\(\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx\) [380]

Optimal result

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

[Out]

-4*b*Ci(b*c/d+b*x)*cos(a-b*c/d)/d^2+4*b*Si(b*c/d+b*x)*sin(a-b*c/d)/d^2+4*sin(b*x+a)/d/(d*x+c)+3*Unintegrable(c
sc(b*x+a)/(d*x+c)^2,x)

Rubi [N/A]

Not integrable

Time = 0.33 (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 = {} \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

(-4*b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2 + (4*Sin[a + b*x])/(d*(c + d*x)) + (4*b*Sin[a - (b*c)/d
]*SinIntegral[(b*c)/d + b*x])/d^2 + 3*Defer[Int][Csc[a + b*x]/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos (a+b x) \cot (a+b x)}{(c+d x)^2}-\frac {\sin (a+b x)}{(c+d x)^2}\right ) \, dx \\ & = 3 \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx-\int \frac {\sin (a+b x)}{(c+d x)^2} \, dx \\ & = \frac {\sin (a+b x)}{d (c+d x)}+3 \int \frac {\csc (a+b x)}{(c+d x)^2} \, dx-3 \int \frac {\sin (a+b x)}{(c+d x)^2} \, dx-\frac {b \int \frac {\cos (a+b x)}{c+d x} \, dx}{d} \\ & = \frac {4 \sin (a+b x)}{d (c+d x)}+3 \int \frac {\csc (a+b x)}{(c+d x)^2} \, dx-\frac {(3 b) \int \frac {\cos (a+b x)}{c+d x} \, dx}{d}-\frac {\left (b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\frac {\left (b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d} \\ & = -\frac {b \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {4 \sin (a+b x)}{d (c+d x)}+\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+3 \int \frac {\csc (a+b x)}{(c+d x)^2} \, dx-\frac {\left (3 b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\frac {\left (3 b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d} \\ & = -\frac {4 b \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {4 \sin (a+b x)}{d (c+d x)}+\frac {4 b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+3 \int \frac {\csc (a+b x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 342, normalized size of antiderivative = 13.68 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.99 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 33.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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