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
Result contains complex when optimal does not.
Time = 0.07 (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
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
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])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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]
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]
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.25
method | result | size |
risch | \(\frac {{\mathrm e}^{i \left (b x +a \right )}}{2 b}+\frac {{\mathrm e}^{-i \left (b x +a \right )}}{2 b}+\frac {{\mathrm e}^{i \left (b x +3 a \right )}+{\mathrm e}^{i \left (b x +a +2 c \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\) | \(143\) |
Input:
int(sin(b*x+a)*tan(b*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/2/b*exp(I*(b*x+a))+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)
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (44) = 88\).
Time = 0.10 (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))
\[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=\int \sin {\left (a + b x \right )} \tan ^{2}{\left (b x + c \right )}\, dx \] Input:
integrate(sin(b*x+a)*tan(b*x+c)**2,x)
Output:
Integral(sin(a + b*x)*tan(b*x + c)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (44) = 88\).
Time = 0.19 (sec) , antiderivative size = 520, normalized size of antiderivative = 11.82 \[ \int \sin (a+b x) \tan ^2(c+b x) \, dx =\text {Too large to display} \] 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)
\[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=\int { \sin \left (b x + a\right ) \tan \left (b x + c\right )^{2} \,d x } \] 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)
Time = 21.28 (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))
\[ \int \sin (a+b x) \tan ^2(c+b x) \, dx=\int \sin \left (b x +a \right ) \tan \left (b x +c \right )^{2}d x \] Input:
int(sin(b*x+a)*tan(b*x+c)^2,x)
Output:
int(sin(a + b*x)*tan(b*x + c)**2,x)