3.135 \(\int \sqrt{a+a \sec (c+d x)} \tan ^5(c+d x) \, dx\)

Optimal. Leaf size=147 \[ \frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d} \]

[Out]

(-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c +
d*x])^(3/2))/(3*a*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^2*d) - (6*(a + a*Sec[c + d*x])^(7/2))/(7*a^3*d) + (
2*(a + a*Sec[c + d*x])^(9/2))/(9*a^4*d)

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Rubi [A]  time = 0.114971, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3880, 88, 50, 63, 207} \[ \frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac{6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}+\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^5,x]

[Out]

(-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c +
d*x])^(3/2))/(3*a*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^2*d) - (6*(a + a*Sec[c + d*x])^(7/2))/(7*a^3*d) + (
2*(a + a*Sec[c + d*x])^(9/2))/(9*a^4*d)

Rule 3880

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

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

Rubi steps

\begin{align*} \int \sqrt{a+a \sec (c+d x)} \tan ^5(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-a+a x)^2 (a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-3 a^2 (a+a x)^{5/2}+\frac{a^2 (a+a x)^{5/2}}{x}+a (a+a x)^{7/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac{6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac{6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac{6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{2 \sqrt{a+a \sec (c+d x)}}{d}+\frac{2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac{6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{2 \sqrt{a+a \sec (c+d x)}}{d}+\frac{2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac{6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{d}\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{d}+\frac{2 \sqrt{a+a \sec (c+d x)}}{d}+\frac{2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac{6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}\\ \end{align*}

Mathematica [A]  time = 0.55141, size = 102, normalized size = 0.69 \[ \frac{2 \sqrt{a (\sec (c+d x)+1)} \left (\sqrt{\sec (c+d x)+1} \left (35 \sec ^4(c+d x)+5 \sec ^3(c+d x)-132 \sec ^2(c+d x)-34 \sec (c+d x)+383\right )-315 \tanh ^{-1}\left (\sqrt{\sec (c+d x)+1}\right )\right )}{315 d \sqrt{\sec (c+d x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^5,x]

[Out]

(2*Sqrt[a*(1 + Sec[c + d*x])]*(-315*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(383 - 34*Sec[c +
d*x] - 132*Sec[c + d*x]^2 + 5*Sec[c + d*x]^3 + 35*Sec[c + d*x]^4)))/(315*d*Sqrt[1 + Sec[c + d*x]])

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Maple [B]  time = 0.268, size = 359, normalized size = 2.4 \begin{align*}{\frac{1}{5040\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 315\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +1260\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +1890\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +1260\,\sqrt{2}\cos \left ( dx+c \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +315\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}+12256\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-1088\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-4224\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+160\,\cos \left ( dx+c \right ) +1120 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^5,x)

[Out]

1/5040/d*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(315*2^(1/2)*cos(d*x+c)^4*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*ar
ctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1260*2^(1/2)*cos(d*x+c)^3*(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(9/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1890*2^(1/2)*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos
(d*x+c)+1))^(9/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1260*2^(1/2)*cos(d*x+c)*(-2*cos(d*x
+c)/(cos(d*x+c)+1))^(9/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+315*2^(1/2)*arctan(1/2*2^(1
/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)+12256*cos(d*x+c)^4-1088*cos(d*x
+c)^3-4224*cos(d*x+c)^2+160*cos(d*x+c)+1120)/cos(d*x+c)^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.42903, size = 810, normalized size = 5.51 \begin{align*} \left [\frac{315 \, \sqrt{a} \cos \left (d x + c\right )^{4} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \,{\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \,{\left (383 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{630 \, d \cos \left (d x + c\right )^{4}}, \frac{315 \, \sqrt{-a} \arctan \left (\frac{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 )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{4} + 2 \,{\left (383 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^5,x, algorithm="fricas")

[Out]

[1/630*(315*sqrt(a)*cos(d*x + c)^4*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt(
(a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(383*cos(d*x + c)^4 - 34*cos(d*x + c)^3 - 132*c
os(d*x + c)^2 + 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4), 1/315*(315*s
qrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x
+ c)^4 + 2*(383*cos(d*x + c)^4 - 34*cos(d*x + c)^3 - 132*cos(d*x + c)^2 + 5*cos(d*x + c) + 35)*sqrt((a*cos(d*
x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \tan ^{5}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**(1/2)*tan(d*x+c)**5,x)

[Out]

Integral(sqrt(a*(sec(c + d*x) + 1))*tan(c + d*x)**5, x)

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Giac [A]  time = 5.0463, size = 261, normalized size = 1.78 \begin{align*} \frac{\sqrt{2}{\left (\frac{315 \, \sqrt{2} a \arctan \left (\frac{\sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \,{\left (315 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{4} a - 210 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} a^{2} + 252 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} a^{3} + 1080 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a^{4} + 560 \, a^{5}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}\right )} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^5,x, algorithm="giac")

[Out]

1/315*sqrt(2)*(315*sqrt(2)*a*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/sqrt(-a) + 2*(31
5*(a*tan(1/2*d*x + 1/2*c)^2 - a)^4*a - 210*(a*tan(1/2*d*x + 1/2*c)^2 - a)^3*a^2 + 252*(a*tan(1/2*d*x + 1/2*c)^
2 - a)^2*a^3 + 1080*(a*tan(1/2*d*x + 1/2*c)^2 - a)*a^4 + 560*a^5)/((a*tan(1/2*d*x + 1/2*c)^2 - a)^4*sqrt(-a*ta
n(1/2*d*x + 1/2*c)^2 + a)))*sgn(cos(d*x + c))/d