3.550 \(\int e^{m x} \csc ^2(x) \, dx\)

Optimal. Leaf size=45 \[ -\frac{4 e^{(m+2 i) x} \, _2F_1\left (2,1-\frac{i m}{2};2-\frac{i m}{2};e^{2 i x}\right )}{m+2 i} \]

[Out]

(-4*E^((2*I + m)*x)*Hypergeometric2F1[2, 1 - (I/2)*m, 2 - (I/2)*m, E^((2*I)*x)])
/(2*I + m)

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Rubi [A]  time = 0.0337854, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{4 e^{(m+2 i) x} \, _2F_1\left (2,1-\frac{i m}{2};2-\frac{i m}{2};e^{2 i x}\right )}{m+2 i} \]

Antiderivative was successfully verified.

[In]  Int[E^(m*x)*Csc[x]^2,x]

[Out]

(-4*E^((2*I + m)*x)*Hypergeometric2F1[2, 1 - (I/2)*m, 2 - (I/2)*m, E^((2*I)*x)])
/(2*I + m)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{m x}}{\sin ^{2}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(m*x)/sin(x)**2,x)

[Out]

Integral(exp(m*x)/sin(x)**2, x)

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Mathematica [A]  time = 0.260766, size = 90, normalized size = 2. \[ \frac{e^{m x} \left (m e^{2 i x} \, _2F_1\left (1,1-\frac{i m}{2};2-\frac{i m}{2};e^{2 i x}\right )+(m+2 i) \left (\, _2F_1\left (1,-\frac{i m}{2};1-\frac{i m}{2};e^{2 i x}\right )-i \cot (x)\right )\right )}{-2+i m} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(m*x)*Csc[x]^2,x]

[Out]

(E^(m*x)*(E^((2*I)*x)*m*Hypergeometric2F1[1, 1 - (I/2)*m, 2 - (I/2)*m, E^((2*I)*
x)] + (2*I + m)*((-I)*Cot[x] + Hypergeometric2F1[1, (-I/2)*m, 1 - (I/2)*m, E^((2
*I)*x)])))/(-2 + I*m)

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \[ \int{\frac{{{\rm e}^{mx}}}{ \left ( \sin \left ( x \right ) \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(m*x)/sin(x)^2,x)

[Out]

int(exp(m*x)/sin(x)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(m*x)/sin(x)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{e^{\left (m x\right )}}{\cos \left (x\right )^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(m*x)/sin(x)^2,x, algorithm="fricas")

[Out]

integral(-e^(m*x)/(cos(x)^2 - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{m x}}{\sin ^{2}{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(m*x)/sin(x)**2,x)

[Out]

Integral(exp(m*x)/sin(x)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\left (m x\right )}}{\sin \left (x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(m*x)/sin(x)^2,x, algorithm="giac")

[Out]

integrate(e^(m*x)/sin(x)^2, x)