3.1.49 \(\int (a \sec ^2(x))^{3/2} \, dx\) [49]

Optimal. Leaf size=46 \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {1}{2} a \sqrt {a \sec ^2(x)} \tan (x) \]

[Out]

1/2*a^(3/2)*arctanh(a^(1/2)*tan(x)/(a*sec(x)^2)^(1/2))+1/2*a*(a*sec(x)^2)^(1/2)*tan(x)

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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {1}{2} a \tan (x) \sqrt {a \sec ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Sec[x]^2)^(3/2),x]

[Out]

(a^(3/2)*ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[a*Sec[x]^2]])/2 + (a*Sqrt[a*Sec[x]^2]*Tan[x])/2

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 )^{3/2} \, dx &=a \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tan (x)}{\sqrt {a \sec ^2(x)}}\right )\\ &=\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {1}{2} a \sqrt {a \sec ^2(x)} \tan (x)\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 55, normalized size = 1.20 \begin {gather*} \frac {1}{2} a \cos (x) \sqrt {a \sec ^2(x)} \left (-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\sec (x) \tan (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Sec[x]^2)^(3/2),x]

[Out]

(a*Cos[x]*Sqrt[a*Sec[x]^2]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]] + Sec[x]*Tan[x]))/2

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Maple [A]
time = 0.22, size = 55, normalized size = 1.20

method	result	size
default	\(\frac {\left (\left (\cos ^{2}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1-\sin \left (x \right )}{\sin \left (x \right )}\right )-\left (\cos ^{2}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1+\sin \left (x \right )}{\sin \left (x \right )}\right )+\sin \left (x \right )\right ) \cos \left (x \right ) \left (\frac {a}{\cos \left (x \right )^{2}}\right )^{\frac {3}{2}}}{2}\)	\(55\)
risch	\(-\frac {i a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}+a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )-a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sec(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(cos(x)^2*ln(-(cos(x)-1-sin(x))/sin(x))-cos(x)^2*ln(-(cos(x)-1+sin(x))/sin(x))+sin(x))*cos(x)*(a/cos(x)^2)
^(3/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (34) = 68\).
time = 0.54, size = 324, normalized size = 7.04 \begin {gather*} -\frac {{\left (8 \, a \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, a \cos \left (x\right ) \sin \left (2 \, x\right ) + 8 \, a \cos \left (2 \, x\right ) \sin \left (x\right ) - 4 \, {\left (a \sin \left (3 \, x\right ) - a \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) + 4 \, {\left (a \cos \left (3 \, x\right ) - a \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 4 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \sin \left (3 \, x\right ) + 4 \, a \sin \left (x\right )\right )} \sqrt {a}}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/4*(8*a*cos(3*x)*sin(2*x) - 8*a*cos(x)*sin(2*x) + 8*a*cos(2*x)*sin(x) - 4*(a*sin(3*x) - a*sin(x))*cos(4*x) -
(a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*sin(4*x)^2 + 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2 + 2*(2*a*cos(2*x) + a)
*cos(4*x) + 4*a*cos(2*x) + a)*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + (a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*sin
(4*x)^2 + 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2 + 2*(2*a*cos(2*x) + a)*cos(4*x) + 4*a*cos(2*x) + a)*log(cos(x
)^2 + sin(x)^2 - 2*sin(x) + 1) + 4*(a*cos(3*x) - a*cos(x))*sin(4*x) - 4*(2*a*cos(2*x) + a)*sin(3*x) + 4*a*sin(
x))*sqrt(a)/(2*(2*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 4*cos(2*x)^2 + sin(4*x)^2 + 4*sin(4*x)*sin(2*x) + 4*si
n(2*x)^2 + 4*cos(2*x) + 1)

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Fricas [A]
time = 3.23, size = 39, normalized size = 0.85 \begin {gather*} -\frac {{\left (a \cos \left (x\right )^{2} \log \left (-\frac {\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right ) - 2 \, a \sin \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{2}}}}{4 \, \cos \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)^2)^(3/2),x, algorithm="fricas")

[Out]

-1/4*(a*cos(x)^2*log(-(sin(x) - 1)/(sin(x) + 1)) - 2*a*sin(x))*sqrt(a/cos(x)^2)/cos(x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \sec ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)**2)**(3/2),x)

[Out]

Integral((a*sec(x)**2)**(3/2), x)

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Giac [A]
time = 0.43, size = 42, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, {\left (\log \left (\sin \left (x\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (x\right )\right ) - \log \left (-\sin \left (x\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (x\right )\right ) - \frac {2 \, \mathrm {sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right )}{\sin \left (x\right )^{2} - 1}\right )} a^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/4*(log(sin(x) + 1)*sgn(cos(x)) - log(-sin(x) + 1)*sgn(cos(x)) - 2*sgn(cos(x))*sin(x)/(sin(x)^2 - 1))*a^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (\frac {a}{{\cos \left (x\right )}^2}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cos(x)^2)^(3/2),x)

[Out]

int((a/cos(x)^2)^(3/2), x)

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