3.2.56 \(\int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx\) [156]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 12.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{m}}{\left (a +i a \sinh \left (f x +e \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(I*d^2*f^2*x^2 + 2*I*c*d*f^2*x + I*c^2*f^2 - I*d^2*m^2 + I*d^2*m + (I*d^2*f*m*x + I*d^2*m^2 + (I*c*d*f - I
*d^2)*m)*e^(2*f*x + 2*e) - (3*d^2*f^2*x^2 + 3*c^2*f^2 - 2*d^2*m^2 - (c*d*f - 2*d^2)*m + (6*c*d*f^2 - d^2*f*m)*
x)*e^(f*x + e))*(d*x + c)^m + 3*(-I*a^2*d^2*f^3*x^2 - 2*I*a^2*c*d*f^3*x - I*a^2*c^2*f^3 - (a^2*d^2*f^3*x^2 + 2
*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(3*f*x + 3*e) + 3*(I*a^2*d^2*f^3*x^2 + 2*I*a^2*c*d*f^3*x + I*a^2*c^2*f^3)*e^(2
*f*x + 2*e) + 3*(a^2*d^2*f^3*x^2 + 2*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(f*x + e))*integral(-2*(I*d^3*f^2*m*x^2 +
2*I*c*d^2*f^2*m*x - I*d^3*m^3 + 3*I*d^3*m^2 + (I*c^2*d*f^2 - 2*I*d^3)*m)*(d*x + c)^m/(-3*I*a^2*d^3*f^3*x^3 - 9
*I*a^2*c*d^2*f^3*x^2 - 9*I*a^2*c^2*d*f^3*x - 3*I*a^2*c^3*f^3 + 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^
2*c^2*d*f^3*x + a^2*c^3*f^3)*e^(f*x + e)), x))/(3*I*a^2*d^2*f^3*x^2 + 6*I*a^2*c*d*f^3*x + 3*I*a^2*c^2*f^3 + 3*
(a^2*d^2*f^3*x^2 + 2*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(3*f*x + 3*e) - 9*(I*a^2*d^2*f^3*x^2 + 2*I*a^2*c*d*f^3*x +
I*a^2*c^2*f^3)*e^(2*f*x + 2*e) - 9*(a^2*d^2*f^3*x^2 + 2*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(f*x + e))

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\left (c + d x\right )^{m}}{\sinh ^{2}{\left (e + f x \right )} - 2 i \sinh {\left (e + f x \right )} - 1}\, dx}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral((c + d*x)**m/(sinh(e + f*x)**2 - 2*I*sinh(e + f*x) - 1), x)/a**2

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^m/(a + a*sinh(e + f*x)*1i)^2,x)

[Out]

int((c + d*x)^m/(a + a*sinh(e + f*x)*1i)^2, x)

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