3.1.47 \(\int (a \sec ^2(x))^{7/2} \, dx\) [47]

Optimal. Leaf size=84 \[ \frac {5}{16} a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {5}{16} a^3 \sqrt {a \sec ^2(x)} \tan (x)+\frac {5}{24} a^2 \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac {1}{6} a \left (a \sec ^2(x)\right )^{5/2} \tan (x) \]

[Out]

5/16*a^(7/2)*arctanh(a^(1/2)*tan(x)/(a*sec(x)^2)^(1/2))+5/24*a^2*(a*sec(x)^2)^(3/2)*tan(x)+1/6*a*(a*sec(x)^2)^
(5/2)*tan(x)+5/16*a^3*(a*sec(x)^2)^(1/2)*tan(x)

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Rubi [A]
time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223, 212} \begin {gather*} \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 \tan (x) \sqrt {a \sec ^2(x)}+\frac {5}{24} a^2 \tan (x) \left (a \sec ^2(x)\right )^{3/2}+\frac {1}{6} a \tan (x) \left (a \sec ^2(x)\right )^{5/2} \end {gather*}

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 201

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 4207

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
]

Rubi steps

\begin {align*} \int \left (a \sec ^2(x)\right )^{7/2} \, dx &=a \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.15, size = 78, normalized size = 0.93 \begin {gather*} \frac {1}{96} \cos ^7(x) \left (a \sec ^2(x)\right )^{7/2} \left (-30 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+30 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {1}{8} \sec ^6(x) (198 \sin (x)+85 \sin (3 x)+15 \sin (5 x))\right ) \end {gather*}

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 = 0.36, size = 74, normalized size = 0.88

method	result	size
default	\(\frac {\left (15 \left (\cos ^{6}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1-\sin \left (x \right )}{\sin \left (x \right )}\right )-15 \left (\cos ^{6}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1+\sin \left (x \right )}{\sin \left (x \right )}\right )+15 \left (\cos ^{4}\left (x \right )\right ) \sin \left (x \right )+10 \left (\cos ^{2}\left (x \right )\right ) \sin \left (x \right )+8 \sin \left (x \right )\right ) \cos \left (x \right ) \left (\frac {a}{\cos \left (x \right )^{2}}\right )^{\frac {7}{2}}}{48}\)	\(74\)
risch	\(-\frac {i a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left (15 \,{\mathrm e}^{10 i x}+85 \,{\mathrm e}^{8 i x}+198 \,{\mathrm e}^{6 i x}-198 \,{\mathrm e}^{4 i x}-85 \,{\mathrm e}^{2 i x}-15\right )}{24 \left ({\mathrm e}^{2 i x}+1\right )^{5}}+\frac {5 a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )}{8}-\frac {5 a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )}{8}\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sec(x)^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/48*(15*cos(x)^6*ln(-(cos(x)-1-sin(x))/sin(x))-15*cos(x)^6*ln(-(cos(x)-1+sin(x))/sin(x))+15*cos(x)^4*sin(x)+1
0*cos(x)^2*sin(x)+8*sin(x))*cos(x)*(a/cos(x)^2)^(7/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2175 vs. \(2 (64) = 128\).
time = 4.49, size = 2175, normalized size = 25.89 \begin {gather*} \text {Too large to display} \end {gather*}

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)

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Fricas [A]
time = 3.56, size = 65, normalized size = 0.77 \begin {gather*} -\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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)**2)**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3061 deep

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Giac [A]
time = 0.45, size = 79, normalized size = 0.94 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {a}{{\cos \left (x\right )}^2}\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cos(x)^2)^(7/2),x)

[Out]

int((a/cos(x)^2)^(7/2), x)

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