3.270 \(\int x^m \sinh (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=73 \[ \frac{(m+1) x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-b^2 n^2}-\frac{b n x^{m+1} \cosh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-b^2 n^2} \]

[Out]

-((b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/((1 + m)^2 - b^2*n^2)) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/(
(1 + m)^2 - b^2*n^2)

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Rubi [A]  time = 0.0237147, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5527} \[ \frac{(m+1) x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-b^2 n^2}-\frac{b n x^{m+1} \cosh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-b^2 n^2} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Sinh[a + b*Log[c*x^n]],x]

[Out]

-((b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/((1 + m)^2 - b^2*n^2)) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/(
(1 + m)^2 - b^2*n^2)

Rule 5527

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

Rubi steps

\begin{align*} \int x^m \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-b^2 n^2}+\frac{(1+m) x^{1+m} \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-b^2 n^2}\\ \end{align*}

Mathematica [A]  time = 0.132853, size = 54, normalized size = 0.74 \[ \frac{x^{m+1} \left ((m+1) \sinh \left (a+b \log \left (c x^n\right )\right )-b n \cosh \left (a+b \log \left (c x^n\right )\right )\right )}{(-b n+m+1) (b n+m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Sinh[a + b*Log[c*x^n]],x]

[Out]

(x^(1 + m)*(-(b*n*Cosh[a + b*Log[c*x^n]]) + (1 + m)*Sinh[a + b*Log[c*x^n]]))/((1 + m - b*n)*(1 + m + b*n))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sinh(a+b*ln(c*x^n)),x)

[Out]

int(x^m*sinh(a+b*ln(c*x^n)),x)

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Maxima [A]  time = 1.11783, size = 86, normalized size = 1.18 \begin{align*} \frac{c^{b} x e^{\left (b \log \left (x^{n}\right ) + m \log \left (x\right ) + a\right )}}{2 \,{\left (b n + m + 1\right )}} + \frac{x e^{\left (-b \log \left (x^{n}\right ) + m \log \left (x\right ) - a\right )}}{2 \,{\left (b c^{b} n - c^{b}{\left (m + 1\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^b*x*e^(b*log(x^n) + m*log(x) + a)/(b*n + m + 1) + 1/2*x*e^(-b*log(x^n) + m*log(x) - a)/(b*c^b*n - c^b*(m
+ 1))

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Fricas [A]  time = 2.08741, size = 304, normalized size = 4.16 \begin{align*} \frac{b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \cosh \left (m \log \left (x\right )\right ) + b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (m \log \left (x\right )\right ) -{\left ({\left (m + 1\right )} x \cosh \left (m \log \left (x\right )\right ) +{\left (m + 1\right )} x \sinh \left (m \log \left (x\right )\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} - m^{2} - 2 \, m - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*n*x*cosh(b*n*log(x) + b*log(c) + a)*cosh(m*log(x)) + b*n*x*cosh(b*n*log(x) + b*log(c) + a)*sinh(m*log(x)) -
((m + 1)*x*cosh(m*log(x)) + (m + 1)*x*sinh(m*log(x)))*sinh(b*n*log(x) + b*log(c) + a))/(b^2*n^2 - m^2 - 2*m -
1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.19303, size = 317, normalized size = 4.34 \begin{align*} \frac{b c^{b} n x x^{b n} x^{m} e^{a}}{2 \,{\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )}} - \frac{c^{b} m x x^{b n} x^{m} e^{a}}{2 \,{\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )}} - \frac{c^{b} x x^{b n} x^{m} e^{a}}{2 \,{\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )}} + \frac{b n x x^{m} e^{\left (-a\right )}}{2 \,{\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )} c^{b} x^{b n}} + \frac{m x x^{m} e^{\left (-a\right )}}{2 \,{\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )} c^{b} x^{b n}} + \frac{x x^{m} e^{\left (-a\right )}}{2 \,{\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )} c^{b} x^{b n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*c^b*n*x*x^(b*n)*x^m*e^a/(b^2*n^2 - m^2 - 2*m - 1) - 1/2*c^b*m*x*x^(b*n)*x^m*e^a/(b^2*n^2 - m^2 - 2*m - 1
) - 1/2*c^b*x*x^(b*n)*x^m*e^a/(b^2*n^2 - m^2 - 2*m - 1) + 1/2*b*n*x*x^m*e^(-a)/((b^2*n^2 - m^2 - 2*m - 1)*c^b*
x^(b*n)) + 1/2*m*x*x^m*e^(-a)/((b^2*n^2 - m^2 - 2*m - 1)*c^b*x^(b*n)) + 1/2*x*x^m*e^(-a)/((b^2*n^2 - m^2 - 2*m
- 1)*c^b*x^(b*n))