3.34 \(\int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx\)

Optimal. Leaf size=88 \[ \frac {2^{-m-2} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{a f}+\frac {(c+d x)^{m+1}}{2 a d (m+1)} \]

[Out]

1/2*(d*x+c)^(1+m)/a/d/(1+m)+2^(-2-m)*exp(-2*e+2*c*f/d)*(d*x+c)^m*GAMMA(1+m,2*f*(d*x+c)/d)/a/f/((f*(d*x+c)/d)^m
)

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Rubi [A]  time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3727, 2181} \[ \frac {2^{-m-2} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 f (c+d x)}{d}\right )}{a f}+\frac {(c+d x)^{m+1}}{2 a d (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^m/(a + a*Coth[e + f*x]),x]

[Out]

(c + d*x)^(1 + m)/(2*a*d*(1 + m)) + (2^(-2 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/
d])/(a*f*((f*(c + d*x))/d)^m)

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3727

Int[((c_.) + (d_.)*(x_))^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(2*a
*d*(m + 1)), x] + Dist[1/(2*a), Int[(c + d*x)^m*E^((2*a*(e + f*x))/b), x], x] /; FreeQ[{a, b, c, d, e, f, m},
x] && EqQ[a^2 + b^2, 0] &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {\int e^{2 i \left (i e+\frac {\pi }{2}+i f x\right )} (c+d x)^m \, dx}{2 a}\\ &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {2^{-2-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a f}\\ \end {align*}

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Mathematica [B]  time = 1.20, size = 186, normalized size = 2.11 \[ \frac {2^{-m-2} (c+d x)^m \text {csch}(e+f x) \left (-\frac {f (c+d x)}{d}\right )^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (\sinh \left (f \left (\frac {c}{d}+x\right )\right )+\cosh \left (f \left (\frac {c}{d}+x\right )\right )\right ) \left (f 2^{m+1} (c+d x) \left (f \left (\frac {c}{d}+x\right )\right )^m \left (\sinh \left (e-\frac {c f}{d}\right )+\cosh \left (e-\frac {c f}{d}\right )\right )+d (m+1) \left (\cosh \left (e-\frac {c f}{d}\right )-\sinh \left (e-\frac {c f}{d}\right )\right ) \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )\right )}{a d f (m+1) (\coth (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^m/(a + a*Coth[e + f*x]),x]

[Out]

(2^(-2 - m)*(c + d*x)^m*(-((f*(c + d*x))/d))^m*Csch[e + f*x]*(d*(1 + m)*Gamma[1 + m, (2*f*(c + d*x))/d]*(Cosh[
e - (c*f)/d] - Sinh[e - (c*f)/d]) + 2^(1 + m)*f*(f*(c/d + x))^m*(c + d*x)*(Cosh[e - (c*f)/d] + Sinh[e - (c*f)/
d]))*(Cosh[f*(c/d + x)] + Sinh[f*(c/d + x)]))/(a*d*f*(1 + m)*(-((f^2*(c + d*x)^2)/d^2))^m*(1 + Coth[e + f*x]))

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fricas [A]  time = 0.40, size = 148, normalized size = 1.68 \[ \frac {{\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (d m + d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) + 2 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) + 2 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{4 \, {\left (a d f m + a d f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m/(a+a*coth(f*x+e)),x, algorithm="fricas")

[Out]

1/4*((d*m + d)*cosh((d*m*log(2*f/d) + 2*d*e - 2*c*f)/d)*gamma(m + 1, 2*(d*f*x + c*f)/d) - (d*m + d)*gamma(m +
1, 2*(d*f*x + c*f)/d)*sinh((d*m*log(2*f/d) + 2*d*e - 2*c*f)/d) + 2*(d*f*x + c*f)*cosh(m*log(d*x + c)) + 2*(d*f
*x + c*f)*sinh(m*log(d*x + c)))/(a*d*f*m + a*d*f)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{a \coth \left (f x + e\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m/(a+a*coth(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*x + c)^m/(a*coth(f*x + e) + a), x)

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maple [F]  time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{a +a \coth \left (f x +e \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m/(a+a*coth(f*x+e)),x)

[Out]

int((d*x+c)^m/(a+a*coth(f*x+e)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{a \coth \left (f x + e\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m/(a+a*coth(f*x+e)),x, algorithm="maxima")

[Out]

integrate((d*x + c)^m/(a*coth(f*x + e) + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^m}{a+a\,\mathrm {coth}\left (e+f\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^m/(a + a*coth(e + f*x)),x)

[Out]

int((c + d*x)^m/(a + a*coth(e + f*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (c + d x\right )^{m}}{\coth {\left (e + f x \right )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**m/(a+a*coth(f*x+e)),x)

[Out]

Integral((c + d*x)**m/(coth(e + f*x) + 1), x)/a

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