\(\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx\) [381]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.40 (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 = {} \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 463, normalized size of antiderivative = 18.52 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 4.52 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\int { \frac {\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 34.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^3} \,d x \]

[In]

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

[Out]

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