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

Optimal. Leaf size=54 \[ \frac{x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac{b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]

[Out]

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

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Rubi [A]  time = 0.0112868, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5517} \[ \frac{x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac{b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*Log[c*x^n]],x]

[Out]

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

Rule 5517

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

Rubi steps

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

Mathematica [A]  time = 0.0586636, size = 41, normalized size = 0.76 \[ \frac{x \left (b n \cosh \left (a+b \log \left (c x^n\right )\right )-\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b^2 n^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*Log[c*x^n]],x]

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int \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(sinh(a+b*ln(c*x^n)),x)

[Out]

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

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Maxima [A]  time = 1.11891, size = 70, normalized size = 1.3 \begin{align*} \frac{c^{b} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{2 \,{\left (b n + 1\right )}} + \frac{x e^{\left (-b \log \left (x^{n}\right ) - a\right )}}{2 \,{\left (b c^{b} n - c^{b}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.04283, size = 123, normalized size = 2.28 \begin{align*} \frac{b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.13202, size = 63, normalized size = 1.17 \begin{align*} \frac{c^{b} x x^{b n} e^{a}}{2 \,{\left (b n + 1\right )}} + \frac{x e^{\left (-a\right )}}{2 \,{\left (b n - 1\right )} c^{b} x^{b n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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