\(\int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx\) [393]

Optimal result

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

[Out]

-CannotIntegrate(sec(b*x+a)*tan(b*x+a)/(d*x+c)^2,x)+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)

Rubi [N/A]

Not integrable

Time = 0.47 (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 = {} \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx \]

[In]

Int[(Sec[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 - Defer[Int][(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^2, x]

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

Mathematica [N/A]

Not integrable

Time = 15.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.73 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 1419, normalized size of antiderivative = 56.76 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 4.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 32.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{{\cos \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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