3.3.31 \(\int \sin (a+b x) \tan ^2(c+b x) \, dx\) [231]

3.3.31.1 Optimal result

Integrand size = 15, antiderivative size = 44 \[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=\frac {\cos (a+b x)}{b}+\frac {\cos (a-c) \sec (c+b x)}{b}+\frac {\text {arctanh}(\sin (c+b x)) \sin (a-c)}{b} \]

output

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

3.3.31.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.48 \[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=\frac {\cos (a) \cos (b x)}{b}+\frac {\cos (a-c) \sec (c+b x)}{b}-\frac {2 i \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (\cos \left (\frac {b x}{2}\right ) \sin (c)+\cos (c) \sin \left (\frac {b x}{2}\right )\right )}{\cos (c) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \sin (a-c)}{b}-\frac {\sin (a) \sin (b x)}{b} \]

input

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

output

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

3.3.31.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {5087, 3042, 3086, 24, 5090, 3042, 3118, 4257}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin (a+b x) \tan ^2(b x+c) \, dx\)

\(\Big \downarrow \) 5087

\(\displaystyle \cos (a-c) \int \sec (c+b x) \tan (c+b x)dx-\int \cos (a+b x) \tan (c+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \cos (a-c) \int \sec (c+b x) \tan (c+b x)dx-\int \cos (a+b x) \tan (c+b x)dx\)

\(\Big \downarrow \) 3086

\(\displaystyle \frac {\cos (a-c) \int 1d\sec (c+b x)}{b}-\int \cos (a+b x) \tan (c+b x)dx\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\cos (a-c) \sec (b x+c)}{b}-\int \cos (a+b x) \tan (c+b x)dx\)

\(\Big \downarrow \) 5090

\(\displaystyle \sin (a-c) \int \sec (c+b x)dx-\int \sin (a+b x)dx+\frac {\cos (a-c) \sec (b x+c)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \sin (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )dx-\int \sin (a+b x)dx+\frac {\cos (a-c) \sec (b x+c)}{b}\)

\(\Big \downarrow \) 3118

\(\displaystyle \sin (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )dx+\frac {\cos (a-c) \sec (b x+c)}{b}+\frac {\cos (a+b x)}{b}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\sin (a-c) \text {arctanh}(\sin (b x+c))}{b}+\frac {\cos (a-c) \sec (b x+c)}{b}+\frac {\cos (a+b x)}{b}\)

input

Int[Sin[a + b*x]*Tan[c + b*x]^2,x]
 

output

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

3.3.31.3.1 Defintions of rubi rules used

rule 24

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[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 3118

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

rule 4257

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

rule 5087

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

rule 5090

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

3.3.31.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.25

method	result	size
risch	\(\frac {{\mathrm e}^{i \left (x b +a \right )}}{2 b}+\frac {{\mathrm e}^{-i \left (x b +a \right )}}{2 b}+\frac {{\mathrm e}^{i \left (x b +3 a \right )}+{\mathrm e}^{i \left (x b +a +2 c \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\)	\(143\)

input

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

output

1/2*exp(I*(b*x+a))/b+1/2/b*exp(-I*(b*x+a))+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)
 

3.3.31.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (44) = 88\).

Time = 0.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 7.16 \[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=-\frac {4 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right )^{2} - 4 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {\sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \sin \left (b x + a\right )\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) - 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} + 4 \, \cos \left (-2 \, a + 2 \, c\right ) + 4}{4 \, {\left (b \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - {\left (b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \cos \left (b x + a\right )\right )}} \]

input

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

output

-1/4*(4*(cos(-2*a + 2*c) + 1)*cos(b*x + a)^2 - 4*cos(b*x + a)*sin(b*x + a) 
*sin(-2*a + 2*c) + sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a)*sin(-2*a + 
2*c) + (cos(-2*a + 2*c)^2 - 1)*sin(b*x + a))*log(-(2*cos(b*x + a)^2*cos(-2 
*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) + 2*sqrt(2)*((cos( 
-2*a + 2*c) + 1)*sin(b*x + a) + cos(b*x + a)*sin(-2*a + 2*c))/sqrt(cos(-2* 
a + 2*c) + 1) - cos(-2*a + 2*c) - 3)/(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2 
*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - cos(-2*a + 2*c) + 1))/sqrt(co 
s(-2*a + 2*c) + 1) + 4*cos(-2*a + 2*c) + 4)/(b*sin(b*x + a)*sin(-2*a + 2*c 
) - (b*cos(-2*a + 2*c) + b)*cos(b*x + a))
 

3.3.31.6 Sympy [F]

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

input

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

output

Integral(sin(a + b*x)*tan(b*x + c)**2, x)
 

3.3.31.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (44) = 88\).

Time = 0.43 (sec) , antiderivative size = 520, normalized size of antiderivative = 11.82 \[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=\frac {{\left (\cos \left (3 \, b x + a + 2 \, c\right ) + \cos \left (b x + a\right )\right )} \cos \left (4 \, b x + 2 \, a + 2 \, c\right ) + {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) + 1\right )} \cos \left (3 \, b x + a + 2 \, c\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) \cos \left (b x + a\right ) + {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) + 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \sin \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \sin \left (-a + c\right ) + \sin \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) + 2 \, \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) \sin \left (-a + c\right ) + \sin \left (b x + a\right )^{2} \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 ) + {\left (\sin \left (3 \, b x + a + 2 \, c\right ) + \sin \left (b x + a\right )\right )} \sin \left (4 \, b x + 2 \, a + 2 \, c\right ) + 3 \, {\left (\sin \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, c\right )\right )} \sin \left (3 \, b x + a + 2 \, c\right ) + 3 \, \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 3 \, \sin \left (2 \, b x + 2 \, c\right ) \sin \left (b x + a\right ) + \cos \left (b x + a\right )}{2 \, {\left (b \cos \left (3 \, b x + a + 2 \, c\right )^{2} + 2 \, b \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) + b \cos \left (b x + a\right )^{2} + b \sin \left (3 \, b x + a + 2 \, c\right )^{2} + 2 \, b \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) + b \sin \left (b x + a\right )^{2}\right )}} \]

input

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

output

1/2*((cos(3*b*x + a + 2*c) + cos(b*x + a))*cos(4*b*x + 2*a + 2*c) + (3*cos 
(2*b*x + 2*a) + 3*cos(2*b*x + 2*c) + 1)*cos(3*b*x + a + 2*c) + 3*cos(2*b*x 
 + 2*a)*cos(b*x + a) + 3*cos(2*b*x + 2*c)*cos(b*x + a) + (cos(3*b*x + a + 
2*c)^2*sin(-a + c) + 2*cos(3*b*x + a + 2*c)*cos(b*x + a)*sin(-a + c) + cos 
(b*x + a)^2*sin(-a + c) + sin(3*b*x + a + 2*c)^2*sin(-a + c) + 2*sin(3*b*x 
 + a + 2*c)*sin(b*x + a)*sin(-a + c) + sin(b*x + a)^2*sin(-a + c))*log((co 
s(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)) 
+ (sin(3*b*x + a + 2*c) + sin(b*x + a))*sin(4*b*x + 2*a + 2*c) + 3*(sin(2* 
b*x + 2*a) + sin(2*b*x + 2*c))*sin(3*b*x + a + 2*c) + 3*sin(2*b*x + 2*a)*s 
in(b*x + a) + 3*sin(2*b*x + 2*c)*sin(b*x + a) + cos(b*x + a))/(b*cos(3*b*x 
 + a + 2*c)^2 + 2*b*cos(3*b*x + a + 2*c)*cos(b*x + a) + b*cos(b*x + a)^2 + 
 b*sin(3*b*x + a + 2*c)^2 + 2*b*sin(3*b*x + a + 2*c)*sin(b*x + a) + b*sin( 
b*x + a)^2)
 

3.3.31.8 Giac [F]

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

input

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

output

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

3.3.31.9 Mupad [B] (verification not implemented)

Time = 27.60 (sec) , antiderivative size = 294, normalized size of antiderivative = 6.68 \[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}+\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}-\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \]

input

int(sin(a + b*x)*tan(c + b*x)^2,x)
 

output

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