3.14 \(\int \sec ^m(c+d x) (-\frac{C m}{1+m}+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=26 \[ \frac{C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (m+1)} \]

[Out]

(C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m))

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Rubi [A]  time = 0.0395476, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {4043} \[ \frac{C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^m*(-((C*m)/(1 + m)) + C*Sec[c + d*x]^2),x]

[Out]

(C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m))

Rule 4043

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[C*m + A*(m + 1), 0]

Rubi steps

\begin{align*} \int \sec ^m(c+d x) \left (-\frac{C m}{1+m}+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.314314, size = 26, normalized size = 1. \[ \frac{C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^m*(-((C*m)/(1 + m)) + C*Sec[c + d*x]^2),x]

[Out]

(C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m))

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Maple [C]  time = 0.385, size = 512, normalized size = 19.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^m*(-C*m/(1+m)+C*sec(d*x+c)^2),x)

[Out]

-I*C/(1+m)/d/(exp(2*I*(d*x+c))+1)*(2^m*exp(I*(Re(d*x)+Re(c)))^m/((exp(2*I*(d*x+c))+1)^m)*exp(-m*Im(d*x)-m*Im(c
))*exp(-1/2*I*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3*m)*exp(1/2*I*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*
(d*x+c))+1))^2*csgn(I*exp(I*(d*x+c)))*m)*exp(1/2*I*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*csgn(I/(ex
p(2*I*(d*x+c))+1))*m)*exp(-1/2*I*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(I/
(exp(2*I*(d*x+c))+1))*m)*exp(2*I*d*x)*exp(2*I*c)-2^m*exp(I*(Re(d*x)+Re(c)))^m/((exp(2*I*(d*x+c))+1)^m)*exp(-1/
2*m*(I*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-I*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*csg
n(I*exp(I*(d*x+c)))-I*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*csgn(I/(exp(2*I*(d*x+c))+1))+I*Pi*csgn(
I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(I/(exp(2*I*(d*x+c))+1))+2*Im(d*x)+2*Im(c)))
)

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Maxima [B]  time = 1.99644, size = 410, normalized size = 15.77 \begin{align*} -\frac{2^{m} C \cos \left (-{\left (d x + c\right )}{\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2^{m} C \cos \left (-{\left (d x + c\right )} m + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) \sin \left (2 \, d x + 2 \, c\right ) +{\left (2^{m} C \cos \left (2 \, d x + 2 \, c\right ) + 2^{m} C\right )} \sin \left (-{\left (d x + c\right )}{\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) -{\left (2^{m} C \cos \left (2 \, d x + 2 \, c\right ) + 2^{m} C\right )} \sin \left (-{\left (d x + c\right )} m + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )}{{\left ({\left (m + 1\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} +{\left (m + 1\right )} \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \,{\left (m + 1\right )} \cos \left (2 \, d x + 2 \, c\right ) + m + 1\right )}{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac{1}{2} \, m} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^m*(-C*m/(1+m)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

-(2^m*C*cos(-(d*x + c)*(m + 2) + m*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))*sin(2*d*x + 2*c) - 2^m*C*c
os(-(d*x + c)*m + m*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))*sin(2*d*x + 2*c) + (2^m*C*cos(2*d*x + 2*c
) + 2^m*C)*sin(-(d*x + c)*(m + 2) + m*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - (2^m*C*cos(2*d*x + 2*
c) + 2^m*C)*sin(-(d*x + c)*m + m*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(((m + 1)*cos(2*d*x + 2*c)^
2 + (m + 1)*sin(2*d*x + 2*c)^2 + 2*(m + 1)*cos(2*d*x + 2*c) + m + 1)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2
+ 2*cos(2*d*x + 2*c) + 1)^(1/2*m)*d)

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Fricas [A]  time = 0.494877, size = 81, normalized size = 3.12 \begin{align*} \frac{C \frac{1}{\cos \left (d x + c\right )}^{m} \sin \left (d x + c\right )}{{\left (d m + d\right )} \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^m*(-C*m/(1+m)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

C*(1/cos(d*x + c))^m*sin(d*x + c)/((d*m + d)*cos(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{C \left (\int - m \sec ^{m}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )} \sec ^{m}{\left (c + d x \right )}\, dx + \int m \sec ^{2}{\left (c + d x \right )} \sec ^{m}{\left (c + d x \right )}\, dx\right )}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**m*(-C*m/(1+m)+C*sec(d*x+c)**2),x)

[Out]

C*(Integral(-m*sec(c + d*x)**m, x) + Integral(sec(c + d*x)**2*sec(c + d*x)**m, x) + Integral(m*sec(c + d*x)**2
*sec(c + d*x)**m, x))/(m + 1)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} - \frac{C m}{m + 1}\right )} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^m*(-C*m/(1+m)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 - C*m/(m + 1))*sec(d*x + c)^m, x)