3.109 \(\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=116 \[ \frac{64 a^3 \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]

[Out]

(64*a^3*Tan[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(21*d)
+ (2*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)

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Rubi [A]  time = 0.155143, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3798, 3793, 3792} \[ \frac{64 a^3 \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(64*a^3*Tan[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(21*d)
+ (2*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)

Rule 3798

Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(Cot[e + f*x]*(a
+ b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*m)/(b*(m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x
] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 3793

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(b*Cot[e + f*x]*(a
+ b*Csc[e + f*x])^(m - 1))/(f*m), x] + Dist[(a*(2*m - 1))/m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x
], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && IntegerQ[2*m]

Rule 3792

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*b*Cot[e + f*x])/
(f*Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx &=\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{5}{7} \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac{2 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{1}{7} (8 a) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac{2 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{1}{21} \left (32 a^2\right ) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{64 a^3 \tan (c+d x)}{21 d \sqrt{a+a \sec (c+d x)}}+\frac{16 a^2 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac{2 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.165702, size = 60, normalized size = 0.52 \[ \frac{2 a^3 \tan (c+d x) \left (3 \sec ^3(c+d x)+12 \sec ^2(c+d x)+23 \sec (c+d x)+46\right )}{21 d \sqrt{a (\sec (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*(46 + 23*Sec[c + d*x] + 12*Sec[c + d*x]^2 + 3*Sec[c + d*x]^3)*Tan[c + d*x])/(21*d*Sqrt[a*(1 + Sec[c + d
*x])])

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Maple [A]  time = 0.135, size = 85, normalized size = 0.7 \begin{align*} -{\frac{2\,{a}^{2} \left ( 46\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-23\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-11\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-9\,\cos \left ( dx+c \right ) -3 \right ) }{21\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21/d*a^2*(46*cos(d*x+c)^4-23*cos(d*x+c)^3-11*cos(d*x+c)^2-9*cos(d*x+c)-3)*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/
2)/cos(d*x+c)^3/sin(d*x+c)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 1.95026, size = 236, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (46 \, a^{2} \cos \left (d x + c\right )^{3} + 23 \, a^{2} \cos \left (d x + c\right )^{2} + 12 \, a^{2} \cos \left (d x + c\right ) + 3 \, 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 )}{21 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(46*a^2*cos(d*x + c)^3 + 23*a^2*cos(d*x + c)^2 + 12*a^2*cos(d*x + c) + 3*a^2)*sqrt((a*cos(d*x + c) + a)/c
os(d*x + c))*sin(d*x + c)/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 5.42586, size = 204, normalized size = 1.76 \begin{align*} -\frac{8 \,{\left (21 \, \sqrt{2} a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (35 \, \sqrt{2} a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 4 \,{\left (2 \, \sqrt{2} a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7 \, \sqrt{2} a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{21 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/21*(21*sqrt(2)*a^6*sgn(cos(d*x + c)) - (35*sqrt(2)*a^6*sgn(cos(d*x + c)) + 4*(2*sqrt(2)*a^6*sgn(cos(d*x + c
))*tan(1/2*d*x + 1/2*c)^2 - 7*sqrt(2)*a^6*sgn(cos(d*x + c)))*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*t
an(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - a)^3*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*d)