\(\int \sec ^2(c+b x) \sin (a+b x) \, dx\) [214]

Optimal result

Integrand size = 15, antiderivative size = 34 \[ \int \sec ^2(c+b x) \sin (a+b x) \, dx=\frac {\cos (a-c) \sec (c+b x)}{b}+\frac {\text {arctanh}(\sin (c+b x)) \sin (a-c)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4676, 2686, 8, 3855} \[ \int \sec ^2(c+b x) \sin (a+b x) \, dx=\frac {\sin (a-c) \text {arctanh}(\sin (b x+c))}{b}+\frac {\cos (a-c) \sec (b x+c)}{b} \]

[In]

Int[Sec[c + b*x]^2*Sin[a + b*x],x]

[Out]

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

Rule 8

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

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*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 \sec (c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec (c+b x) \, dx \\ & = \frac {\text {arctanh}(\sin (c+b x)) \sin (a-c)}{b}+\frac {\cos (a-c) \text {Subst}(\int 1 \, dx,x,\sec (c+b x))}{b} \\ & = \frac {\cos (a-c) \sec (c+b x)}{b}+\frac {\text {arctanh}(\sin (c+b x)) \sin (a-c)}{b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.59 \[ \int \sec ^2(c+b x) \sin (a+b x) \, dx=\frac {\cos (a-c) \sec (c+b x)}{b}-\frac {2 i \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (\cos \left (\frac {b x}{2}\right ) \sin (c)+\cos (c) \sin \left (\frac {b x}{2}\right )\right )}{\cos (c) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \sin (a-c)}{b} \]

[In]

Integrate[Sec[c + b*x]^2*Sin[a + b*x],x]

[Out]

(Cos[a - c]*Sec[c + b*x])/b - ((2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/
(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.95 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.38

method	result	size
risch	\(\frac {{\mathrm e}^{i \left (x b +3 a \right )}+{\mathrm e}^{i \left (x b +a +2 c \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\)	\(115\)
default	\(\frac {\frac {4 \left (2 \sin \left (a \right ) \cos \left (c \right )-2 \cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+8 \cos \left (a \right ) \cos \left (c \right )+8 \sin \left (a \right ) \sin \left (c \right )}{\left (-4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}-4 \cos \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (\cos \left (c \right ) \cos \left (a \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+\sin \left (c \right ) \sin \left (a \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \sin \left (a \right ) \cos \left (c \right )+2 \tan \left (\frac {a}{2}+\frac {x b}{2}\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 )}+\frac {4 \left (2 \sin \left (a \right ) \cos \left (c \right )-2 \cos \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\frac {2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-2 \sin \left (a \right ) \cos \left (c \right )+2 \cos \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\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}}}\right )}{\left (-4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}-4 \cos \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \sqrt {-\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}\)	\(346\)

[In]

int(sec(b*x+c)^2*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b/(exp(2*I*(b*x+a+c))+exp(2*I*a))*(exp(I*(b*x+3*a))+exp(I*(b*x+a+2*c)))-ln(exp(I*(b*x+a))-I*exp(I*(a-c)))/b*
sin(a-c)+ln(exp(I*(b*x+a))+I*exp(I*(a-c)))/b*sin(a-c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (34) = 68\).

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \sec ^2(c+b x) \sin (a+b x) \, dx=-\frac {\cos \left (b x + c\right ) \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - \cos \left (b x + c\right ) \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 2 \, \cos \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (34) = 68\).

Time = 0.42 (sec) , antiderivative size = 387, normalized size of antiderivative = 11.38 \[ \int \sec ^2(c+b x) \sin (a+b x) \, dx=\frac {2 \, {\left (\cos \left (b x + 2 \, a\right ) + \cos \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) + 2 \, \cos \left (b x + 2 \, a\right ) \cos \left (a\right ) + 2 \, \cos \left (b x + 2 \, c\right ) \cos \left (a\right ) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) + 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \sin \left (-a + c\right ) + \sin \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) + 2 \, \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) \sin \left (-a + c\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} \sin \left (-a + c\right )\right )} \log \left (\frac {\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} - 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} + 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}{\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} + 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} - 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}\right ) + 2 \, {\left (\sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) + 2 \, \sin \left (b x + 2 \, a\right ) \sin \left (a\right ) + 2 \, \sin \left (b x + 2 \, c\right ) \sin \left (a\right )}{2 \, {\left (b \cos \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) + b \sin \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} b\right )}} \]

[In]

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

[Out]

1/2*(2*(cos(b*x + 2*a) + cos(b*x + 2*c))*cos(2*b*x + a + 2*c) + 2*cos(b*x + 2*a)*cos(a) + 2*cos(b*x + 2*c)*cos
(a) + (cos(2*b*x + a + 2*c)^2*sin(-a + c) + 2*cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(2*b*x + a + 2*c)^2
*sin(-a + c) + 2*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c) + (cos(a)^2 + sin(a)^2)*sin(-a + c))*log((cos(b*x + 2
*c)^2 + cos(c)^2 - 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 + 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)/(cos(b*x +
2*c)^2 + cos(c)^2 + 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 - 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)) + 2*(si
n(b*x + 2*a) + sin(b*x + 2*c))*sin(2*b*x + a + 2*c) + 2*sin(b*x + 2*a)*sin(a) + 2*sin(b*x + 2*c)*sin(a))/(b*co
s(2*b*x + a + 2*c)^2 + 2*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(2*b*x + a + 2*c)^2 + 2*b*sin(2*b*x + a + 2*c)*s
in(a) + (cos(a)^2 + sin(a)^2)*b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (34) = 68\).

Time = 0.31 (sec) , antiderivative size = 248, normalized size of antiderivative = 7.29 \[ \int \sec ^2(c+b x) \sin (a+b x) \, dx=\frac {2 \, {\left (\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} {\left (\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{2} - 1\right )}}\right )}}{b} \]

[In]

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

[Out]

2*((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))*log(abs(tan(1/2*b*x + 1/2*c)
+ 1))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - (tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*ta
n(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*c) - 1))/(tan(1/2*a)^2*tan(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)/((tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*(tan(1/2*b*x + 1/2*c)^2 - 1)))/b

Mupad [B] (verification not implemented)

Time = 26.08 (sec) , antiderivative size = 254, normalized size of antiderivative = 7.47 \[ \int \sec ^2(c+b x) \sin (a+b x) \, dx=\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}+\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}-\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \]

[In]

int(sin(a + b*x)/cos(c + b*x)^2,x)

[Out]

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