\(\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx\) [379]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=-\frac {4 \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d}-\frac {4 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}+3 \text {Int}\left (\frac {\csc (a+b x)}{c+d x},x\right ) \]

[Out]

-4*cos(a-b*c/d)*Si(b*c/d+b*x)/d-4*Ci(b*c/d+b*x)*sin(a-b*c/d)/d+3*Unintegrable(csc(b*x+a)/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

[In]

Int[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]

[Out]

(-4*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d - (4*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d + 3*Def
er[Int][Csc[a + b*x]/(c + d*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos (a+b x) \cot (a+b x)}{c+d x}-\frac {\sin (a+b x)}{c+d x}\right ) \, dx \\ & = 3 \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx-\int \frac {\sin (a+b x)}{c+d x} \, dx \\ & = 3 \int \frac {\csc (a+b x)}{c+d x} \, dx-3 \int \frac {\sin (a+b x)}{c+d x} \, dx-\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx \\ & = -\frac {\operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}+3 \int \frac {\csc (a+b x)}{c+d x} \, dx-\left (3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx-\left (3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx \\ & = -\frac {4 \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d}-\frac {4 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}+3 \int \frac {\csc (a+b x)}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

[In]

Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]

[Out]

Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

\[\int \frac {\csc \left (x b +a \right )^{2} \sin \left (3 x b +3 a \right )}{d x +c}d x\]

[In]

int(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x)

[Out]

int(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c} \,d x } \]

[In]

integrate(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^2*sin(3*b*x + 3*a)/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 125.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\sin {\left (3 a + 3 b x \right )} \csc ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

integrate(csc(b*x+a)**2*sin(3*b*x+3*a)/(d*x+c),x)

[Out]

Integral(sin(3*a + 3*b*x)*csc(a + b*x)**2/(c + d*x), x)

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 230, normalized size of antiderivative = 9.20 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c} \,d x } \]

[In]

integrate(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x, algorithm="maxima")

[Out]

-(2*(-I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) + I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d
) - 3*d*integrate(sin(b*x + a)/((d*x + c)*cos(b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x + 2*(d*x + c)*cos(b*
x + a) + c), x) - 3*d*integrate(sin(b*x + a)/((d*x + c)*cos(b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x - 2*(d
*x + c)*cos(b*x + a) + c), x) - 2*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*
c)/d))*sin(-(b*c - a*d)/d))/d

Giac [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c} \,d x } \]

[In]

integrate(csc(b*x+a)^2*sin(3*b*x+3*a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^2*sin(3*b*x + 3*a)/(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 30.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{{\sin \left (a+b\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

int(sin(3*a + 3*b*x)/(sin(a + b*x)^2*(c + d*x)),x)

[Out]

int(sin(3*a + 3*b*x)/(sin(a + b*x)^2*(c + d*x)), x)