\(\int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx\) [165]

Optimal result

Integrand size = 23, antiderivative size = 290 \[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^3 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a+a \sec (c+d x))^{15/2}} \]

[Out]

-2*a^(5/2)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+2*a^3*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-2/3*a
^4*tan(d*x+c)^3/d/(a+a*sec(d*x+c))^(3/2)+2/5*a^5*tan(d*x+c)^5/d/(a+a*sec(d*x+c))^(5/2)+62/7*a^6*tan(d*x+c)^7/d
/(a+a*sec(d*x+c))^(7/2)+98/9*a^7*tan(d*x+c)^9/d/(a+a*sec(d*x+c))^(9/2)+62/11*a^8*tan(d*x+c)^11/d/(a+a*sec(d*x+
c))^(11/2)+18/13*a^9*tan(d*x+c)^13/d/(a+a*sec(d*x+c))^(13/2)+2/15*a^10*tan(d*x+c)^15/d/(a+a*sec(d*x+c))^(15/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3972, 472, 209} \[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac {2 a^3 \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]

[In]

Int[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^6,x]

[Out]

(-2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*Tan[c + d*x])/(d*Sqrt[a + a*Se
c[c + d*x]]) - (2*a^4*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^5*Tan[c + d*x]^5)/(5*d*(a + a*Se
c[c + d*x])^(5/2)) + (62*a^6*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (98*a^7*Tan[c + d*x]^9)/(9*d*(
a + a*Sec[c + d*x])^(9/2)) + (62*a^8*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2)) + (18*a^9*Tan[c + d*x
]^13)/(13*d*(a + a*Sec[c + d*x])^(13/2)) + (2*a^10*Tan[c + d*x]^15)/(15*d*(a + a*Sec[c + d*x])^(15/2))

Rule 209

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

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 3972

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[-2*(a^(m/2 +
n + 1/2)/d), Subst[Int[x^m*((2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2 a^6\right ) \text {Subst}\left (\int \frac {x^6 \left (2+a x^2\right )^5}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {\left (2 a^6\right ) \text {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}+31 x^6+49 a x^8+31 a^2 x^{10}+9 a^3 x^{12}+a^4 x^{14}-\frac {1}{a^3 \left (1+a x^2\right )}\right ) \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = \frac {2 a^3 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a+a \sec (c+d x))^{15/2}}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^3 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a+a \sec (c+d x))^{15/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.60 \[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=\frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^7(c+d x) \sqrt {a (1+\sec (c+d x))} \left (-2882880 \sqrt {2} \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^{\frac {15}{2}}(c+d x)+604890 \sin \left (\frac {1}{2} (c+d x)\right )-87230 \sin \left (\frac {3}{2} (c+d x)\right )+450450 \sin \left (\frac {5}{2} (c+d x)\right )-137670 \sin \left (\frac {7}{2} (c+d x)\right )+210210 \sin \left (\frac {9}{2} (c+d x)\right )+75450 \sin \left (\frac {11}{2} (c+d x)\right )+90090 \sin \left (\frac {13}{2} (c+d x)\right )+16066 \sin \left (\frac {15}{2} (c+d x)\right )\right )}{2882880 d} \]

[In]

Integrate[(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^6,x]

[Out]

(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^7*Sqrt[a*(1 + Sec[c + d*x])]*(-2882880*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)
/2]]*Cos[c + d*x]^(15/2) + 604890*Sin[(c + d*x)/2] - 87230*Sin[(3*(c + d*x))/2] + 450450*Sin[(5*(c + d*x))/2]
- 137670*Sin[(7*(c + d*x))/2] + 210210*Sin[(9*(c + d*x))/2] + 75450*Sin[(11*(c + d*x))/2] + 90090*Sin[(13*(c +
d*x))/2] + 16066*Sin[(15*(c + d*x))/2]))/(2882880*d)

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.92

\[-\frac {2 a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (45045 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+45045 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-16066 \sin \left (d x +c \right )-53078 \tan \left (d x +c \right )-17286 \sec \left (d x +c \right ) \tan \left (d x +c \right )+30640 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}+26810 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}-2898 \sec \left (d x +c \right )^{4} \tan \left (d x +c \right )-10164 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{5}-3003 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{6}\right )}{45045 d \left (\cos \left (d x +c \right )+1\right )}\]

[In]

int((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^6,x)

[Out]

-2/45045/d*a^2*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*(45045*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d
*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)+45045*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a
rctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-16066*sin(d*x+c)-53078*tan(d*x+c)-17286*s
ec(d*x+c)*tan(d*x+c)+30640*tan(d*x+c)*sec(d*x+c)^2+26810*tan(d*x+c)*sec(d*x+c)^3-2898*sec(d*x+c)^4*tan(d*x+c)-
10164*tan(d*x+c)*sec(d*x+c)^5-3003*tan(d*x+c)*sec(d*x+c)^6)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.64 \[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=\left [\frac {45045 \, {\left (a^{2} \cos \left (d x + c\right )^{8} + a^{2} \cos \left (d x + c\right )^{7}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (16066 \, a^{2} \cos \left (d x + c\right )^{7} + 53078 \, a^{2} \cos \left (d x + c\right )^{6} + 17286 \, a^{2} \cos \left (d x + c\right )^{5} - 30640 \, a^{2} \cos \left (d x + c\right )^{4} - 26810 \, a^{2} \cos \left (d x + c\right )^{3} + 2898 \, a^{2} \cos \left (d x + c\right )^{2} + 10164 \, a^{2} \cos \left (d x + c\right ) + 3003 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{8} + d \cos \left (d x + c\right )^{7}\right )}}, \frac {2 \, {\left (45045 \, {\left (a^{2} \cos \left (d x + c\right )^{8} + a^{2} \cos \left (d x + c\right )^{7}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (16066 \, a^{2} \cos \left (d x + c\right )^{7} + 53078 \, a^{2} \cos \left (d x + c\right )^{6} + 17286 \, a^{2} \cos \left (d x + c\right )^{5} - 30640 \, a^{2} \cos \left (d x + c\right )^{4} - 26810 \, a^{2} \cos \left (d x + c\right )^{3} + 2898 \, a^{2} \cos \left (d x + c\right )^{2} + 10164 \, a^{2} \cos \left (d x + c\right ) + 3003 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{45045 \, {\left (d \cos \left (d x + c\right )^{8} + d \cos \left (d x + c\right )^{7}\right )}}\right ] \]

[In]

integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^6,x, algorithm="fricas")

[Out]

[1/45045*(45045*(a^2*cos(d*x + c)^8 + a^2*cos(d*x + c)^7)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((
a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(160
66*a^2*cos(d*x + c)^7 + 53078*a^2*cos(d*x + c)^6 + 17286*a^2*cos(d*x + c)^5 - 30640*a^2*cos(d*x + c)^4 - 26810
*a^2*cos(d*x + c)^3 + 2898*a^2*cos(d*x + c)^2 + 10164*a^2*cos(d*x + c) + 3003*a^2)*sqrt((a*cos(d*x + c) + a)/c
os(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^8 + d*cos(d*x + c)^7), 2/45045*(45045*(a^2*cos(d*x + c)^8 + a^2*cos
(d*x + c)^7)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (16
066*a^2*cos(d*x + c)^7 + 53078*a^2*cos(d*x + c)^6 + 17286*a^2*cos(d*x + c)^5 - 30640*a^2*cos(d*x + c)^4 - 2681
0*a^2*cos(d*x + c)^3 + 2898*a^2*cos(d*x + c)^2 + 10164*a^2*cos(d*x + c) + 3003*a^2)*sqrt((a*cos(d*x + c) + a)/
cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^8 + d*cos(d*x + c)^7)]

Sympy [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))**(5/2)*tan(d*x+c)**6,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^6,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \tan \left (d x + c\right )^{6} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^6,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]

[In]

int(tan(c + d*x)^6*(a + a/cos(c + d*x))^(5/2),x)

[Out]

int(tan(c + d*x)^6*(a + a/cos(c + d*x))^(5/2), x)