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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{(c+d x)^m}{(a+i a \sinh (e+f x))^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0572106, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx \]

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*}

Mathematica [A]  time = 15.7454, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx \]

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 = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{ \left ( a+ia\sinh \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{m}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \frac{{\left (-2 i \, d^{2} f^{2} x^{2} - 4 i \, c d f^{2} x - 2 i \, c^{2} f^{2} + 2 i \, d^{2} m^{2} - 2 i \, d^{2} m +{\left (-2 i \, d^{2} f m x - 2 i \, d^{2} m^{2} +{\left (-2 i \, c d f + 2 i \, d^{2}\right )} m\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \,{\left (3 \, d^{2} f^{2} x^{2} + 3 \, c^{2} f^{2} - 2 \, d^{2} m^{2} -{\left (c d f - 2 \, d^{2}\right )} m +{\left (6 \, c d f^{2} - d^{2} f m\right )} x\right )} e^{\left (f x + e\right )}\right )}{\left (d x + c\right )}^{m} +{\left (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 \,{\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} f^{3}\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (-9 i \, a^{2} d^{2} f^{3} x^{2} - 18 i \, a^{2} c d f^{3} x - 9 i \, a^{2} c^{2} f^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 9 \,{\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} f^{3}\right )} e^{\left (f x + e\right )}\right )}{\rm integral}\left (\frac{{\left (-2 i \, d^{3} f^{2} m x^{2} - 4 i \, c d^{2} f^{2} m x + 2 i \, d^{3} m^{3} - 6 i \, d^{3} m^{2} +{\left (-2 i \, c^{2} d f^{2} + 4 i \, d^{3}\right )} m\right )}{\left (d x + c\right )}^{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 \,{\left (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}\right )} e^{\left (f x + e\right )}}, x\right )}{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 \,{\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} f^{3}\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (-9 i \, a^{2} d^{2} f^{3} x^{2} - 18 i \, a^{2} c d f^{3} x - 9 i \, a^{2} c^{2} f^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 9 \,{\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} f^{3}\right )} e^{\left (f x + e\right )}} \end{align*}

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 - 4*I*c*d*f^2*x - 2*I*c^2*f^2 + 2*I*d^2*m^2 - 2*I*d^2*m + (-2*I*d^2*f*m*x - 2*I*d^2*m^2 + (
-2*I*c*d*f + 2*I*d^2)*m)*e^(2*f*x + 2*e) + 2*(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 + 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 - 18*I*a^2*c*d*f^3*
x - 9*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))*integr
al((-2*I*d^3*f^2*m*x^2 - 4*I*c*d^2*f^2*m*x + 2*I*d^3*m^3 - 6*I*d^3*m^2 + (-2*I*c^2*d*f^2 + 4*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 - 18*I*a^2*c*d*f^3*x - 9*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{m}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

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)