3.100 \(\int (b \sec (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=98 \[ \frac{6 b^3 \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}-\frac{6 b^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d} \]

[Out]

(-6*b^4*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (6*b^3*Sqrt[b*Sec[c + d*x]]
*Sin[c + d*x])/(5*d) + (2*b*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)

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Rubi [A]  time = 0.0484491, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3771, 2639} \[ \frac{6 b^3 \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}-\frac{6 b^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[c + d*x])^(7/2),x]

[Out]

(-6*b^4*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (6*b^3*Sqrt[b*Sec[c + d*x]]
*Sin[c + d*x])/(5*d) + (2*b*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)

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 2639

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

Rubi steps

\begin{align*} \int (b \sec (c+d x))^{7/2} \, dx &=\frac{2 b (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{1}{5} \left (3 b^2\right ) \int (b \sec (c+d x))^{3/2} \, dx\\ &=\frac{6 b^3 \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac{1}{5} \left (3 b^4\right ) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx\\ &=\frac{6 b^3 \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac{\left (3 b^4\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=-\frac{6 b^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{6 b^3 \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.0847728, size = 62, normalized size = 0.63 \[ \frac{b (b \sec (c+d x))^{5/2} \left (7 \sin (c+d x)+3 \sin (3 (c+d x))-12 \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{10 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sec[c + d*x])^(7/2),x]

[Out]

(b*(b*Sec[c + d*x])^(5/2)*(-12*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 7*Sin[c + d*x] + 3*Sin[3*(c + d*
x)]))/(10*d)

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Maple [C]  time = 0.222, size = 354, normalized size = 3.6 \begin{align*} -{\frac{2\, \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 3\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -3\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \left ( \cos \left ( dx+c \right ) \right ) ^{3}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) +3\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -3\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) +3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1 \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(d*x+c))^(7/2),x)

[Out]

-2/5/d*(-1+cos(d*x+c))^2*(3*I*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)*cos(d*x+c)^3*(1/(cos(d*x+c)+1))^(1/2)*
(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-3*I*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+c),I)*cos(d*x+c)^3*(1/(co
s(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+3*I*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),I)*
(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^2*sin(d*x+c)-3*I*EllipticE(I*(-1+cos(d*x
+c))/sin(d*x+c),I)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^2*sin(d*x+c)+3*cos(d*
x+c)^3-2*cos(d*x+c)^2-1)*cos(d*x+c)*(cos(d*x+c)+1)^2*(b/cos(d*x+c))^(7/2)/sin(d*x+c)^5

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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{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c))^(7/2), 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^{3} \sec \left (d x + c\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c))*b^3*sec(d*x + c)^3, 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((b*sec(d*x+c))**(7/2),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{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c))^(7/2), x)