*Sqrt[2]*Cos[e + f*x]^2*Log[1 + S 
qrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(5/2) + 12*ArcTanh[ 
(-Tan[e + f*x]^2)^(1/4)]*(Cos[e + f*x]^2*(-Tan[e + f*x])^(3/4)*Tan[e + f*x 
]^(7/4) + 2*Sin[e + f*x]^2*(-Tan[e + f*x]^2)^(1/4)) + 12*ArcTan[(-Tan[e + 
f*x]^2)^(1/4)]*Cos[e + f*x]^2*((-Tan[e + f*x])^(3/4)*Tan[e + f*x]^(7/4) + 
2*(-Tan[e + f*x]^2)^(5/4))))/(12*d^2*f*(Cos[e + f*x] + Sin[e + f*x])^2*Sqr 
t[d*Tan[e + f*x]])
 

Rubi [A] (warning: unable to verify)

Time = 0.68 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.20, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4025, 27, 3042, 3955, 3042, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {(a \tan (e+f x)+a)^2}{(d \tan (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \tan (e+f x)+a)^2}{(d \tan (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 4025

\(\displaystyle \frac {\int \frac {2 a^2 d}{(d \tan (e+f x))^{3/2}}dx}{d^2}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a^2 \int \frac {1}{(d \tan (e+f x))^{3/2}}dx}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 \int \frac {1}{(d \tan (e+f x))^{3/2}}dx}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3955

\(\displaystyle \frac {2 a^2 \left (-\frac {\int \sqrt {d \tan (e+f x)}dx}{d^2}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 \left (-\frac {\int \sqrt {d \tan (e+f x)}dx}{d^2}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {2 a^2 \left (-\frac {\int \frac {\sqrt {d \tan (e+f x)}}{\tan ^2(e+f x) d^2+d^2}d(d \tan (e+f x))}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \int \frac {d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \int \frac {d^2 \tan ^2(e+f x)+d}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2 a^2 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{d f}-\frac {2}{d f \sqrt {d \tan (e+f x)}}\right )}{d}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}\)

Input:

Int[(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(5/2),x]
 

Output:

(-2*a^2)/(3*d*f*(d*Tan[e + f*x])^(3/2)) + (2*a^2*((-2*((-(ArcTan[1 - Sqrt[ 
2]*Sqrt[d]*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d])) + ArcTan[1 + Sqrt[2]*Sqrt[d]*T 
an[e + f*x]]/(Sqrt[2]*Sqrt[d]))/2 + (Log[d - Sqrt[2]*d^(3/2)*Tan[e + f*x] 
+ d^2*Tan[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]) - Log[d + Sqrt[2]*d^(3/2)*Tan[e 
+ f*x] + d^2*Tan[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]))/2))/(d*f) - 2/(d*f*Sqrt[ 
d*Tan[e + f*x]])))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3955

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] 
)^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2   Int[(b*Tan[c + d*x])^(n + 2), x] 
, x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4025

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 
 1)/(b*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^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta 
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] 
 && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
 

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (d^{2}\right )^{\frac {1}{4}}}-\frac {1}{3 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}\right )}{f d}\)	\(174\)
default	\(\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (d^{2}\right )^{\frac {1}{4}}}-\frac {1}{3 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}\right )}{f d}\)	\(174\)
parts	\(\frac {2 a^{2} 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^{2} \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^{3}}+\frac {2 a^{2} \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}\)	\(459\)

Input:

int((a+a*tan(f*x+e))^2/(d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.10 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=-\frac {6 \, \sqrt {2} a^{2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} + 1\right ) \tan \left (f x + e\right )^{2} + 6 \, \sqrt {2} a^{2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} - 1\right ) \tan \left (f x + e\right )^{2} - 3 \, \sqrt {2} a^{2} \sqrt {d} \log \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} + \tan \left (f x + e\right ) + 1\right ) \tan \left (f x + e\right )^{2} + 3 \, \sqrt {2} a^{2} \sqrt {d} \log \left (-\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} + \tan \left (f x + e\right ) + 1\right ) \tan \left (f x + e\right )^{2} + 4 \, {\left (6 \, a^{2} \tan \left (f x + e\right ) + a^{2}\right )} \sqrt {d \tan \left (f x + e\right )}}{6 \, d^{3} f \tan \left (f x + e\right )^{2}} \] Input:

integrate((a+a*tan(f*x+e))^2/(d*tan(f*x+e))^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=a^{2} \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {2 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \] Input:

integrate((a+a*tan(f*x+e))**2/(d*tan(f*x+e))**(5/2),x)
 

Output:

a**2*(Integral((d*tan(e + f*x))**(-5/2), x) + Integral(2*tan(e + f*x)/(d*t 
an(e + f*x))**(5/2), x) + Integral(tan(e + f*x)**2/(d*tan(e + f*x))**(5/2) 
, x))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=-\frac {\frac {3 \, a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )}}{d} + \frac {4 \, {\left (6 \, a^{2} d \tan \left (f x + e\right ) + a^{2} d\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d}}{6 \, d f} \] Input:

integrate((a+a*tan(f*x+e))^2/(d*tan(f*x+e))^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*tan(f*x+e))^2/(d*tan(f*x+e))^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.52 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{5/2}\,f}-\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{5/2}\,f}-\frac {4\,a^2\,\mathrm {tan}\left (e+f\,x\right )+\frac {2\,a^2}{3}}{d\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \] Input:

int((a + a*tan(e + f*x))^2/(d*tan(e + f*x))^(5/2),x)
 

Output:

(2*(-1)^(1/4)*a^2*atanh(((-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2)))/(d^( 
5/2)*f) - (2*(-1)^(1/4)*a^2*atan(((-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/ 
2)))/(d^(5/2)*f) - (4*a^2*tan(e + f*x) + (2*a^2)/3)/(d*f*(d*tan(e + f*x))^ 
(3/2))
 

Reduce [F]

\[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {d}\, a^{2} \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{3}}d x +2 \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{2}}d x \right )+\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right )}{d^{3}} \] Input:

int((a+a*tan(f*x+e))^2/(d*tan(f*x+e))^(5/2),x)
 

Output:

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