3.186 \(\int \frac {(c+d x)^m}{(a+b \sinh (e+f x))^2} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(c+d x)^m}{(a+b \sinh (e+f x))^2},x\right ) \]

[Out]

Unintegrable((d*x+c)^m/(a+b*sinh(f*x+e))^2,x)

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Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(c+d x)^m}{(a+b \sinh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + d*x)^m/(a + b*Sinh[e + f*x])^2,x]

[Out]

Defer[Int][(c + d*x)^m/(a + b*Sinh[e + f*x])^2, x]

Rubi steps

\begin {align*} \int \frac {(c+d x)^m}{(a+b \sinh (e+f x))^2} \, dx &=\int \frac {(c+d x)^m}{(a+b \sinh (e+f x))^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 5.38, size = 0, normalized size = 0.00 \[ \int \frac {(c+d x)^m}{(a+b \sinh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + d*x)^m/(a + b*Sinh[e + f*x])^2,x]

[Out]

Integrate[(c + d*x)^m/(a + b*Sinh[e + f*x])^2, x]

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fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{m}}{b^{2} \sinh \left (f x + e\right )^{2} + 2 \, a b \sinh \left (f x + e\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m/(a+b*sinh(f*x+e))^2,x, algorithm="fricas")

[Out]

integral((d*x + c)^m/(b^2*sinh(f*x + e)^2 + 2*a*b*sinh(f*x + e) + a^2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m/(a+b*sinh(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^m/(b*sinh(f*x + e) + a)^2, x)

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maple [A]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{\left (a +b \sinh \left (f x +e \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m/(a+b*sinh(f*x+e))^2,x)

[Out]

int((d*x+c)^m/(a+b*sinh(f*x+e))^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m/(a+b*sinh(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((d*x + c)^m/(b*sinh(f*x + e) + a)^2, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (c+d\,x\right )}^m}{{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^m/(a + b*sinh(e + f*x))^2,x)

[Out]

int((c + d*x)^m/(a + b*sinh(e + f*x))^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x\right )^{m}}{\left (a + b \sinh {\left (e + f x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**m/(a+b*sinh(f*x+e))**2,x)

[Out]

Integral((c + d*x)**m/(a + b*sinh(e + f*x))**2, x)

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