3.1.41 \(\int \sec ^2(x)^{3/2} \, dx\) [41]

Optimal. Leaf size=22 \[ \frac {1}{2} \sinh ^{-1}(\tan (x))+\frac {1}{2} \sqrt {\sec ^2(x)} \tan (x) \]

[Out]

1/2*arcsinh(tan(x))+1/2*(sec(x)^2)^(1/2)*tan(x)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4207, 201, 221} \begin {gather*} \frac {1}{2} \tan (x) \sqrt {\sec ^2(x)}+\frac {1}{2} \sinh ^{-1}(\tan (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[Tan[x]]/2 + (Sqrt[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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(22)=44\).
time = 0.07, size = 52, normalized size = 2.36 \begin {gather*} \frac {1}{2} \cos (x) \sqrt {\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[(Sec[x]^2)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(16)=32\).
time = 0.20, size = 55, normalized size = 2.50

method	result	size
default	\(-\frac {\left (\left (\cos ^{2}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1+\sin \left (x \right )}{\sin \left (x \right )}\right )-\left (\cos ^{2}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1-\sin \left (x \right )}{\sin \left (x \right )}\right )-\sin \left (x \right )\right ) \cos \left (x \right ) \sqrt {2}\, \sqrt {\frac {1}{\cos \left (2 x \right )+1}}}{\cos \left (2 x \right )+1}\)	\(55\)
risch	\(-\frac {i \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}-\sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )+\sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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)*(1/cos(x)^2
)^(3/2)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 18, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, \sqrt {\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + \frac {1}{2} \, \operatorname {arsinh}\left (\tan \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(tan(x)^2 + 1)*tan(x) + 1/2*arcsinh(tan(x))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).
time = 2.74, size = 34, normalized size = 1.55 \begin {gather*} -\frac {\cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, \sin \left (x\right )}{4 \, \cos \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((sec(x)**2)**(3/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (16) = 32\).
time = 0.43, size = 44, normalized size = 2.00 \begin {gather*} \frac {\log \left (\sin \left (x\right ) + 1\right )}{4 \, \mathrm {sgn}\left (\cos \left (x\right )\right )} - \frac {\log \left (-\sin \left (x\right ) + 1\right )}{4 \, \mathrm {sgn}\left (\cos \left (x\right )\right )} - \frac {\sin \left (x\right )}{2 \, {\left (\sin \left (x\right )^{2} - 1\right )} \mathrm {sgn}\left (\cos \left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int {\left (\frac {1}{{\cos \left (x\right )}^2}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cos(x)^2)^(3/2),x)

[Out]

int((1/cos(x)^2)^(3/2), x)

________________________________________________________________________________________