3.3 \(\int \sec ^3(a+b x) \, dx\)

Optimal. Leaf size=34 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {\tan (a+b x) \sec (a+b x)}{2 b} \]

[Out]

1/2*arctanh(sin(b*x+a))/b+1/2*sec(b*x+a)*tan(b*x+a)/b

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3770} \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {\tan (a+b x) \sec (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^3,x]

[Out]

ArcTanh[Sin[a + b*x]]/(2*b) + (Sec[a + b*x]*Tan[a + b*x])/(2*b)

Rule 3768

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \sec ^3(a+b x) \, dx &=\frac {\sec (a+b x) \tan (a+b x)}{2 b}+\frac {1}{2} \int \sec (a+b x) \, dx\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {\sec (a+b x) \tan (a+b x)}{2 b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {\tan (a+b x) \sec (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^3,x]

[Out]

ArcTanh[Sin[a + b*x]]/(2*b) + (Sec[a + b*x]*Tan[a + b*x])/(2*b)

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fricas [B]  time = 0.81, size = 61, normalized size = 1.79 \[ \frac {\cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) - \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, \sin \left (b x + a\right )}{4 \, b \cos \left (b x + a\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.94, size = 48, normalized size = 1.41 \[ -\frac {\frac {2 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} - \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^3,x, algorithm="giac")

[Out]

-1/4*(2*sin(b*x + a)/(sin(b*x + a)^2 - 1) - log(abs(sin(b*x + a) + 1)) + log(abs(sin(b*x + a) - 1)))/b

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maple [A]  time = 0.56, size = 38, normalized size = 1.12 \[ \frac {\sec \left (b x +a \right ) \tan \left (b x +a \right )}{2 b}+\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^3,x)

[Out]

1/2*sec(b*x+a)*tan(b*x+a)/b+1/2/b*ln(sec(b*x+a)+tan(b*x+a))

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maxima [A]  time = 0.47, size = 46, normalized size = 1.35 \[ -\frac {\frac {2 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} - \log \left (\sin \left (b x + a\right ) + 1\right ) + \log \left (\sin \left (b x + a\right ) - 1\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.11, size = 36, normalized size = 1.06 \[ \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{2\,b}-\frac {\sin \left (a+b\,x\right )}{2\,b\,\left ({\sin \left (a+b\,x\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(a + b*x)^3,x)

[Out]

atanh(sin(a + b*x))/(2*b) - sin(a + b*x)/(2*b*(sin(a + b*x)^2 - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^{3}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**3,x)

[Out]

Integral(sec(a + b*x)**3, x)

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