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

Optimal. Leaf size=98 \[ \frac{10 b^2 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{21 d}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b d}+\frac{10 b \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 d} \]

[Out]

(10*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (10*b*(b*Sec[c + d*x])^(3/
2)*Sin[c + d*x])/(21*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b*d)

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Rubi [A]  time = 0.0588525, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3768, 3771, 2641} \[ \frac{10 b^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 d}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b d}+\frac{10 b \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (10*b*(b*Sec[c + d*x])^(3/
2)*Sin[c + d*x])/(21*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b*d)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (b \sec (c+d x))^{5/2} \, dx &=\frac{\int (b \sec (c+d x))^{9/2} \, dx}{b^2}\\ &=\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b d}+\frac{5}{7} \int (b \sec (c+d x))^{5/2} \, dx\\ &=\frac{10 b (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b d}+\frac{1}{21} \left (5 b^2\right ) \int \sqrt{b \sec (c+d x)} \, dx\\ &=\frac{10 b (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b d}+\frac{1}{21} \left (5 b^2 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{10 b^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 d}+\frac{10 b (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b d}\\ \end{align*}

Mathematica [A]  time = 0.179666, size = 61, normalized size = 0.62 \[ \frac{(b \sec (c+d x))^{5/2} \left (10 \cos ^{\frac{5}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+5 \sin (2 (c+d x))+6 \tan (c+d x)\right )}{21 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*Sec[c + d*x])^(5/2)*(10*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 5*Sin[2*(c + d*x)] + 6*Tan[c + d*x]
))/(21*d)

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Maple [C]  time = 0.191, size = 152, normalized size = 1.6 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{21\,d\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( 5\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-3\,\cos \left ( dx+c \right ) +3 \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21/d*(cos(d*x+c)+1)^2*(-1+cos(d*x+c))*(5*I*cos(d*x+c)^3*sin(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(co
s(d*x+c)+1))^(1/2)*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)-5*cos(d*x+c)^3+5*cos(d*x+c)^2-3*cos(d*x+c)+3)*(b/
cos(d*x+c))^(5/2)/cos(d*x+c)/sin(d*x+c)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sec \left (d x + c\right )} b^{2} \sec \left (d x + c\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c))*b^2*sec(d*x + c)^4, x)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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