3.47 \(\int (a \sec ^2(x))^{7/2} \, dx\)
Optimal. Leaf size=84 \[ \frac{5}{16} a^3 \tan (x) \sqrt{a \sec ^2(x)}+\frac{5}{24} a^2 \tan (x) \left (a \sec ^2(x)\right )^{3/2}+\frac{5}{16} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a \sec ^2(x)}}\right )+\frac{1}{6} a \tan (x) \left (a \sec ^2(x)\right )^{5/2} \]
[Out]
(5*a^(7/2)*ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[a*Sec[x]^2]])/16 + (5*a^3*Sqrt[a*Sec[x]^2]*Tan[x])/16 + (5*a^2*(a*Sec
[x]^2)^(3/2)*Tan[x])/24 + (a*(a*Sec[x]^2)^(5/2)*Tan[x])/6
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Rubi [A] time = 0.0413025, antiderivative size = 84, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used =
{4122, 195, 217, 206} \[ \frac{5}{16} a^3 \tan (x) \sqrt{a \sec ^2(x)}+\frac{5}{24} a^2 \tan (x) \left (a \sec ^2(x)\right )^{3/2}+\frac{5}{16} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a \sec ^2(x)}}\right )+\frac{1}{6} a \tan (x) \left (a \sec ^2(x)\right )^{5/2} \]
Antiderivative was successfully verified.
[In]
Int[(a*Sec[x]^2)^(7/2),x]
[Out]
(5*a^(7/2)*ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[a*Sec[x]^2]])/16 + (5*a^3*Sqrt[a*Sec[x]^2]*Tan[x])/16 + (5*a^2*(a*Sec
[x]^2)^(3/2)*Tan[x])/24 + (a*(a*Sec[x]^2)^(5/2)*Tan[x])/6
Rule 4122
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p
]
Rule 195
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \left (a \sec ^2(x)\right )^{7/2} \, dx &=a \operatorname{Subst}\left (\int \left (a+a x^2\right )^{5/2} \, dx,x,\tan (x)\right )\\ &=\frac{1}{6} a \left (a \sec ^2(x)\right )^{5/2} \tan (x)+\frac{1}{6} \left (5 a^2\right ) \operatorname{Subst}\left (\int \left (a+a x^2\right )^{3/2} \, dx,x,\tan (x)\right )\\ &=\frac{5}{24} a^2 \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac{1}{6} a \left (a \sec ^2(x)\right )^{5/2} \tan (x)+\frac{1}{8} \left (5 a^3\right ) \operatorname{Subst}\left (\int \sqrt{a+a x^2} \, dx,x,\tan (x)\right )\\ &=\frac{5}{16} a^3 \sqrt{a \sec ^2(x)} \tan (x)+\frac{5}{24} a^2 \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac{1}{6} a \left (a \sec ^2(x)\right )^{5/2} \tan (x)+\frac{1}{16} \left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+a x^2}} \, dx,x,\tan (x)\right )\\ &=\frac{5}{16} a^3 \sqrt{a \sec ^2(x)} \tan (x)+\frac{5}{24} a^2 \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac{1}{6} a \left (a \sec ^2(x)\right )^{5/2} \tan (x)+\frac{1}{16} \left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}\right )\\ &=\frac{5}{16} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a \sec ^2(x)}}\right )+\frac{5}{16} a^3 \sqrt{a \sec ^2(x)} \tan (x)+\frac{5}{24} a^2 \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac{1}{6} a \left (a \sec ^2(x)\right )^{5/2} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.124033, size = 78, normalized size = 0.93 \[ \frac{1}{96} \cos ^7(x) \left (a \sec ^2(x)\right )^{7/2} \left (\frac{1}{8} (198 \sin (x)+85 \sin (3 x)+15 \sin (5 x)) \sec ^6(x)-30 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+30 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(a*Sec[x]^2)^(7/2),x]
[Out]
(Cos[x]^7*(a*Sec[x]^2)^(7/2)*(-30*Log[Cos[x/2] - Sin[x/2]] + 30*Log[Cos[x/2] + Sin[x/2]] + (Sec[x]^6*(198*Sin[
x] + 85*Sin[3*x] + 15*Sin[5*x]))/8))/96
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Maple [A] time = 0.145, size = 74, normalized size = 0.9 \begin{align*}{\frac{\cos \left ( x \right ) }{48} \left ( 15\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) -\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \left ( \cos \left ( x \right ) \right ) ^{6}-15\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \left ( \cos \left ( x \right ) \right ) ^{6}+15\, \left ( \cos \left ( x \right ) \right ) ^{4}\sin \left ( x \right ) +10\, \left ( \cos \left ( x \right ) \right ) ^{2}\sin \left ( x \right ) +8\,\sin \left ( x \right ) \right ) \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*sec(x)^2)^(7/2),x)
[Out]
1/48*(15*ln(-(-1+cos(x)-sin(x))/sin(x))*cos(x)^6-15*ln(-(-1+cos(x)+sin(x))/sin(x))*cos(x)^6+15*cos(x)^4*sin(x)
+10*cos(x)^2*sin(x)+8*sin(x))*cos(x)*(a/cos(x)^2)^(7/2)
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Maxima [B] time = 16.7386, size = 2936, normalized size = 34.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*sec(x)^2)^(7/2),x, algorithm="maxima")
[Out]
1/96*(2040*a^3*cos(3*x)*sin(2*x) + 360*a^3*cos(x)*sin(2*x) - 360*a^3*cos(2*x)*sin(x) - 60*a^3*sin(x) + 4*(15*a
^3*sin(11*x) + 85*a^3*sin(9*x) + 198*a^3*sin(7*x) - 198*a^3*sin(5*x) - 85*a^3*sin(3*x) - 15*a^3*sin(x))*cos(12
*x) - 60*(6*a^3*sin(10*x) + 15*a^3*sin(8*x) + 20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*cos(11*x) +
24*(85*a^3*sin(9*x) + 198*a^3*sin(7*x) - 198*a^3*sin(5*x) - 85*a^3*sin(3*x) - 15*a^3*sin(x))*cos(10*x) - 340*(
15*a^3*sin(8*x) + 20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*cos(9*x) + 60*(198*a^3*sin(7*x) - 198*a^
3*sin(5*x) - 85*a^3*sin(3*x) - 15*a^3*sin(x))*cos(8*x) - 792*(20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a^3*sin(2*
x))*cos(7*x) - 80*(198*a^3*sin(5*x) + 85*a^3*sin(3*x) + 15*a^3*sin(x))*cos(6*x) + 2376*(5*a^3*sin(4*x) + 2*a^3
*sin(2*x))*cos(5*x) - 300*(17*a^3*sin(3*x) + 3*a^3*sin(x))*cos(4*x) + 15*(a^3*cos(12*x)^2 + 36*a^3*cos(10*x)^2
+ 225*a^3*cos(8*x)^2 + 400*a^3*cos(6*x)^2 + 225*a^3*cos(4*x)^2 + 36*a^3*cos(2*x)^2 + a^3*sin(12*x)^2 + 36*a^3
*sin(10*x)^2 + 225*a^3*sin(8*x)^2 + 400*a^3*sin(6*x)^2 + 225*a^3*sin(4*x)^2 + 180*a^3*sin(4*x)*sin(2*x) + 36*a
^3*sin(2*x)^2 + 12*a^3*cos(2*x) + a^3 + 2*(6*a^3*cos(10*x) + 15*a^3*cos(8*x) + 20*a^3*cos(6*x) + 15*a^3*cos(4*
x) + 6*a^3*cos(2*x) + a^3)*cos(12*x) + 12*(15*a^3*cos(8*x) + 20*a^3*cos(6*x) + 15*a^3*cos(4*x) + 6*a^3*cos(2*x
) + a^3)*cos(10*x) + 30*(20*a^3*cos(6*x) + 15*a^3*cos(4*x) + 6*a^3*cos(2*x) + a^3)*cos(8*x) + 40*(15*a^3*cos(4
*x) + 6*a^3*cos(2*x) + a^3)*cos(6*x) + 30*(6*a^3*cos(2*x) + a^3)*cos(4*x) + 2*(6*a^3*sin(10*x) + 15*a^3*sin(8*
x) + 20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(12*x) + 12*(15*a^3*sin(8*x) + 20*a^3*sin(6*x) + 1
5*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(10*x) + 30*(20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(8*x)
+ 120*(5*a^3*sin(4*x) + 2*a^3*sin(2*x))*sin(6*x))*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) - 15*(a^3*cos(12*x)^
2 + 36*a^3*cos(10*x)^2 + 225*a^3*cos(8*x)^2 + 400*a^3*cos(6*x)^2 + 225*a^3*cos(4*x)^2 + 36*a^3*cos(2*x)^2 + a^
3*sin(12*x)^2 + 36*a^3*sin(10*x)^2 + 225*a^3*sin(8*x)^2 + 400*a^3*sin(6*x)^2 + 225*a^3*sin(4*x)^2 + 180*a^3*si
n(4*x)*sin(2*x) + 36*a^3*sin(2*x)^2 + 12*a^3*cos(2*x) + a^3 + 2*(6*a^3*cos(10*x) + 15*a^3*cos(8*x) + 20*a^3*co
s(6*x) + 15*a^3*cos(4*x) + 6*a^3*cos(2*x) + a^3)*cos(12*x) + 12*(15*a^3*cos(8*x) + 20*a^3*cos(6*x) + 15*a^3*co
s(4*x) + 6*a^3*cos(2*x) + a^3)*cos(10*x) + 30*(20*a^3*cos(6*x) + 15*a^3*cos(4*x) + 6*a^3*cos(2*x) + a^3)*cos(8
*x) + 40*(15*a^3*cos(4*x) + 6*a^3*cos(2*x) + a^3)*cos(6*x) + 30*(6*a^3*cos(2*x) + a^3)*cos(4*x) + 2*(6*a^3*sin
(10*x) + 15*a^3*sin(8*x) + 20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(12*x) + 12*(15*a^3*sin(8*x)
+ 20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a^3*sin(2*x))*sin(10*x) + 30*(20*a^3*sin(6*x) + 15*a^3*sin(4*x) + 6*a
^3*sin(2*x))*sin(8*x) + 120*(5*a^3*sin(4*x) + 2*a^3*sin(2*x))*sin(6*x))*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1
) - 4*(15*a^3*cos(11*x) + 85*a^3*cos(9*x) + 198*a^3*cos(7*x) - 198*a^3*cos(5*x) - 85*a^3*cos(3*x) - 15*a^3*cos
(x))*sin(12*x) + 60*(6*a^3*cos(10*x) + 15*a^3*cos(8*x) + 20*a^3*cos(6*x) + 15*a^3*cos(4*x) + 6*a^3*cos(2*x) +
a^3)*sin(11*x) - 24*(85*a^3*cos(9*x) + 198*a^3*cos(7*x) - 198*a^3*cos(5*x) - 85*a^3*cos(3*x) - 15*a^3*cos(x))*
sin(10*x) + 340*(15*a^3*cos(8*x) + 20*a^3*cos(6*x) + 15*a^3*cos(4*x) + 6*a^3*cos(2*x) + a^3)*sin(9*x) - 60*(19
8*a^3*cos(7*x) - 198*a^3*cos(5*x) - 85*a^3*cos(3*x) - 15*a^3*cos(x))*sin(8*x) + 792*(20*a^3*cos(6*x) + 15*a^3*
cos(4*x) + 6*a^3*cos(2*x) + a^3)*sin(7*x) + 80*(198*a^3*cos(5*x) + 85*a^3*cos(3*x) + 15*a^3*cos(x))*sin(6*x) -
792*(15*a^3*cos(4*x) + 6*a^3*cos(2*x) + a^3)*sin(5*x) + 300*(17*a^3*cos(3*x) + 3*a^3*cos(x))*sin(4*x) - 340*(
6*a^3*cos(2*x) + a^3)*sin(3*x))*sqrt(a)/(2*(6*cos(10*x) + 15*cos(8*x) + 20*cos(6*x) + 15*cos(4*x) + 6*cos(2*x)
+ 1)*cos(12*x) + cos(12*x)^2 + 12*(15*cos(8*x) + 20*cos(6*x) + 15*cos(4*x) + 6*cos(2*x) + 1)*cos(10*x) + 36*c
os(10*x)^2 + 30*(20*cos(6*x) + 15*cos(4*x) + 6*cos(2*x) + 1)*cos(8*x) + 225*cos(8*x)^2 + 40*(15*cos(4*x) + 6*c
os(2*x) + 1)*cos(6*x) + 400*cos(6*x)^2 + 30*(6*cos(2*x) + 1)*cos(4*x) + 225*cos(4*x)^2 + 36*cos(2*x)^2 + 2*(6*
sin(10*x) + 15*sin(8*x) + 20*sin(6*x) + 15*sin(4*x) + 6*sin(2*x))*sin(12*x) + sin(12*x)^2 + 12*(15*sin(8*x) +
20*sin(6*x) + 15*sin(4*x) + 6*sin(2*x))*sin(10*x) + 36*sin(10*x)^2 + 30*(20*sin(6*x) + 15*sin(4*x) + 6*sin(2*x
))*sin(8*x) + 225*sin(8*x)^2 + 120*(5*sin(4*x) + 2*sin(2*x))*sin(6*x) + 400*sin(6*x)^2 + 225*sin(4*x)^2 + 180*
sin(4*x)*sin(2*x) + 36*sin(2*x)^2 + 12*cos(2*x) + 1)
________________________________________________________________________________________
Fricas [A] time = 1.47792, size = 186, normalized size = 2.21 \begin{align*} -\frac{{\left (15 \, a^{3} \cos \left (x\right )^{6} \log \left (-\frac{\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right ) - 2 \,{\left (15 \, a^{3} \cos \left (x\right )^{4} + 10 \, a^{3} \cos \left (x\right )^{2} + 8 \, a^{3}\right )} \sin \left (x\right )\right )} \sqrt{\frac{a}{\cos \left (x\right )^{2}}}}{96 \, \cos \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*sec(x)^2)^(7/2),x, algorithm="fricas")
[Out]
-1/96*(15*a^3*cos(x)^6*log(-(sin(x) - 1)/(sin(x) + 1)) - 2*(15*a^3*cos(x)^4 + 10*a^3*cos(x)^2 + 8*a^3)*sin(x))
*sqrt(a/cos(x)^2)/cos(x)^5
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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((a*sec(x)**2)**(7/2),x)
[Out]
Timed out
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Giac [A] time = 1.30087, size = 107, normalized size = 1.27 \begin{align*} \frac{1}{96} \,{\left (15 \, a^{3} \log \left (\sin \left (x\right ) + 1\right ) \mathrm{sgn}\left (\cos \left (x\right )\right ) - 15 \, a^{3} \log \left (-\sin \left (x\right ) + 1\right ) \mathrm{sgn}\left (\cos \left (x\right )\right ) - \frac{2 \,{\left (15 \, a^{3} \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right )^{5} - 40 \, a^{3} \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right )^{3} + 33 \, a^{3} \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right )\right )}}{{\left (\sin \left (x\right )^{2} - 1\right )}^{3}}\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*sec(x)^2)^(7/2),x, algorithm="giac")
[Out]
1/96*(15*a^3*log(sin(x) + 1)*sgn(cos(x)) - 15*a^3*log(-sin(x) + 1)*sgn(cos(x)) - 2*(15*a^3*sgn(cos(x))*sin(x)^
5 - 40*a^3*sgn(cos(x))*sin(x)^3 + 33*a^3*sgn(cos(x))*sin(x))/(sin(x)^2 - 1)^3)*sqrt(a)