3.4.33 \(\int (-\cos (x)+\sec (x))^{7/2} \, dx\) [333]

3.4.33.1 Optimal result

Integrand size = 11, antiderivative size = 73 \[ \int (-\cos (x)+\sec (x))^{7/2} \, dx=-\frac {256}{35} \csc (x) \sqrt {\sin (x) \tan (x)}+\frac {64}{35} \sec (x) \tan (x) \sqrt {\sin (x) \tan (x)}-\frac {8}{7} \sin (x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {2}{7} \sin ^3(x) \tan ^2(x) \sqrt {\sin (x) \tan (x)} \]

output

-256/35*csc(x)*(sin(x)*tan(x))^(1/2)+64/35*sec(x)*(sin(x)*tan(x))^(1/2)*ta 
n(x)-8/7*sin(x)*(sin(x)*tan(x))^(1/2)*tan(x)^2-2/7*sin(x)^3*(sin(x)*tan(x) 
)^(1/2)*tan(x)^2
 

3.4.33.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51 \[ \int (-\cos (x)+\sec (x))^{7/2} \, dx=\frac {1}{70} \sec (x) \sqrt {\sin (x) \tan (x)} (-512 \cot (x)-5 \cos (x) (-23 \sin (x)+\sin (3 x))+28 \tan (x)) \]

input

Integrate[(-Cos[x] + Sec[x])^(7/2),x]
 

output

(Sec[x]*Sqrt[Sin[x]*Tan[x]]*(-512*Cot[x] - 5*Cos[x]*(-23*Sin[x] + Sin[3*x] 
) + 28*Tan[x]))/70
 

3.4.33.3 Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {3042, 4897, 3042, 4900, 3042, 3078, 3042, 3078, 3042, 3074, 3042, 3069}

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.

\(\displaystyle \int (\sec (x)-\cos (x))^{7/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (\sec (x)-\cos (x))^{7/2}dx\)

\(\Big \downarrow \) 4897

\(\displaystyle \int (\sin (x) \tan (x))^{7/2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (\sin (x) \tan (x))^{7/2}dx\)

\(\Big \downarrow \) 4900

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \int \sin ^{\frac {7}{2}}(x) \tan ^{\frac {7}{2}}(x)dx}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \int \sin (x)^{7/2} \tan (x)^{7/2}dx}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3078

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {12}{7} \int \sin ^{\frac {3}{2}}(x) \tan ^{\frac {7}{2}}(x)dx-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {12}{7} \int \sin (x)^{3/2} \tan (x)^{7/2}dx-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3078

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {12}{7} \left (\frac {8}{3} \int \frac {\tan ^{\frac {7}{2}}(x)}{\sqrt {\sin (x)}}dx-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {12}{7} \left (\frac {8}{3} \int \frac {\tan (x)^{7/2}}{\sqrt {\sin (x)}}dx-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3074

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {12}{7} \left (\frac {8}{3} \left (\frac {2 \tan ^{\frac {5}{2}}(x)}{5 \sqrt {\sin (x)}}-\frac {4}{5} \int \frac {\tan ^{\frac {3}{2}}(x)}{\sqrt {\sin (x)}}dx\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {12}{7} \left (\frac {8}{3} \left (\frac {2 \tan ^{\frac {5}{2}}(x)}{5 \sqrt {\sin (x)}}-\frac {4}{5} \int \frac {\tan (x)^{3/2}}{\sqrt {\sin (x)}}dx\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

\(\Big \downarrow \) 3069

\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {12}{7} \left (\frac {8}{3} \left (\frac {2 \tan ^{\frac {5}{2}}(x)}{5 \sqrt {\sin (x)}}-\frac {8 \sqrt {\tan (x)}}{5 \sqrt {\sin (x)}}\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\)

input

Int[(-Cos[x] + Sec[x])^(7/2),x]
 

output

(Sqrt[Sin[x]*Tan[x]]*((-2*Sin[x]^(7/2)*Tan[x]^(5/2))/7 + (12*((-2*Sin[x]^( 
3/2)*Tan[x]^(5/2))/3 + (8*((-8*Sqrt[Tan[x]])/(5*Sqrt[Sin[x]]) + (2*Tan[x]^ 
(5/2))/(5*Sqrt[Sin[x]])))/3))/7))/(Sqrt[Sin[x]]*Sqrt[Tan[x]])
 

3.4.33.3.1 Defintions of rubi rules used

rule 3042

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

rule 3069

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f* 
m)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]
 

rule 3074

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[b*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(n 
 - 1))), x] - Simp[b^2*((m + n - 1)/(n - 1))   Int[(a*Sin[e + f*x])^m*(b*Ta 
n[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && In 
tegersQ[2*m, 2*n] &&  !(GtQ[m, 1] &&  !IntegerQ[(m - 1)/2])
 

rule 3078

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/( 
f*m)), x] + Simp[a^2*((m + n - 1)/m)   Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[ 
e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1 
] && EqQ[n, 1/2])) && IntegersQ[2*m, 2*n]
 

rule 4897

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
 

rule 4900

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri 
g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar 
t[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p]))   Int[uu*vv^(m*p)*ww^(n*p), x] 
, x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  ! 
InertTrigFreeQ[w])
 

3.4.33.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(285\) vs. \(2(57)=114\).

Time = 11.32 (sec) , antiderivative size = 286, normalized size of antiderivative = 3.92

method	result	size
default	\(\frac {\sec \left (x \right )^{2} \csc \left (x \right ) \left (20 \cos \left (x \right )^{6}+105 \ln \left (\frac {4 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \cos \left (x \right )+4 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2}{\cos \left (x \right )+1}\right ) \cos \left (x \right )^{3} \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-105 \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right ) \cos \left (x \right )^{3} \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-140 \cos \left (x \right )^{4}+105 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \ln \left (\frac {4 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \cos \left (x \right )+4 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2}{\cos \left (x \right )+1}\right ) \cos \left (x \right )^{2}-105 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right ) \cos \left (x \right )^{2}-420 \cos \left (x \right )^{2}+28\right ) \sqrt {\sin \left (x \right ) \tan \left (x \right )}}{70}\)	\(286\)

input

int((-cos(x)+sec(x))^(7/2),x,method=_RETURNVERBOSE)
 

output

1/70*sec(x)^2*csc(x)*(20*cos(x)^6+105*ln(2*(2*cos(x)*(-cos(x)/(cos(x)+1)^2 
)^(1/2)+2*(-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)+1))*cos(x)^3*(-co 
s(x)/(cos(x)+1)^2)^(1/2)-105*ln((2*cos(x)*(-cos(x)/(cos(x)+1)^2)^(1/2)+2*( 
-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)+1))*cos(x)^3*(-cos(x)/(cos(x 
)+1)^2)^(1/2)-140*cos(x)^4+105*(-cos(x)/(cos(x)+1)^2)^(1/2)*ln(2*(2*cos(x) 
*(-cos(x)/(cos(x)+1)^2)^(1/2)+2*(-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(co 
s(x)+1))*cos(x)^2-105*(-cos(x)/(cos(x)+1)^2)^(1/2)*ln((2*cos(x)*(-cos(x)/( 
cos(x)+1)^2)^(1/2)+2*(-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)+1))*co 
s(x)^2-420*cos(x)^2+28)*(sin(x)*tan(x))^(1/2)
 

3.4.33.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60 \[ \int (-\cos (x)+\sec (x))^{7/2} \, dx=\frac {2 \, {\left (5 \, \cos \left (x\right )^{6} - 35 \, \cos \left (x\right )^{4} - 105 \, \cos \left (x\right )^{2} + 7\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{35 \, \cos \left (x\right )^{2} \sin \left (x\right )} \]

input

integrate((-cos(x)+sec(x))^(7/2),x, algorithm="fricas")
 

output

2/35*(5*cos(x)^6 - 35*cos(x)^4 - 105*cos(x)^2 + 7)*sqrt(-(cos(x)^2 - 1)/co 
s(x))/(cos(x)^2*sin(x))
 

3.4.33.6 Sympy [F(-1)]

Timed out. \[ \int (-\cos (x)+\sec (x))^{7/2} \, dx=\text {Timed out} \]

input

integrate((-cos(x)+sec(x))**(7/2),x)
 

output

Timed out
 

3.4.33.7 Maxima [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12 \[ \int (-\cos (x)+\sec (x))^{7/2} \, dx=\frac {128 \, {\left (\frac {7 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {7 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac {2 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} - 2\right )}}{35 \, {\left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \]

input

integrate((-cos(x)+sec(x))^(7/2),x, algorithm="maxima")
 

output

128/35*(7*sin(x)^4/(cos(x) + 1)^4 - 7*sin(x)^10/(cos(x) + 1)^10 + 2*sin(x) 
^14/(cos(x) + 1)^14 - 2)/((sin(x)/(cos(x) + 1) + 1)^(7/2)*(-sin(x)/(cos(x) 
 + 1) + 1)^(7/2)*(sin(x)^2/(cos(x) + 1)^2 + 1)^(7/2))
 

3.4.33.8 Giac [F]

\[ \int (-\cos (x)+\sec (x))^{7/2} \, dx=\int { {\left (-\cos \left (x\right ) + \sec \left (x\right )\right )}^{\frac {7}{2}} \,d x } \]

input

integrate((-cos(x)+sec(x))^(7/2),x, algorithm="giac")
 

output

integrate((-cos(x) + sec(x))^(7/2), x)
 

3.4.33.9 Mupad [F(-1)]

Timed out. \[ \int (-\cos (x)+\sec (x))^{7/2} \, dx=\int {\left (\frac {1}{\cos \left (x\right )}-\cos \left (x\right )\right )}^{7/2} \,d x \]

input

int((1/cos(x) - cos(x))^(7/2),x)
 

output

int((1/cos(x) - cos(x))^(7/2), x)