\(\int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx\) [91]

Optimal result

Integrand size = 23, antiderivative size = 153 \[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=-\frac {b \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tanh (e+f x)}{a-b},\frac {a+b \tanh (e+f x)}{a+b}\right ) (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^{1+n} \left (1-\frac {a+b \tanh (e+f x)}{a-b}\right )^{-m/2} \left (1-\frac {a+b \tanh (e+f x)}{a+b}\right )^{-m/2}}{\left (a^2-b^2\right ) f (1+n)} \] Output:

-b*AppellF1(1+n,1-1/2*m,1-1/2*m,2+n,(a+b*tanh(f*x+e))/(a-b),(a+b*tanh(f*x+ 
e))/(a+b))*(d*sech(f*x+e))^m*(a+b*tanh(f*x+e))^(1+n)/(a^2-b^2)/f/(1+n)/((1 
-(a+b*tanh(f*x+e))/(a-b))^(1/2*m))/((1-(a+b*tanh(f*x+e))/(a+b))^(1/2*m))
 

Mathematica [F]

\[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=\int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx \] Input:

Integrate[(d*Sech[e + f*x])^m*(a + b*Tanh[e + f*x])^n,x]
 

Output:

Integrate[(d*Sech[e + f*x])^m*(a + b*Tanh[e + f*x])^n, x]
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3042, 3995, 150}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (d \sec (i e+i f x))^m (a-i b \tan (i e+i f x))^ndx\)

\(\Big \downarrow \) 3995

\(\displaystyle -\frac {b (d \text {sech}(e+f x))^m \left (1-\frac {a+b \tanh (e+f x)}{a-b}\right )^{-m/2} \left (1-\frac {a+b \tanh (e+f x)}{a+b}\right )^{-m/2} \int (a+b \tanh (e+f x))^n \left (1-\frac {a+b \tanh (e+f x)}{a-b}\right )^{\frac {m-2}{2}} \left (1-\frac {a+b \tanh (e+f x)}{a+b}\right )^{\frac {m-2}{2}}d(a+b \tanh (e+f x))}{f \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 150

\(\displaystyle -\frac {b (d \text {sech}(e+f x))^m \left (1-\frac {a+b \tanh (e+f x)}{a-b}\right )^{-m/2} \left (1-\frac {a+b \tanh (e+f x)}{a+b}\right )^{-m/2} (a+b \tanh (e+f x))^{n+1} \operatorname {AppellF1}\left (n+1,\frac {2-m}{2},\frac {2-m}{2},n+2,\frac {a+b \tanh (e+f x)}{a-b},\frac {a+b \tanh (e+f x)}{a+b}\right )}{f (n+1) \left (a^2-b^2\right )}\)

Input:

Int[(d*Sech[e + f*x])^m*(a + b*Tanh[e + f*x])^n,x]
 

Output:

-((b*AppellF1[1 + n, (2 - m)/2, (2 - m)/2, 2 + n, (a + b*Tanh[e + f*x])/(a 
 - b), (a + b*Tanh[e + f*x])/(a + b)]*(d*Sech[e + f*x])^m*(a + b*Tanh[e + 
f*x])^(1 + n))/((a^2 - b^2)*f*(1 + n)*(1 - (a + b*Tanh[e + f*x])/(a - b))^ 
(m/2)*(1 - (a + b*Tanh[e + f*x])/(a + b))^(m/2)))
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3995

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[d^(2*IntPart[m/2])*(a^2 + b^2)^(IntPart[m/2] 
- 1)*((d*Sec[e + f*x])^(2*FracPart[m/2])/(f*b^(2*IntPart[m/2] - 1)*(1 - (a 
+ b*Tan[e + f*x])/(a - Rt[-b^2, 2]))^FracPart[m/2]*(1 - (a + b*Tan[e + f*x] 
)/(a + Rt[-b^2, 2]))^FracPart[m/2]))   Subst[Int[x^n*(1 - x/(a - Rt[-b^2, 2 
]))^(m/2 - 1)*(1 - x/(a + Rt[-b^2, 2]))^(m/2 - 1), x], x, a + b*Tan[e + f*x 
]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] &&  !Integer 
Q[m] &&  !IntegerQ[n]
 

Maple [F]

\[\int \left (d \,\operatorname {sech}\left (f x +e \right )\right )^{m} \left (a +b \tanh \left (f x +e \right )\right )^{n}d x\]

Input:

int((d*sech(f*x+e))^m*(a+b*tanh(f*x+e))^n,x)
 

Output:

int((d*sech(f*x+e))^m*(a+b*tanh(f*x+e))^n,x)
 

Fricas [F]

\[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=\int { \left (d \operatorname {sech}\left (f x + e\right )\right )^{m} {\left (b \tanh \left (f x + e\right ) + a\right )}^{n} \,d x } \] Input:

integrate((d*sech(f*x+e))^m*(a+b*tanh(f*x+e))^n,x, algorithm="fricas")
 

Output:

integral((d*sech(f*x + e))^m*(b*tanh(f*x + e) + a)^n, x)
 

Sympy [F]

\[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=\int \left (d \operatorname {sech}{\left (e + f x \right )}\right )^{m} \left (a + b \tanh {\left (e + f x \right )}\right )^{n}\, dx \] Input:

integrate((d*sech(f*x+e))**m*(a+b*tanh(f*x+e))**n,x)
 

Output:

Integral((d*sech(e + f*x))**m*(a + b*tanh(e + f*x))**n, x)
 

Maxima [F]

\[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=\int { \left (d \operatorname {sech}\left (f x + e\right )\right )^{m} {\left (b \tanh \left (f x + e\right ) + a\right )}^{n} \,d x } \] Input:

integrate((d*sech(f*x+e))^m*(a+b*tanh(f*x+e))^n,x, algorithm="maxima")
 

Output:

integrate((d*sech(f*x + e))^m*(b*tanh(f*x + e) + a)^n, x)
 

Giac [F]

\[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=\int { \left (d \operatorname {sech}\left (f x + e\right )\right )^{m} {\left (b \tanh \left (f x + e\right ) + a\right )}^{n} \,d x } \] Input:

integrate((d*sech(f*x+e))^m*(a+b*tanh(f*x+e))^n,x, algorithm="giac")
 

Output:

integrate((d*sech(f*x + e))^m*(b*tanh(f*x + e) + a)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=\int {\left (\frac {d}{\mathrm {cosh}\left (e+f\,x\right )}\right )}^m\,{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^n \,d x \] Input:

int((d/cosh(e + f*x))^m*(a + b*tanh(e + f*x))^n,x)
 

Output:

int((d/cosh(e + f*x))^m*(a + b*tanh(e + f*x))^n, x)
 

Reduce [F]

\[ \int (d \text {sech}(e+f x))^m (a+b \tanh (e+f x))^n \, dx=d^{m} \left (\int \mathrm {sech}\left (f x +e \right )^{m} \left (\tanh \left (f x +e \right ) b +a \right )^{n}d x \right ) \] Input:

int((d*sech(f*x+e))^m*(a+b*tanh(f*x+e))^n,x)
 

Output:

d**m*int(sech(e + f*x)**m*(tanh(e + f*x)*b + a)**n,x)