\(\int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx\) [171]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\text {Int}\left (\frac {1}{(c+d x) (a+b \sin (e+f x))^2},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)/(a+b*sin(f*x+e))^2,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx \]

[In]

Int[1/((c + d*x)*(a + b*Sin[e + f*x])^2),x]

[Out]

Defer[Int][1/((c + d*x)*(a + b*Sin[e + f*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 23.75 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx \]

[In]

Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])^2),x]

[Out]

Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d x +c \right ) \left (a +b \sin \left (f x +e \right )\right )^{2}}d x\]

[In]

int(1/(d*x+c)/(a+b*sin(f*x+e))^2,x)

[Out]

int(1/(d*x+c)/(a+b*sin(f*x+e))^2,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.20 \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(d*x+c)/(a+b*sin(f*x+e))**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 7.20 (sec) , antiderivative size = 1601, normalized size of antiderivative = 80.05 \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

(2*a*b*cos(2*f*x + 2*e)*cos(f*x + e) + 2*a*b*cos(f*x + e) - ((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f + ((a
^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*cos(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f
)*cos(f*x + e)^2 + 4*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*cos(f*x + e)*sin(2*f*x + 2*e) + ((a^2*b^2 -
b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*sin(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*sin(f*
x + e)^2 - 2*((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f + 2*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*si
n(f*x + e))*cos(2*f*x + 2*e) + 4*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*sin(f*x + e))*integrate(-2*(a*b
*d*cos(f*x + e) + 2*(a^2*d*f*x + a^2*c*f)*cos(f*x + e)^2 + 2*(a^2*d*f*x + a^2*c*f)*sin(f*x + e)^2 + (a*b*d*cos
(f*x + e) - (a*b*d*f*x + a*b*c*f)*sin(f*x + e))*cos(2*f*x + 2*e) + (a*b*d*sin(f*x + e) + b^2*d + (a*b*d*f*x +
a*b*c*f)*cos(f*x + e))*sin(2*f*x + 2*e) + (a*b*d*f*x + a*b*c*f)*sin(f*x + e))/((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(
a^2*b^2 - b^4)*c*d*f*x + (a^2*b^2 - b^4)*c^2*f + ((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(a^2*b^2 - b^4)*c*d*f*x + (a^2
*b^2 - b^4)*c^2*f)*cos(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d^2*f*x^2 + 2*(a^4 - a^2*b^2)*c*d*f*x + (a^4 - a^2*
b^2)*c^2*f)*cos(f*x + e)^2 + 4*((a^3*b - a*b^3)*d^2*f*x^2 + 2*(a^3*b - a*b^3)*c*d*f*x + (a^3*b - a*b^3)*c^2*f)
*cos(f*x + e)*sin(2*f*x + 2*e) + ((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(a^2*b^2 - b^4)*c*d*f*x + (a^2*b^2 - b^4)*c^2*
f)*sin(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d^2*f*x^2 + 2*(a^4 - a^2*b^2)*c*d*f*x + (a^4 - a^2*b^2)*c^2*f)*sin(
f*x + e)^2 - 2*((a^2*b^2 - b^4)*d^2*f*x^2 + 2*(a^2*b^2 - b^4)*c*d*f*x + (a^2*b^2 - b^4)*c^2*f + 2*((a^3*b - a*
b^3)*d^2*f*x^2 + 2*(a^3*b - a*b^3)*c*d*f*x + (a^3*b - a*b^3)*c^2*f)*sin(f*x + e))*cos(2*f*x + 2*e) + 4*((a^3*b
- a*b^3)*d^2*f*x^2 + 2*(a^3*b - a*b^3)*c*d*f*x + (a^3*b - a*b^3)*c^2*f)*sin(f*x + e)), x) + 2*(a*b*sin(f*x +
e) + b^2)*sin(2*f*x + 2*e))/((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f + ((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 -
b^4)*c*f)*cos(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*cos(f*x + e)^2 + 4*((a^3*b - a
*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*cos(f*x + e)*sin(2*f*x + 2*e) + ((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*
f)*sin(2*f*x + 2*e)^2 + 4*((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f)*sin(f*x + e)^2 - 2*((a^2*b^2 - b^4)*d*
f*x + (a^2*b^2 - b^4)*c*f + 2*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*sin(f*x + e))*cos(2*f*x + 2*e) + 4
*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*sin(f*x + e))

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*sin(f*x+e))^2,x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(b*sin(f*x + e) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \sin (e+f x))^2} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((a + b*sin(e + f*x))^2*(c + d*x)),x)

[Out]

int(1/((a + b*sin(e + f*x))^2*(c + d*x)), x)