\(\int (e x)^m \sinh (a+b x^n) \, dx\) [82]

Optimal result

Integrand size = 14, antiderivative size = 99 \[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=-\frac {e^a (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{2 e n}+\frac {e^{-a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{2 e n} \]

[Out]

-1/2*exp(a)*(e*x)^(1+m)*GAMMA((1+m)/n,-b*x^n)/e/n/((-b*x^n)^((1+m)/n))+1/2*(e*x)^(1+m)*GAMMA((1+m)/n,b*x^n)/e/
exp(a)/n/((b*x^n)^((1+m)/n))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5468, 2250} \[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=\frac {e^{-a} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},b x^n\right )}{2 e n}-\frac {e^a (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n\right )}{2 e n} \]

[In]

Int[(e*x)^m*Sinh[a + b*x^n],x]

[Out]

-1/2*(E^a*(e*x)^(1 + m)*Gamma[(1 + m)/n, -(b*x^n)])/(e*n*(-(b*x^n))^((1 + m)/n)) + ((e*x)^(1 + m)*Gamma[(1 + m
)/n, b*x^n])/(2*e*E^a*n*(b*x^n)^((1 + m)/n))

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 5468

Int[((e_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x]
- Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int e^{-a-b x^n} (e x)^m \, dx\right )+\frac {1}{2} \int e^{a+b x^n} (e x)^m \, dx \\ & = -\frac {e^a (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{2 e n}+\frac {e^{-a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{2 e n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=\frac {-e^a x (e x)^m \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )+e^{-a} x (e x)^m \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{2 n} \]

[In]

Integrate[(e*x)^m*Sinh[a + b*x^n],x]

[Out]

(-((E^a*x*(e*x)^m*Gamma[(1 + m)/n, -(b*x^n)])/(-(b*x^n))^((1 + m)/n)) + (x*(e*x)^m*Gamma[(1 + m)/n, b*x^n])/(E
^a*(b*x^n)^((1 + m)/n)))/(2*n)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 1.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.16

method	result	size
meijerg	\(\frac {\left (e x \right )^{m} x \operatorname {hypergeom}\left (\left [\frac {m}{2 n}+\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {m}{2 n}+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{1+m}+\frac {\left (e x \right )^{m} x^{n +1} b \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {m}{2 n}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {m}{2 n}+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )}{n +m +1}\)	\(115\)

[In]

int((e*x)^m*sinh(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

(e*x)^m/(1+m)*x*hypergeom([1/2/n*m+1/2/n],[1/2,1+1/2/n*m+1/2/n],1/4*x^(2*n)*b^2)*sinh(a)+(e*x)^m/(n+m+1)*x^(n+
1)*b*hypergeom([1/2+1/2/n*m+1/2/n],[3/2,3/2+1/2/n*m+1/2/n],1/4*x^(2*n)*b^2)*cosh(a)

Fricas [F]

\[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (b x^{n} + a\right ) \,d x } \]

[In]

integrate((e*x)^m*sinh(a+b*x^n),x, algorithm="fricas")

[Out]

integral((e*x)^m*sinh(b*x^n + a), x)

Sympy [F]

\[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=\int \left (e x\right )^{m} \sinh {\left (a + b x^{n} \right )}\, dx \]

[In]

integrate((e*x)**m*sinh(a+b*x**n),x)

[Out]

Integral((e*x)**m*sinh(a + b*x**n), x)

Maxima [F]

\[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (b x^{n} + a\right ) \,d x } \]

[In]

integrate((e*x)^m*sinh(a+b*x^n),x, algorithm="maxima")

[Out]

integrate((e*x)^m*sinh(b*x^n + a), x)

Giac [F]

\[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (b x^{n} + a\right ) \,d x } \]

[In]

integrate((e*x)^m*sinh(a+b*x^n),x, algorithm="giac")

[Out]

integrate((e*x)^m*sinh(b*x^n + a), x)

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \sinh \left (a+b x^n\right ) \, dx=\int \mathrm {sinh}\left (a+b\,x^n\right )\,{\left (e\,x\right )}^m \,d x \]

[In]

int(sinh(a + b*x^n)*(e*x)^m,x)

[Out]

int(sinh(a + b*x^n)*(e*x)^m, x)