\(\int \sec (c+b x) \sin (a+b x) \, dx\) [213]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \sec (c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \log (\cos (c+b x))}{b}+x \sin (a-c) \]

[Out]

-cos(a-c)*ln(cos(b*x+c))/b+x*sin(a-c)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4676, 3556, 8} \[ \int \sec (c+b x) \sin (a+b x) \, dx=x \sin (a-c)-\frac {\cos (a-c) \log (\cos (b x+c))}{b} \]

[In]

Int[Sec[c + b*x]*Sin[a + b*x],x]

[Out]

-((Cos[a - c]*Log[Cos[c + b*x]])/b) + x*Sin[a - c]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4676

Int[Sec[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Cos[v - w], Int[Tan[w]*Sec[w]^(n - 1), x], x] + Dist[Sin[v - w],
Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]

Rubi steps \begin{align*} \text {integral}& = \cos (a-c) \int \tan (c+b x) \, dx+\sin (a-c) \int 1 \, dx \\ & = -\frac {\cos (a-c) \log (\cos (c+b x))}{b}+x \sin (a-c) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \sec (c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \log (\cos (c+b x))}{b}+x \sin (a-c) \]

[In]

Integrate[Sec[c + b*x]*Sin[a + b*x],x]

[Out]

-((Cos[a - c]*Log[Cos[c + b*x]])/b) + x*Sin[a - c]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56

method	result	size
risch	\(2 i \cos \left (a -c \right ) x -i x \,{\mathrm e}^{i \left (a -c \right )}+\frac {2 i \cos \left (a -c \right ) a}{b}-\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}\)	\(69\)
default	\(\frac {\frac {\frac {\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \ln \left (1+\tan \left (x b +a \right )^{2}\right )}{2}+\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\tan \left (x b +a \right )\right )}{\left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right ) \left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right )}-\frac {\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (x b +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (x b +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}{\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\cos \left (a \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}}{b}\)	\(161\)

[In]

int(sec(b*x+c)*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

2*I*cos(a-c)*x-I*x*exp(I*(a-c))+2*I/b*cos(a-c)*a-ln(exp(2*I*(b*x+a))+exp(2*I*(a-c)))/b*cos(a-c)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \sec (c+b x) \sin (a+b x) \, dx=-\frac {b x \sin \left (-a + c\right ) + \cos \left (-a + c\right ) \log \left (-\cos \left (b x + c\right )\right )}{b} \]

[In]

integrate(sec(b*x+c)*sin(b*x+a),x, algorithm="fricas")

[Out]

-(b*x*sin(-a + c) + cos(-a + c)*log(-cos(b*x + c)))/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (20) = 40\).

Time = 4.32 (sec) , antiderivative size = 435, normalized size of antiderivative = 16.11 \[ \int \sec (c+b x) \sin (a+b x) \, dx=\left (\begin {cases} - x & \text {for}\: c = \frac {\pi }{2} \\x & \text {for}\: c = - \frac {\pi }{2} \\0 & \text {for}\: b = 0 \\- \frac {2 b x \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {\tan {\left (\frac {c}{2} \right )}}{\tan {\left (\frac {c}{2} \right )} - 1} - \frac {1}{\tan {\left (\frac {c}{2} \right )} - 1} \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {\tan {\left (\frac {c}{2} \right )}}{\tan {\left (\frac {c}{2} \right )} - 1} - \frac {1}{\tan {\left (\frac {c}{2} \right )} - 1} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} + \frac {\tan {\left (\frac {c}{2} \right )}}{\tan {\left (\frac {c}{2} \right )} + 1} - \frac {1}{\tan {\left (\frac {c}{2} \right )} + 1} \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} + \frac {\tan {\left (\frac {c}{2} \right )}}{\tan {\left (\frac {c}{2} \right )} + 1} - \frac {1}{\tan {\left (\frac {c}{2} \right )} + 1} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} & \text {otherwise} \end {cases}\right ) \cos {\left (a \right )} + \left (\begin {cases} - \frac {\log {\left (\sin {\left (b x \right )} \right )}}{b} & \text {for}\: c = \frac {\pi }{2} \\\frac {\log {\left (\sin {\left (b x \right )} \right )}}{b} & \text {for}\: c = - \frac {\pi }{2} \\\frac {x}{\cos {\left (c \right )}} & \text {for}\: b = 0 \\- \frac {b x \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {b x}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {2 \log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {2 \log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {\tan {\left (\frac {c}{2} \right )}}{\tan {\left (\frac {c}{2} \right )} - 1} - \frac {1}{\tan {\left (\frac {c}{2} \right )} - 1} \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {2 \log {\left (\tan {\left (\frac {b x}{2} \right )} + \frac {\tan {\left (\frac {c}{2} \right )}}{\tan {\left (\frac {c}{2} \right )} + 1} - \frac {1}{\tan {\left (\frac {c}{2} \right )} + 1} \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} & \text {otherwise} \end {cases}\right ) \sin {\left (a \right )} \]

[In]

integrate(sec(b*x+c)*sin(b*x+a),x)

[Out]

Piecewise((-x, Eq(c, pi/2)), (x, Eq(c, -pi/2)), (0, Eq(b, 0)), (-2*b*x*tan(c/2)/(b*tan(c/2)**2 + b) - log(tan(
b*x/2)**2 + 1)*tan(c/2)**2/(b*tan(c/2)**2 + b) + log(tan(b*x/2)**2 + 1)/(b*tan(c/2)**2 + b) + log(tan(b*x/2) -
tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**2/(b*tan(c/2)**2 + b) - log(tan(b*x/2) - tan(c/2)/(tan(
c/2) - 1) - 1/(tan(c/2) - 1))/(b*tan(c/2)**2 + b) + log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1
))*tan(c/2)**2/(b*tan(c/2)**2 + b) - log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))/(b*tan(c/2)*
*2 + b), True))*cos(a) + Piecewise((-log(sin(b*x))/b, Eq(c, pi/2)), (log(sin(b*x))/b, Eq(c, -pi/2)), (x/cos(c)
, Eq(b, 0)), (-b*x*tan(c/2)**2/(b*tan(c/2)**2 + b) + b*x/(b*tan(c/2)**2 + b) + 2*log(tan(b*x/2)**2 + 1)*tan(c/
2)/(b*tan(c/2)**2 + b) - 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)/(b*tan(c/2)**
2 + b) - 2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)/(b*tan(c/2)**2 + b), True))*s
in(a)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (27) = 54\).

Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \sec (c+b x) \sin (a+b x) \, dx=-\frac {2 \, b x \sin \left (-a + c\right ) + \cos \left (-a + c\right ) \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, c\right ) + \cos \left (2 \, c\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, c\right ) + \sin \left (2 \, c\right )^{2}\right )}{2 \, b} \]

[In]

integrate(sec(b*x+c)*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/2*(2*b*x*sin(-a + c) + cos(-a + c)*log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*c) + cos(2*c)^2 + sin(2*b*x)^2 - 2
*sin(2*b*x)*sin(2*c) + sin(2*c)^2))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (27) = 54\).

Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 5.85 \[ \int \sec (c+b x) \sin (a+b x) \, dx=\frac {\frac {4 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} {\left (b x + c\right )}}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left (\tan \left (b x + c\right )^{2} + 1\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1}}{2 \, b} \]

[In]

integrate(sec(b*x+c)*sin(b*x+a),x, algorithm="giac")

[Out]

1/2*(4*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*(b*x + c)/(tan(1/2*a)^2*t
an(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) + (tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a)*tan(
1/2*c) - tan(1/2*c)^2 + 1)*log(tan(b*x + c)^2 + 1)/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 +
1))/b

Mupad [B] (verification not implemented)

Time = 23.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.15 \[ \int \sec (c+b x) \sin (a+b x) \, dx=x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )+x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}}{2}\right )}{b} \]

[In]

int(sin(a + b*x)/cos(c + b*x),x)

[Out]

x*((exp(c*1i - a*1i)*1i)/2 - (exp(a*1i - c*1i)*1i)/2) + x*((exp(c*1i - a*1i)*1i)/2 + (exp(a*1i - c*1i)*1i)/2)
- (log(exp(a*2i - c*2i) + exp(a*2i + b*x*2i))*(exp(c*1i - a*1i)/2 + exp(a*1i - c*1i)/2))/b