\(\int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx\) [392]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.41 (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 = {} \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

[In]

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

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

Mathematica [N/A]

Not integrable

Time = 11.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 1275, normalized size of antiderivative = 51.00 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.90 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 29.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{{\cos \left (a+b\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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