3.67 \(\int \frac{1}{(a \sec ^4(x))^{5/2}} \, dx\)

Optimal. Leaf size=132 \[ \frac{63 x \sec ^2(x)}{256 a^2 \sqrt{a \sec ^4(x)}}+\frac{63 \tan (x)}{256 a^2 \sqrt{a \sec ^4(x)}}+\frac{\sin (x) \cos ^7(x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{9 \sin (x) \cos ^5(x)}{80 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \sin (x) \cos ^3(x)}{160 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \sin (x) \cos (x)}{128 a^2 \sqrt{a \sec ^4(x)}} \]

[Out]

(63*x*Sec[x]^2)/(256*a^2*Sqrt[a*Sec[x]^4]) + (21*Cos[x]*Sin[x])/(128*a^2*Sqrt[a*Sec[x]^4]) + (21*Cos[x]^3*Sin[
x])/(160*a^2*Sqrt[a*Sec[x]^4]) + (9*Cos[x]^5*Sin[x])/(80*a^2*Sqrt[a*Sec[x]^4]) + (Cos[x]^7*Sin[x])/(10*a^2*Sqr
t[a*Sec[x]^4]) + (63*Tan[x])/(256*a^2*Sqrt[a*Sec[x]^4])

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Rubi [A]  time = 0.0508577, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 2635, 8} \[ \frac{63 x \sec ^2(x)}{256 a^2 \sqrt{a \sec ^4(x)}}+\frac{63 \tan (x)}{256 a^2 \sqrt{a \sec ^4(x)}}+\frac{\sin (x) \cos ^7(x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{9 \sin (x) \cos ^5(x)}{80 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \sin (x) \cos ^3(x)}{160 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \sin (x) \cos (x)}{128 a^2 \sqrt{a \sec ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sec[x]^4)^(-5/2),x]

[Out]

(63*x*Sec[x]^2)/(256*a^2*Sqrt[a*Sec[x]^4]) + (21*Cos[x]*Sin[x])/(128*a^2*Sqrt[a*Sec[x]^4]) + (21*Cos[x]^3*Sin[
x])/(160*a^2*Sqrt[a*Sec[x]^4]) + (9*Cos[x]^5*Sin[x])/(80*a^2*Sqrt[a*Sec[x]^4]) + (Cos[x]^7*Sin[x])/(10*a^2*Sqr
t[a*Sec[x]^4]) + (63*Tan[x])/(256*a^2*Sqrt[a*Sec[x]^4])

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx &=\frac{\sec ^2(x) \int \cos ^{10}(x) \, dx}{a^2 \sqrt{a \sec ^4(x)}}\\ &=\frac{\cos ^7(x) \sin (x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{\left (9 \sec ^2(x)\right ) \int \cos ^8(x) \, dx}{10 a^2 \sqrt{a \sec ^4(x)}}\\ &=\frac{9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt{a \sec ^4(x)}}+\frac{\cos ^7(x) \sin (x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{\left (63 \sec ^2(x)\right ) \int \cos ^6(x) \, dx}{80 a^2 \sqrt{a \sec ^4(x)}}\\ &=\frac{21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt{a \sec ^4(x)}}+\frac{9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt{a \sec ^4(x)}}+\frac{\cos ^7(x) \sin (x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{\left (21 \sec ^2(x)\right ) \int \cos ^4(x) \, dx}{32 a^2 \sqrt{a \sec ^4(x)}}\\ &=\frac{21 \cos (x) \sin (x)}{128 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt{a \sec ^4(x)}}+\frac{9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt{a \sec ^4(x)}}+\frac{\cos ^7(x) \sin (x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{\left (63 \sec ^2(x)\right ) \int \cos ^2(x) \, dx}{128 a^2 \sqrt{a \sec ^4(x)}}\\ &=\frac{21 \cos (x) \sin (x)}{128 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt{a \sec ^4(x)}}+\frac{9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt{a \sec ^4(x)}}+\frac{\cos ^7(x) \sin (x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{63 \tan (x)}{256 a^2 \sqrt{a \sec ^4(x)}}+\frac{\left (63 \sec ^2(x)\right ) \int 1 \, dx}{256 a^2 \sqrt{a \sec ^4(x)}}\\ &=\frac{63 x \sec ^2(x)}{256 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \cos (x) \sin (x)}{128 a^2 \sqrt{a \sec ^4(x)}}+\frac{21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt{a \sec ^4(x)}}+\frac{9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt{a \sec ^4(x)}}+\frac{\cos ^7(x) \sin (x)}{10 a^2 \sqrt{a \sec ^4(x)}}+\frac{63 \tan (x)}{256 a^2 \sqrt{a \sec ^4(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0824578, size = 55, normalized size = 0.42 \[ \frac{(2520 x+2100 \sin (2 x)+600 \sin (4 x)+150 \sin (6 x)+25 \sin (8 x)+2 \sin (10 x)) \cos ^2(x) \sqrt{a \sec ^4(x)}}{10240 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sec[x]^4)^(-5/2),x]

[Out]

(Cos[x]^2*Sqrt[a*Sec[x]^4]*(2520*x + 2100*Sin[2*x] + 600*Sin[4*x] + 150*Sin[6*x] + 25*Sin[8*x] + 2*Sin[10*x]))
/(10240*a^3)

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Maple [A]  time = 0.221, size = 57, normalized size = 0.4 \begin{align*}{\frac{128\, \left ( \cos \left ( x \right ) \right ) ^{9}\sin \left ( x \right ) +144\, \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) +168\, \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) +210\, \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) +315\,\cos \left ( x \right ) \sin \left ( x \right ) +315\,x}{1280\, \left ( \cos \left ( x \right ) \right ) ^{10}} \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{4}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sec(x)^4)^(5/2),x)

[Out]

1/1280*(128*cos(x)^9*sin(x)+144*cos(x)^7*sin(x)+168*cos(x)^5*sin(x)+210*cos(x)^3*sin(x)+315*cos(x)*sin(x)+315*
x)/cos(x)^10/(a/cos(x)^4)^(5/2)

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Maxima [A]  time = 1.71142, size = 119, normalized size = 0.9 \begin{align*} \frac{315 \, \tan \left (x\right )^{9} + 1470 \, \tan \left (x\right )^{7} + 2688 \, \tan \left (x\right )^{5} + 2370 \, \tan \left (x\right )^{3} + 965 \, \tan \left (x\right )}{1280 \,{\left (a^{\frac{5}{2}} \tan \left (x\right )^{10} + 5 \, a^{\frac{5}{2}} \tan \left (x\right )^{8} + 10 \, a^{\frac{5}{2}} \tan \left (x\right )^{6} + 10 \, a^{\frac{5}{2}} \tan \left (x\right )^{4} + 5 \, a^{\frac{5}{2}} \tan \left (x\right )^{2} + a^{\frac{5}{2}}\right )}} + \frac{63 \, x}{256 \, a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^4)^(5/2),x, algorithm="maxima")

[Out]

1/1280*(315*tan(x)^9 + 1470*tan(x)^7 + 2688*tan(x)^5 + 2370*tan(x)^3 + 965*tan(x))/(a^(5/2)*tan(x)^10 + 5*a^(5
/2)*tan(x)^8 + 10*a^(5/2)*tan(x)^6 + 10*a^(5/2)*tan(x)^4 + 5*a^(5/2)*tan(x)^2 + a^(5/2)) + 63/256*x/a^(5/2)

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Fricas [A]  time = 1.45152, size = 177, normalized size = 1.34 \begin{align*} \frac{{\left (315 \, x \cos \left (x\right )^{2} +{\left (128 \, \cos \left (x\right )^{11} + 144 \, \cos \left (x\right )^{9} + 168 \, \cos \left (x\right )^{7} + 210 \, \cos \left (x\right )^{5} + 315 \, \cos \left (x\right )^{3}\right )} \sin \left (x\right )\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4}}}}{1280 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^4)^(5/2),x, algorithm="fricas")

[Out]

1/1280*(315*x*cos(x)^2 + (128*cos(x)^11 + 144*cos(x)^9 + 168*cos(x)^7 + 210*cos(x)^5 + 315*cos(x)^3)*sin(x))*s
qrt(a/cos(x)^4)/a^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sec ^{4}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)**4)**(5/2),x)

[Out]

Integral((a*sec(x)**4)**(-5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^4)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError