3.41 \(\int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx\)

Optimal. Leaf size=156 \[ \frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{93 a^5 \tan (c+d x) \sec (c+d x)}{16 d} \]

[Out]

(93*a^5*ArcTanh[Sin[c + d*x]])/(16*d) + (16*a^5*Tan[c + d*x])/d + (93*a^5*Sec[c + d*x]*Tan[c + d*x])/(16*d) +
(85*a^5*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (5*a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (28*a^5*Tan[c + d*x]
^3)/(3*d) + (13*a^5*Tan[c + d*x]^5)/(5*d) + (a^5*Tan[c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.197865, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3791, 3768, 3770, 3767} \[ \frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{93 a^5 \tan (c+d x) \sec (c+d x)}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^5,x]

[Out]

(93*a^5*ArcTanh[Sin[c + d*x]])/(16*d) + (16*a^5*Tan[c + d*x])/d + (93*a^5*Sec[c + d*x]*Tan[c + d*x])/(16*d) +
(85*a^5*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (5*a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (28*a^5*Tan[c + d*x]
^3)/(3*d) + (13*a^5*Tan[c + d*x]^5)/(5*d) + (a^5*Tan[c + d*x]^7)/(7*d)

Rule 3791

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
 && IGtQ[m, 0] && RationalQ[n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
 1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx &=\int \left (a^5 \sec ^3(c+d x)+5 a^5 \sec ^4(c+d x)+10 a^5 \sec ^5(c+d x)+10 a^5 \sec ^6(c+d x)+5 a^5 \sec ^7(c+d x)+a^5 \sec ^8(c+d x)\right ) \, dx\\ &=a^5 \int \sec ^3(c+d x) \, dx+a^5 \int \sec ^8(c+d x) \, dx+\left (5 a^5\right ) \int \sec ^4(c+d x) \, dx+\left (5 a^5\right ) \int \sec ^7(c+d x) \, dx+\left (10 a^5\right ) \int \sec ^5(c+d x) \, dx+\left (10 a^5\right ) \int \sec ^6(c+d x) \, dx\\ &=\frac{a^5 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{5 a^5 \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{2} a^5 \int \sec (c+d x) \, dx+\frac{1}{6} \left (25 a^5\right ) \int \sec ^5(c+d x) \, dx+\frac{1}{2} \left (15 a^5\right ) \int \sec ^3(c+d x) \, dx-\frac{a^5 \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac{\left (5 a^5\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac{\left (10 a^5\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{17 a^5 \sec (c+d x) \tan (c+d x)}{4 d}+\frac{85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{1}{8} \left (25 a^5\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{4} \left (15 a^5\right ) \int \sec (c+d x) \, dx\\ &=\frac{17 a^5 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{1}{16} \left (25 a^5\right ) \int \sec (c+d x) \, dx\\ &=\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{a^5 \tan ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 1.34252, size = 229, normalized size = 1.47 \[ -\frac{a^5 (\cos (c+d x)+1)^5 \sec ^{10}\left (\frac{1}{2} (c+d x)\right ) \sec ^7(c+d x) \left (624960 \cos ^7(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-162400 \sin (2 c+d x)+118825 \sin (c+2 d x)+118825 \sin (3 c+2 d x)+305088 \sin (2 c+3 d x)-16800 \sin (4 c+3 d x)+62860 \sin (3 c+4 d x)+62860 \sin (5 c+4 d x)+107296 \sin (4 c+5 d x)+9765 \sin (5 c+6 d x)+9765 \sin (7 c+6 d x)+15328 \sin (6 c+7 d x)+374080 \sin (d x))\right )}{3440640 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^5,x]

[Out]

-(a^5*(1 + Cos[c + d*x])^5*Sec[(c + d*x)/2]^10*Sec[c + d*x]^7*(624960*Cos[c + d*x]^7*(Log[Cos[(c + d*x)/2] - S
in[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(374080*Sin[d*x] - 162400*Sin[2*c + d*x]
 + 118825*Sin[c + 2*d*x] + 118825*Sin[3*c + 2*d*x] + 305088*Sin[2*c + 3*d*x] - 16800*Sin[4*c + 3*d*x] + 62860*
Sin[3*c + 4*d*x] + 62860*Sin[5*c + 4*d*x] + 107296*Sin[4*c + 5*d*x] + 9765*Sin[5*c + 6*d*x] + 9765*Sin[7*c + 6
*d*x] + 15328*Sin[6*c + 7*d*x])))/(3440640*d)

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Maple [A]  time = 0.046, size = 168, normalized size = 1.1 \begin{align*}{\frac{93\,{a}^{5}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{93\,{a}^{5}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{958\,{a}^{5}\tan \left ( dx+c \right ) }{105\,d}}+{\frac{479\,{a}^{5}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{105\,d}}+{\frac{85\,{a}^{5} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{24\,d}}+{\frac{76\,{a}^{5}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35\,d}}+{\frac{5\,{a}^{5} \left ( \sec \left ( dx+c \right ) \right ) ^{5}\tan \left ( dx+c \right ) }{6\,d}}+{\frac{{a}^{5}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(a+a*sec(d*x+c))^5,x)

[Out]

93/16*a^5*sec(d*x+c)*tan(d*x+c)/d+93/16/d*a^5*ln(sec(d*x+c)+tan(d*x+c))+958/105*a^5*tan(d*x+c)/d+479/105/d*a^5
*tan(d*x+c)*sec(d*x+c)^2+85/24*a^5*sec(d*x+c)^3*tan(d*x+c)/d+76/35/d*a^5*tan(d*x+c)*sec(d*x+c)^4+5/6*a^5*sec(d
*x+c)^5*tan(d*x+c)/d+1/7/d*a^5*tan(d*x+c)*sec(d*x+c)^6

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Maxima [B]  time = 1.12852, size = 424, normalized size = 2.72 \begin{align*} \frac{96 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{5} + 2240 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{5} + 5600 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{5} - 175 \, a^{5}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2100 \, a^{5}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{5}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{3360 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^5,x, algorithm="maxima")

[Out]

1/3360*(96*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*a^5 + 2240*(3*tan(d*x
+ c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*a^5 + 5600*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^5 - 175*a^5*(2*(1
5*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2
- 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 2100*a^5*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/
(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 840*a^5*(2*sin(
d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)))/d

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Fricas [A]  time = 1.80474, size = 413, normalized size = 2.65 \begin{align*} \frac{9765 \, a^{5} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9765 \, a^{5} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (15328 \, a^{5} \cos \left (d x + c\right )^{6} + 9765 \, a^{5} \cos \left (d x + c\right )^{5} + 7664 \, a^{5} \cos \left (d x + c\right )^{4} + 5950 \, a^{5} \cos \left (d x + c\right )^{3} + 3648 \, a^{5} \cos \left (d x + c\right )^{2} + 1400 \, a^{5} \cos \left (d x + c\right ) + 240 \, a^{5}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^5,x, algorithm="fricas")

[Out]

1/3360*(9765*a^5*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 9765*a^5*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 2*(15
328*a^5*cos(d*x + c)^6 + 9765*a^5*cos(d*x + c)^5 + 7664*a^5*cos(d*x + c)^4 + 5950*a^5*cos(d*x + c)^3 + 3648*a^
5*cos(d*x + c)^2 + 1400*a^5*cos(d*x + c) + 240*a^5)*sin(d*x + c))/(d*cos(d*x + c)^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{5} \left (\int \sec ^{3}{\left (c + d x \right )}\, dx + \int 5 \sec ^{4}{\left (c + d x \right )}\, dx + \int 10 \sec ^{5}{\left (c + d x \right )}\, dx + \int 10 \sec ^{6}{\left (c + d x \right )}\, dx + \int 5 \sec ^{7}{\left (c + d x \right )}\, dx + \int \sec ^{8}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3*(a+a*sec(d*x+c))**5,x)

[Out]

a**5*(Integral(sec(c + d*x)**3, x) + Integral(5*sec(c + d*x)**4, x) + Integral(10*sec(c + d*x)**5, x) + Integr
al(10*sec(c + d*x)**6, x) + Integral(5*sec(c + d*x)**7, x) + Integral(sec(c + d*x)**8, x))

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Giac [A]  time = 1.47336, size = 230, normalized size = 1.47 \begin{align*} \frac{9765 \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 9765 \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (9765 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 65100 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 184233 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 285696 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 260183 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 132020 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 43995 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{7}}}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^5,x, algorithm="giac")

[Out]

1/1680*(9765*a^5*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 9765*a^5*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(9765*a^
5*tan(1/2*d*x + 1/2*c)^13 - 65100*a^5*tan(1/2*d*x + 1/2*c)^11 + 184233*a^5*tan(1/2*d*x + 1/2*c)^9 - 285696*a^5
*tan(1/2*d*x + 1/2*c)^7 + 260183*a^5*tan(1/2*d*x + 1/2*c)^5 - 132020*a^5*tan(1/2*d*x + 1/2*c)^3 + 43995*a^5*ta
n(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d