∫ sin2 (c+dx)_____ (e+fx)2(a+a sin(c+dx)) dx [190]

3.2.90 \(\int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [190]

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]

[Out]

Unintegrable(sin(d*x+c)^2/(f*x+e)^2/(a+a*sin(d*x+c)),x)

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Rubi [A]
time = 0.05, 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 {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 6.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.31, size = 0, normalized size = 0.00 \[\int \frac {\sin ^{2}\left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^2/(f*x+e)^2/(a+a*sin(d*x+c)),x)

[Out]

int(sin(d*x+c)^2/(f*x+e)^2/(a+a*sin(d*x+c)),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(sin(d*x+c)^2/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(((I*e*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*e*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*d*cos((c*f -

d*e)/f) + (e*exp_integral_e(2, (I*d*f*x + I*d*e)/f) + e*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*d*sin((c*f -

d*e)/f) + (d*f*(I*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*cos((c*

f - d*e)/f) + d*f*(exp_integral_e(2, (I*d*f*x + I*d*e)/f) + exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*sin((c*f

- d*e)/f) - 2*d*f)*x - 2*d*e)*cos(d*x + c)^2 + (I*e*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*e*exp_integral_

e(2, -(I*d*f*x + I*d*e)/f))*d*cos((c*f - d*e)/f) + ((I*e*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*e*exp_inte

gral_e(2, -(I*d*f*x + I*d*e)/f))*d*cos((c*f - d*e)/f) + (e*exp_integral_e(2, (I*d*f*x + I*d*e)/f) + e*exp_inte

gral_e(2, -(I*d*f*x + I*d*e)/f))*d*sin((c*f - d*e)/f) + (d*f*(I*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*exp

_integral_e(2, -(I*d*f*x + I*d*e)/f))*cos((c*f - d*e)/f) + d*f*(exp_integral_e(2, (I*d*f*x + I*d*e)/f) + exp_i

ntegral_e(2, -(I*d*f*x + I*d*e)/f))*sin((c*f - d*e)/f) - 2*d*f)*x - 2*d*e)*sin(d*x + c)^2 + (e*exp_integral_e(

2, (I*d*f*x + I*d*e)/f) + e*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*d*sin((c*f - d*e)/f) + (d*f*(I*exp_integr

al_e(2, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*cos((c*f - d*e)/f) + d*f*(exp_integr

al_e(2, (I*d*f*x + I*d*e)/f) + exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*sin((c*f - d*e)/f) - 2*d*f)*x + 4*f*co

s(d*x + c) - 2*d*e + 8*(a*d*f^4*x^2 + 2*a*d*f^3*x*e + a*d*f^2*e^2 + (a*d*f^4*x^2 + 2*a*d*f^3*x*e + a*d*f^2*e^2

)*cos(d*x + c)^2 + (a*d*f^4*x^2 + 2*a*d*f^3*x*e + a*d*f^2*e^2)*sin(d*x + c)^2 + 2*(a*d*f^4*x^2 + 2*a*d*f^3*x*e

+ a*d*f^2*e^2)*sin(d*x + c))*integrate(cos(d*x + c)/(a*d*f^3*x^3 + 3*a*d*f^2*x^2*e + 3*a*d*f*x*e^2 + (a*d*f^3

*x^3 + 3*a*d*f^2*x^2*e + 3*a*d*f*x*e^2 + a*d*e^3)*cos(d*x + c)^2 + a*d*e^3 + (a*d*f^3*x^3 + 3*a*d*f^2*x^2*e +

3*a*d*f*x*e^2 + a*d*e^3)*sin(d*x + c)^2 + 2*(a*d*f^3*x^3 + 3*a*d*f^2*x^2*e + 3*a*d*f*x*e^2 + a*d*e^3)*sin(d*x

+ c)), x) - 2*((-I*e*exp_integral_e(2, (I*d*f*x + I*d*e)/f) + I*e*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*d*c

os((c*f - d*e)/f) - (e*exp_integral_e(2, (I*d*f*x + I*d*e)/f) + e*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*d*s

in((c*f - d*e)/f) + (d*f*(-I*exp_integral_e(2, (I*d*f*x + I*d*e)/f) + I*exp_integral_e(2, -(I*d*f*x + I*d*e)/f

))*cos((c*f - d*e)/f) - d*f*(exp_integral_e(2, (I*d*f*x + I*d*e)/f) + exp_integral_e(2, -(I*d*f*x + I*d*e)/f))

*sin((c*f - d*e)/f) + 2*d*f)*x + 2*d*e)*sin(d*x + c))/(a*d*f^3*x^2 + 2*a*d*f^2*x*e + a*d*f*e^2 + (a*d*f^3*x^2

+ 2*a*d*f^2*x*e + a*d*f*e^2)*cos(d*x + c)^2 + (a*d*f^3*x^2 + 2*a*d*f^2*x*e + a*d*f*e^2)*sin(d*x + c)^2 + 2*(a*

d*f^3*x^2 + 2*a*d*f^2*x*e + a*d*f*e^2)*sin(d*x + c))

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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(sin(d*x+c)^2/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral(-(cos(d*x + c)^2 - 1)/(a*f^2*x^2 + 2*a*f*x*e + a*e^2 + (a*f^2*x^2 + 2*a*f*x*e + a*e^2)*sin(d*x + c)),

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**2/(f*x+e)**2/(a+a*sin(d*x+c)),x)

[Out]

Integral(sin(c + d*x)**2/(e**2*sin(c + d*x) + e**2 + 2*e*f*x*sin(c + d*x) + 2*e*f*x + f**2*x**2*sin(c + d*x) +

f**2*x**2), x)/a

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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(sin(d*x+c)^2/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(sin(d*x + c)^2/((f*x + e)^2*(a*sin(d*x + c) + a)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\sin \left (c+d\,x\right )}^2}{{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^2/((e + f*x)^2*(a + a*sin(c + d*x))),x)

[Out]

int(sin(c + d*x)^2/((e + f*x)^2*(a + a*sin(c + d*x))), x)

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