3.3.51 \(\int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx\) [251]

3.3.51.1 Optimal result
3.3.51.2 Mathematica [A] (verified)
3.3.51.3 Rubi [A] (verified)
3.3.51.4 Maple [B] (warning: unable to verify)
3.3.51.5 Fricas [C] (verification not implemented)
3.3.51.6 Sympy [F(-1)]
3.3.51.7 Maxima [A] (verification not implemented)
3.3.51.8 Giac [A] (verification not implemented)
3.3.51.9 Mupad [F(-1)]

3.3.51.1 Optimal result

Integrand size = 21, antiderivative size = 225 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {3 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f} \]

output
-3/8*d^(5/2)*arctan(1-2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/f*2^(1/2)+3/8* 
d^(5/2)*arctan(1+2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/f*2^(1/2)+3/16*d^(5 
/2)*ln(d^(1/2)-2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))/f*2^(1/2)- 
3/16*d^(5/2)*ln(d^(1/2)+2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))/f 
*2^(1/2)-1/2*d*cos(f*x+e)^2*(d*tan(f*x+e))^(3/2)/f
 
3.3.51.2 Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.48 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {d^2 \left (3 \arcsin (\cos (e+f x)-\sin (e+f x)) \csc (e+f x)+3 \csc (e+f x) \log \left (\cos (e+f x)+\sin (e+f x)+\sqrt {\sin (2 (e+f x))}\right )+2 \sqrt {\sin (2 (e+f x))}\right ) \sqrt {\sin (2 (e+f x))} \sqrt {d \tan (e+f x)}}{8 f} \]

input
Integrate[Cos[e + f*x]^2*(d*Tan[e + f*x])^(5/2),x]
 
output
-1/8*(d^2*(3*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Csc[e + f*x] + 3*Csc[e + 
f*x]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]] + 2*Sqrt[Si 
n[2*(e + f*x)]])*Sqrt[Sin[2*(e + f*x)]]*Sqrt[d*Tan[e + f*x]])/f
 
3.3.51.3 Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3087, 252, 266, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3087

\(\displaystyle \frac {\int \frac {(d \tan (e+f x))^{5/2}}{\left (\tan ^2(e+f x)+1\right )^2}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

input
Int[Cos[e + f*x]^2*(d*Tan[e + f*x])^(5/2),x]
 
output
((3*d^3*((-(ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sq 
rt[d])) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt 
[d]))/2 + (Log[d + d*Tan[e + f*x] - Sqrt[2]*Sqrt[d]*Sqrt[d*Tan[e + f*x]]]/ 
(2*Sqrt[2]*Sqrt[d]) - Log[d + d*Tan[e + f*x] + Sqrt[2]*Sqrt[d]*Sqrt[d*Tan[ 
e + f*x]]]/(2*Sqrt[2]*Sqrt[d]))/2))/2 - (d*(d*Tan[e + f*x])^(3/2))/(2*(1 + 
 Tan[e + f*x]^2)))/f
 

3.3.51.3.1 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 252
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

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 3087
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S 
ymbol] :> Simp[1/f   Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + 
f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n - 1) 
/2] && LtQ[0, n, m - 1])
 
3.3.51.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(550\) vs. \(2(169)=338\).

Time = 2.48 (sec) , antiderivative size = 551, normalized size of antiderivative = 2.45

method result size
default \(\frac {\left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right ) \left (4 \sqrt {2}\, \cos \left (f x +e \right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+4 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-3 \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )-2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )+3 \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )\right ) \sqrt {d \tan \left (f x +e \right )}\, d^{2} \sqrt {2}}{16 f \left (\cos \left (f x +e \right )-1\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) \(551\)

input
int(cos(f*x+e)^2*(d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 
output
1/16/f*sin(f*x+e)^2*cos(f*x+e)*(4*2^(1/2)*cos(f*x+e)*(-sin(f*x+e)*cos(f*x+ 
e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+4*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(c 
os(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-3*ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(f*x+e) 
-2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot( 
f*x+e)+csc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+c 
sc(f*x+e)+2)/(cos(f*x+e)-1))+3*ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(f*x+e)+2*s 
in(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+ 
e)+csc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+csc(f 
*x+e)+2)/(cos(f*x+e)-1))-6*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f* 
x+e)+1)^2)^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1))-6*arctan((2^(1/2 
)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1) 
/(cos(f*x+e)-1)))*(d*tan(f*x+e))^(1/2)*d^2/(cos(f*x+e)-1)/(cos(f*x+e)+1)^2 
/(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*2^(1/2)
 
3.3.51.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 969, normalized size of antiderivative = 4.31 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]

input
integrate(cos(f*x+e)^2*(d*tan(f*x+e))^(5/2),x, algorithm="fricas")
 
output
-1/32*(16*d^2*sqrt(d*sin(f*x + e)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) 
- 3*I*(-d^10/f^4)^(1/4)*f*log(27/2*d^8*cos(f*x + e)*sin(f*x + e) + 27/4*(2 
*d^3*f^2*cos(f*x + e)^2 - d^3*f^2)*sqrt(-d^10/f^4) - 27/2*(I*(-d^10/f^4)^( 
1/4)*d^5*f*cos(f*x + e)*sin(f*x + e) + I*(-d^10/f^4)^(3/4)*f^3*cos(f*x + e 
)^2)*sqrt(d*sin(f*x + e)/cos(f*x + e))) + 3*I*(-d^10/f^4)^(1/4)*f*log(27/2 
*d^8*cos(f*x + e)*sin(f*x + e) + 27/4*(2*d^3*f^2*cos(f*x + e)^2 - d^3*f^2) 
*sqrt(-d^10/f^4) - 27/2*(-I*(-d^10/f^4)^(1/4)*d^5*f*cos(f*x + e)*sin(f*x + 
 e) - I*(-d^10/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(d*sin(f*x + e)/cos(f*x 
+ e))) + 3*(-d^10/f^4)^(1/4)*f*log(27/2*d^8*cos(f*x + e)*sin(f*x + e) - 27 
/4*(2*d^3*f^2*cos(f*x + e)^2 - d^3*f^2)*sqrt(-d^10/f^4) + 27/2*((-d^10/f^4 
)^(1/4)*d^5*f*cos(f*x + e)*sin(f*x + e) - (-d^10/f^4)^(3/4)*f^3*cos(f*x + 
e)^2)*sqrt(d*sin(f*x + e)/cos(f*x + e))) - 3*(-d^10/f^4)^(1/4)*f*log(27/2* 
d^8*cos(f*x + e)*sin(f*x + e) - 27/4*(2*d^3*f^2*cos(f*x + e)^2 - d^3*f^2)* 
sqrt(-d^10/f^4) - 27/2*((-d^10/f^4)^(1/4)*d^5*f*cos(f*x + e)*sin(f*x + e) 
- (-d^10/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(d*sin(f*x + e)/cos(f*x + e))) 
 + 3*(-d^10/f^4)^(1/4)*f*log(27*d^8 + 54*((-d^10/f^4)^(1/4)*d^5*f*cos(f*x 
+ e)^2 - (-d^10/f^4)^(3/4)*f^3*cos(f*x + e)*sin(f*x + e))*sqrt(d*sin(f*x + 
 e)/cos(f*x + e))) - 3*(-d^10/f^4)^(1/4)*f*log(27*d^8 - 54*((-d^10/f^4)^(1 
/4)*d^5*f*cos(f*x + e)^2 - (-d^10/f^4)^(3/4)*f^3*cos(f*x + e)*sin(f*x + e) 
)*sqrt(d*sin(f*x + e)/cos(f*x + e))) - 3*I*(-d^10/f^4)^(1/4)*f*log(27*d...
 
3.3.51.6 Sympy [F(-1)]

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

input
integrate(cos(f*x+e)**2*(d*tan(f*x+e))**(5/2),x)
 
output
Timed out
 
3.3.51.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.86 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=\frac {3 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {8 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d^{4}}{d^{2} \tan \left (f x + e\right )^{2} + d^{2}}}{16 \, d f} \]

input
integrate(cos(f*x+e)^2*(d*tan(f*x+e))^(5/2),x, algorithm="maxima")
 
output
1/16*(3*d^4*(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)) - 8*(d*tan(f* 
x + e))^(3/2)*d^4/(d^2*tan(f*x + e)^2 + d^2))/(d*f)
 
3.3.51.8 Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.03 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {1}{16} \, {\left (\frac {8 \, \sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right )}{{\left (d^{2} \tan \left (f x + e\right )^{2} + d^{2}\right )} f} - \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d f} - \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d f} + \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d f} - \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d f}\right )} d^{2} \]

input
integrate(cos(f*x+e)^2*(d*tan(f*x+e))^(5/2),x, algorithm="giac")
 
output
-1/16*(8*sqrt(d*tan(f*x + e))*d^2*tan(f*x + e)/((d^2*tan(f*x + e)^2 + d^2) 
*f) - 6*sqrt(2)*abs(d)^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2* 
sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/(d*f) - 6*sqrt(2)*abs(d)^(3/2)*arctan( 
-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d))) 
/(d*f) + 3*sqrt(2)*abs(d)^(3/2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f* 
x + e))*sqrt(abs(d)) + abs(d))/(d*f) - 3*sqrt(2)*abs(d)^(3/2)*log(d*tan(f* 
x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/(d*f))*d^2
 
3.3.51.9 Mupad [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \]

input
int(cos(e + f*x)^2*(d*tan(e + f*x))^(5/2),x)
 
output
int(cos(e + f*x)^2*(d*tan(e + f*x))^(5/2), x)