3.163 \(\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-2-m} \, dx\)

Optimal. Leaf size=104 \[ \frac{\tan (e+f x) (a \sec (e+f x)+a)^{m+1} (c-c \sec (e+f x))^{-m-2}}{a f \left (4 m^2+8 m+3\right )}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-2}}{f (2 m+1)} \]

[Out]

-(((a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-2 - m)*Tan[e + f*x])/(f*(1 + 2*m))) + ((a + a*Sec[e + f*x])^(
1 + m)*(c - c*Sec[e + f*x])^(-2 - m)*Tan[e + f*x])/(a*f*(3 + 8*m + 4*m^2))

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Rubi [A]  time = 0.224782, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {3951, 3950} \[ \frac{\tan (e+f x) (a \sec (e+f x)+a)^{m+1} (c-c \sec (e+f x))^{-m-2}}{a f \left (4 m^2+8 m+3\right )}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-2}}{f (2 m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-2 - m),x]

[Out]

-(((a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-2 - m)*Tan[e + f*x])/(f*(1 + 2*m))) + ((a + a*Sec[e + f*x])^(
1 + m)*(c - c*Sec[e + f*x])^(-2 - m)*Tan[e + f*x])/(a*f*(3 + 8*m + 4*m^2))

Rule 3951

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
^(n_.), x_Symbol] :> Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] +
Dist[(m + n + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2
*m + 1, 0] &&  !LtQ[n, 0] &&  !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])

Rule 3950

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
^(n_.), x_Symbol] :> Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] /
; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[2*m
+ 1, 0]

Rubi steps

\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-2-m} \, dx &=-\frac{(a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-2-m} \tan (e+f x)}{f (1+2 m)}-\frac{\int \sec (e+f x) (a+a \sec (e+f x))^{1+m} (c-c \sec (e+f x))^{-2-m} \, dx}{a (1+2 m)}\\ &=-\frac{(a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-2-m} \tan (e+f x)}{f (1+2 m)}+\frac{(a+a \sec (e+f x))^{1+m} (c-c \sec (e+f x))^{-2-m} \tan (e+f x)}{a f \left (3+8 m+4 m^2\right )}\\ \end{align*}

Mathematica [C]  time = 3.14231, size = 250, normalized size = 2.4 \[ \frac{i 2^{m+3} \left (1+e^{i (e+f x)}\right ) \left (-i e^{-\frac{1}{2} i (e+f x)} \left (-1+e^{i (e+f x)}\right )\right )^{-2 m} \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{-m} \left (\frac{\left (1+e^{i (e+f x)}\right )^2}{1+e^{2 i (e+f x)}}\right )^m \left ((m+1) e^{2 i (e+f x)}-e^{i (e+f x)}+m+1\right ) \sin ^{2 (m+2)}\left (\frac{1}{2} (e+f x)\right ) \sec ^{m+2}(e+f x) (\sec (e+f x)+1)^{-m} (a (\sec (e+f x)+1))^m (c-c \sec (e+f x))^{-m-2}}{f (2 m+1) (2 m+3) \left (-1+e^{i (e+f x)}\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-2 - m),x]

[Out]

(I*2^(3 + m)*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))^2/(1 + E^((2*I)*(e + f*x))))^m*(1 - E^(I*(e + f*x))
+ m + E^((2*I)*(e + f*x))*(1 + m))*Sec[e + f*x]^(2 + m)*(a*(1 + Sec[e + f*x]))^m*(c - c*Sec[e + f*x])^(-2 - m)
*Sin[(e + f*x)/2]^(2*(2 + m)))/((-1 + E^(I*(e + f*x)))^3*(((-I)*(-1 + E^(I*(e + f*x))))/E^((I/2)*(e + f*x)))^(
2*m)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*f*(1 + 2*m)*(3 + 2*m)*(1 + Sec[e + f*x])^m)

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Maple [F]  time = 0.668, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( fx+e \right ) \left ( a+a\sec \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sec \left ( fx+e \right ) \right ) ^{-2-m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(-2-m),x)

[Out]

int(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(-2-m),x)

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Maxima [A]  time = 1.5165, size = 144, normalized size = 1.38 \begin{align*} -\frac{{\left (\left (-a\right )^{m}{\left (2 \, m + 1\right )} - \frac{\left (-a\right )^{m}{\left (2 \, m + 3\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} c^{-m - 2}{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{2 \,{\left (4 \, m^{2} + 8 \, m + 3\right )} f \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{2 \, m} \sin \left (f x + e\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(-2-m),x, algorithm="maxima")

[Out]

-1/2*((-a)^m*(2*m + 1) - (-a)^m*(2*m + 3)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*c^(-m - 2)*(cos(f*x + e) + 1)^3
/((4*m^2 + 8*m + 3)*f*(sin(f*x + e)/(cos(f*x + e) + 1))^(2*m)*sin(f*x + e)^3)

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Fricas [A]  time = 0.50293, size = 227, normalized size = 2.18 \begin{align*} -\frac{{\left (2 \,{\left (m + 1\right )} \cos \left (f x + e\right ) - 1\right )} \left (\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}\right )^{m} \left (\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}\right )^{-m - 2} \sin \left (f x + e\right )}{{\left (4 \, f m^{2} + 8 \, f m + 3 \, f\right )} \cos \left (f x + e\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(-2-m),x, algorithm="fricas")

[Out]

-(2*(m + 1)*cos(f*x + e) - 1)*((a*cos(f*x + e) + a)/cos(f*x + e))^m*((c*cos(f*x + e) - c)/cos(f*x + e))^(-m -
2)*sin(f*x + e)/((4*f*m^2 + 8*f*m + 3*f)*cos(f*x + e)^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))**m*(c-c*sec(f*x+e))**(-2-m),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(-2-m),x, algorithm="giac")

[Out]

integrate((a*sec(f*x + e) + a)^m*(-c*sec(f*x + e) + c)^(-m - 2)*sec(f*x + e), x)