\(\int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx\) [387]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\frac {4 b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {2 \sin (2 a+2 b x)}{d (c+d x)}-\frac {4 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\text {Int}\left (\frac {\tan (a+b x)}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos (a+b x) \sin (a+b x)}{(c+d x)^2}-\frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2}\right ) \, dx \\ & = 3 \int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx-\int \frac {\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \\ & = 3 \int \frac {\sin (2 a+2 b x)}{2 (c+d x)^2} \, dx+\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx-\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \\ & = \frac {3}{2} \int \frac {\sin (2 a+2 b x)}{(c+d x)^2} \, dx+\int \frac {\sin (2 a+2 b x)}{2 (c+d x)^2} \, dx-\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \\ & = -\frac {3 \sin (2 a+2 b x)}{2 d (c+d x)}+\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{(c+d x)^2} \, dx+\frac {(3 b) \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx}{d}-\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \\ & = -\frac {2 \sin (2 a+2 b x)}{d (c+d x)}+\frac {b \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx}{d}+\frac {\left (3 b \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}-\frac {\left (3 b \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}-\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \\ & = \frac {3 b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {2 \sin (2 a+2 b x)}{d (c+d x)}-\frac {3 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {\left (b \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}-\frac {\left (b \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}-\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \\ & = \frac {4 b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {2 \sin (2 a+2 b x)}{d (c+d x)}-\frac {4 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\int \frac {\tan (a+b x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 242, normalized size of antiderivative = 10.52 \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 30.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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