3.796 \(\int e^{\sin (x)} \sec ^2(x) (x \cos ^3(x)-\sin (x)) \, dx\)

Optimal. Leaf size=13 \[ e^{\sin (x)} (x \cos (x)-1) \sec (x) \]

[Out]

E^Sin[x]*(-1 + x*Cos[x])*Sec[x]

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Rubi [F]  time = 0.640077, 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 = {} \[ \int e^{\sin (x)} \sec ^2(x) \left (x \cos ^3(x)-\sin (x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^Sin[x]*Sec[x]^2*(x*Cos[x]^3 - Sin[x]),x]

[Out]

Defer[Int][E^Sin[x]*x*Cos[x], x] - Defer[Int][E^Sin[x]*Sec[x]*Tan[x], x]

Rubi steps

\begin{align*} \int e^{\sin (x)} \sec ^2(x) \left (x \cos ^3(x)-\sin (x)\right ) \, dx &=\int \left (e^{\sin (x)} x \cos (x)-e^{\sin (x)} \sec (x) \tan (x)\right ) \, dx\\ &=\int e^{\sin (x)} x \cos (x) \, dx-\int e^{\sin (x)} \sec (x) \tan (x) \, dx\\ \end{align*}

Mathematica [A]  time = 0.276428, size = 13, normalized size = 1. \[ e^{\sin (x)} (x \cos (x)-1) \sec (x) \]

Antiderivative was successfully verified.

[In]

Integrate[E^Sin[x]*Sec[x]^2*(x*Cos[x]^3 - Sin[x]),x]

[Out]

E^Sin[x]*(-1 + x*Cos[x])*Sec[x]

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Maple [C]  time = 0.13, size = 30, normalized size = 2.3 \begin{align*}{\frac{ \left ( x{{\rm e}^{2\,ix}}+x-2\,{{\rm e}^{ix}} \right ){{\rm e}^{\sin \left ( x \right ) }}}{1+{{\rm e}^{2\,ix}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(sin(x))*sec(x)^2*(x*cos(x)^3-sin(x)),x)

[Out]

(x*exp(2*I*x)+x-2*exp(I*x))/(1+exp(2*I*x))*exp(sin(x))

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Maxima [B]  time = 2.6413, size = 119, normalized size = 9.15 \begin{align*} \frac{x \cos \left (2 \, x\right )^{2} e^{\sin \left (x\right )} + x e^{\sin \left (x\right )} \sin \left (2 \, x\right )^{2} - 2 \, e^{\sin \left (x\right )} \sin \left (2 \, x\right ) \sin \left (x\right ) + 2 \,{\left (x e^{\sin \left (x\right )} - \cos \left (x\right ) e^{\sin \left (x\right )}\right )} \cos \left (2 \, x\right ) + x e^{\sin \left (x\right )} - 2 \, \cos \left (x\right ) e^{\sin \left (x\right )}}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(sin(x))*sec(x)^2*(x*cos(x)^3-sin(x)),x, algorithm="maxima")

[Out]

(x*cos(2*x)^2*e^sin(x) + x*e^sin(x)*sin(2*x)^2 - 2*e^sin(x)*sin(2*x)*sin(x) + 2*(x*e^sin(x) - cos(x)*e^sin(x))
*cos(2*x) + x*e^sin(x) - 2*cos(x)*e^sin(x))/(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)

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Fricas [A]  time = 2.01477, size = 43, normalized size = 3.31 \begin{align*} \frac{{\left (x \cos \left (x\right ) - 1\right )} e^{\sin \left (x\right )}}{\cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(sin(x))*sec(x)^2*(x*cos(x)^3-sin(x)),x, algorithm="fricas")

[Out]

(x*cos(x) - 1)*e^sin(x)/cos(x)

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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(exp(sin(x))*sec(x)**2*(x*cos(x)**3-sin(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.16169, size = 1072, normalized size = 82.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(sin(x))*sec(x)^2*(x*cos(x)^3-sin(x)),x, algorithm="giac")

[Out]

(x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)^2*tan(1/2*x)^8 + e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2
*x)^2*tan(1/2*x)^8 - 16*x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)^2*tan(1/2*x)^6 + 12*x*e^(2*tan(1/2*x)
/(tan(1/2*x)^2 + 1))*tan(3/2*x)*tan(1/2*x)^7 - x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^8 - 14*e^(2*ta
n(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)^2*tan(1/2*x)^6 + 12*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)*tan
(1/2*x)^7 - e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^8 + 30*x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/
2*x)^2*tan(1/2*x)^4 - 52*x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)*tan(1/2*x)^5 + 16*x*e^(2*tan(1/2*x)/
(tan(1/2*x)^2 + 1))*tan(1/2*x)^6 - 28*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)*tan(1/2*x)^5 + 14*e^(2*ta
n(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^6 - 16*x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)^2*tan(1/2*x)^2
+ 52*x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)*tan(1/2*x)^3 - 30*x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))
*tan(1/2*x)^4 + 14*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)^2*tan(1/2*x)^2 - 28*e^(2*tan(1/2*x)/(tan(1/2
*x)^2 + 1))*tan(3/2*x)*tan(1/2*x)^3 + x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)^2 - 12*x*e^(2*tan(1/2*x
)/(tan(1/2*x)^2 + 1))*tan(3/2*x)*tan(1/2*x) + 16*x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^2 - e^(2*tan
(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)^2 + 12*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(3/2*x)*tan(1/2*x) - 14*e
^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1))*tan(1/2*x)^2 - x*e^(2*tan(1/2*x)/(tan(1/2*x)^2 + 1)) + e^(2*tan(1/2*x)/(tan
(1/2*x)^2 + 1)))/(tan(3/2*x)^2*tan(1/2*x)^8 + 2*tan(3/2*x)^2*tan(1/2*x)^6 + tan(1/2*x)^8 + 2*tan(1/2*x)^6 - 2*
tan(3/2*x)^2*tan(1/2*x)^2 - tan(3/2*x)^2 - 2*tan(1/2*x)^2 - 1)