3.155 \(\int \frac {(c+d x)^m}{a+i a \sinh (e+f x)} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(c+d x)^m}{a+i a \sinh (e+f x)},x\right ) \]

[Out]

Unintegrable((d*x+c)^m/(a+I*a*sinh(f*x+e)),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 = {} \[ \int \frac {(c+d x)^m}{a+i a \sinh (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 4.18, size = 0, normalized size = 0.00 \[ \int \frac {(c+d x)^m}{a+i a \sinh (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ \frac {{\left (a f e^{\left (f x + e\right )} - i \, a f\right )} {\rm integral}\left (-\frac {2 i \, {\left (d x + c\right )}^{m} d m}{-i \, a d f x - i \, a c f + {\left (a d f x + a c f\right )} e^{\left (f x + e\right )}}, x\right ) + 2 i \, {\left (d x + c\right )}^{m}}{a f e^{\left (f x + e\right )} - i \, a f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{a +i a \sinh \left (f x +e \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^m/(a + a*sinh(e + f*x)*1i),x)

[Out]

int((c + d*x)^m/(a + a*sinh(e + f*x)*1i), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*Integral((c + d*x)**m/(sinh(e + f*x) - I), x)/a

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