3.164 \(\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-1-m} \, dx\)

Optimal. Leaf size=47 \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-1}}{f (2 m+1)} \]

[Out]

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

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Rubi [A]  time = 0.101124, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {3950} \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-1}}{f (2 m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 - m),x]

[Out]

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

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))^{-1-m} \, dx &=-\frac{(a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-1-m} \tan (e+f x)}{f (1+2 m)}\\ \end{align*}

Mathematica [C]  time = 1.14629, size = 208, normalized size = 4.43 \[ -\frac{2^{m+1} e^{-\frac{1}{2} i (e+f x)} \left (1+e^{i (e+f x)}\right ) \left (-i e^{-\frac{1}{2} i (e+f x)} \left (-1+e^{i (e+f x)}\right )\right )^{-2 m-1} \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{-m} \left (\frac{\left (1+e^{i (e+f x)}\right )^2}{1+e^{2 i (e+f x)}}\right )^m \sin ^{2 (m+1)}\left (\frac{1}{2} (e+f x)\right ) \sec ^{m+1}(e+f x) (\sec (e+f x)+1)^{-m} (a (\sec (e+f x)+1))^m (c-c \sec (e+f x))^{-m-1}}{2 f m+f} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 - m),x]

[Out]

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

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Maple [F]  time = 0.689, 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 ) ^{-1-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))^(-1-m),x)

[Out]

int(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(-1-m),x)

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Maxima [A]  time = 1.51068, size = 84, normalized size = 1.79 \begin{align*} \frac{\left (-a\right )^{m} c^{-m - 1}{\left (\cos \left (f x + e\right ) + 1\right )}}{f{\left (2 \, m + 1\right )} \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{2 \, m} \sin \left (f x + e\right )} \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))^(-1-m),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.494884, size = 169, normalized size = 3.6 \begin{align*} -\frac{\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 - 1} \sin \left (f x + e\right )}{{\left (2 \, f m + f\right )} \cos \left (f x + e\right )} \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))^(-1-m),x, algorithm="fricas")

[Out]

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

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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))**(-1-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 - 1} \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))^(-1-m),x, algorithm="giac")

[Out]

integrate((a*sec(f*x + e) + a)^m*(-c*sec(f*x + e) + c)^(-m - 1)*sec(f*x + e), x)