\(\int \sin (a+b x) \tan (a+b x) \, dx\) [62]

Optimal result

Integrand size = 13, antiderivative size = 23 \[ \int \sin (a+b x) \tan (a+b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\sin (a+b x)}{b} \]

[Out]

arctanh(sin(b*x+a))/b-sin(b*x+a)/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, 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 = {2672, 327, 212} \[ \int \sin (a+b x) \tan (a+b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\sin (a+b x)}{b} \]

[In]

Int[Sin[a + b*x]*Tan[a + b*x],x]

[Out]

ArcTanh[Sin[a + b*x]]/b - Sin[a + b*x]/b

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (a+b x)\right )}{b} \\ & = -\frac {\sin (a+b x)}{b}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\sin (a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \sin (a+b x) \tan (a+b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\sin (a+b x)}{b} \]

[In]

Integrate[Sin[a + b*x]*Tan[a + b*x],x]

[Out]

ArcTanh[Sin[a + b*x]]/b - Sin[a + b*x]/b

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {-\sin \left (b x +a \right )+\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\)	\(28\)
default	\(\frac {-\sin \left (b x +a \right )+\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\)	\(28\)
parallelrisch	\(\frac {-\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )-\sin \left (b x +a \right )}{b}\)	\(40\)
norman	\(-\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}\)	\(64\)
risch	\(\frac {i {\mathrm e}^{i \left (b x +a \right )}}{2 b}-\frac {i {\mathrm e}^{-i \left (b x +a \right )}}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{b}\)	\(67\)

[In]

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

[Out]

1/b*(-sin(b*x+a)+ln(sec(b*x+a)+tan(b*x+a)))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \sin (a+b x) \tan (a+b x) \, dx=\frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) - 2 \, \sin \left (b x + a\right )}{2 \, b} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).

Time = 18.82 (sec) , antiderivative size = 3160, normalized size of antiderivative = 137.39 \[ \int \sin (a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((log(tan(a + b*x) + sec(a + b*x))/b, Ne(b, 0)), (x*(tan(a)*sec(a) + sec(a)**2)/(tan(a) + sec(a)), Tr
ue))/2 + 2*Piecewise((-sin(b*x)/b, Eq(a, pi/2)), (sin(b*x)/b, Eq(a, -pi/2)), (0, Eq(b, 0)), (-2*log(tan(b*x/2)
- tan(a/2)/(tan(a/2) - 1) - 1/(tan(a/2) - 1))*tan(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(
a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - tan(a/2)
/(tan(a/2) - 1) - 1/(tan(a/2) - 1))*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2
*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - tan(a/2)/(tan(a/2) - 1) - 1/(tan(
a/2) - 1))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2
+ 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - tan(a/2)/(tan(a/2) - 1) - 1/(tan(a/2) - 1))*tan
(a/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b
*x/2)**2 + b) + 2*log(tan(b*x/2) + tan(a/2)/(tan(a/2) + 1) - 1/(tan(a/2) + 1))*tan(a/2)**3*tan(b*x/2)**2/(b*ta
n(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 +
b) + 2*log(tan(b*x/2) + tan(a/2)/(tan(a/2) + 1) - 1/(tan(a/2) + 1))*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**2 +
b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) + t
an(a/2)/(tan(a/2) + 1) - 1/(tan(a/2) + 1))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4
+ 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) + tan(a/2)/(tan(a
/2) + 1) - 1/(tan(a/2) + 1))*tan(a/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2
)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*tan(a/2)**4/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2
*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*tan(a/2)**3*tan(b*x/2)/(b*tan(a/2)**
4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*t
an(a/2)*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)
**2 + b*tan(b*x/2)**2 + b) - 2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 +
2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b), True))*sin(a)*cos(a) + Piecewise((x/cos(a), Eq(b, 0)), (-log(tan(b*x/2
))/b, Eq(a, pi/2)), (log(tan(b*x/2))/b, Eq(a, -pi/2)), (log(tan(b*x/2) - tan(a/2)/(tan(a/2) - 1) - 1/(tan(a/2)
- 1))/b - log(tan(b*x/2) + tan(a/2)/(tan(a/2) + 1) - 1/(tan(a/2) + 1))/b, True))*cos(a)**2 - Piecewise((x/cos
(a), Eq(b, 0)), (-log(tan(b*x/2))/b, Eq(a, pi/2)), (log(tan(b*x/2))/b, Eq(a, -pi/2)), (log(tan(b*x/2) - tan(a/
2)/(tan(a/2) - 1) - 1/(tan(a/2) - 1))/b - log(tan(b*x/2) + tan(a/2)/(tan(a/2) + 1) - 1/(tan(a/2) + 1))/b, True
))/2 - 2*Piecewise((-log(tan(b*x/2))*tan(b*x/2)**2/(b*tan(b*x/2)**2 + b) - log(tan(b*x/2))/(b*tan(b*x/2)**2 +
b) - 2/(b*tan(b*x/2)**2 + b), Eq(a, pi/2)), (log(tan(b*x/2))*tan(b*x/2)**2/(b*tan(b*x/2)**2 + b) + log(tan(b*x
/2))/(b*tan(b*x/2)**2 + b) + 2/(b*tan(b*x/2)**2 + b), Eq(a, -pi/2)), (x/cos(a), Eq(b, 0)), (4*log(tan(b*x/2) -
tan(a/2)/(tan(a/2) - 1) - 1/(tan(a/2) - 1))*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/
2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 4*log(tan(b*x/2) - tan(a/2)/(
tan(a/2) - 1) - 1/(tan(a/2) - 1))*tan(a/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*t
an(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*log(tan(b*x/2) + tan(a/2)/(tan(a/2) + 1) - 1/(tan(a/
2) + 1))*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**
2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*log(tan(b*x/2) + tan(a/2)/(tan(a/2) + 1) - 1/(tan(a/2) + 1))*ta
n(a/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*t
an(b*x/2)**2 + b) - 2*tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*ta
n(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)
**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*tan(a/2)/(b*tan(a/2)**4*tan(b
*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*tan(b*x/
2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/
2)**2 + b), True))*cos(a)**2 + Piecewise((-log(tan(b*x/2))*tan(b*x/2)**2/(b*tan(b*x/2)**2 + b) - log(tan(b*x/2
))/(b*tan(b*x/2)**2 + b) - 2/(b*tan(b*x/2)**2 + b), Eq(a, pi/2)), (log(tan(b*x/2))*tan(b*x/2)**2/(b*tan(b*x/2)
**2 + b) + log(tan(b*x/2))/(b*tan(b*x/2)**2 + b) + 2/(b*tan(b*x/2)**2 + b), Eq(a, -pi/2)), (x/cos(a), Eq(b, 0)
), (4*log(tan(b*x/2) - tan(a/2)/(tan(a/2) - 1) - 1/(tan(a/2) - 1))*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**4*ta
n(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 4*log(t
an(b*x/2) - tan(a/2)/(tan(a/2) - 1) - 1/(tan(a/2) - 1))*tan(a/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)*
*4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*log(tan(b*x/2) + tan(a/2)/(tan
(a/2) + 1) - 1/(tan(a/2) + 1))*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*ta
n(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*log(tan(b*x/2) + tan(a/2)/(tan(a/2) + 1)
- 1/(tan(a/2) + 1))*tan(a/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 +
2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**
4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*tan(a/2)**3/(b*tan(a/2)**4*tan(
b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*tan(a/2
)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2
)**2 + b) + 2*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*ta
n(a/2)**2 + b*tan(b*x/2)**2 + b), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \sin (a+b x) \tan (a+b x) \, dx=\frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right ) - 2 \, \sin \left (b x + a\right )}{2 \, b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \sin (a+b x) \tan (a+b x) \, dx=\frac {\log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right ) - 2 \, \sin \left (b x + a\right )}{2 \, b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \sin (a+b x) \tan (a+b x) \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{b}-\frac {\sin \left (a+b\,x\right )}{b} \]

[In]

int(sin(a + b*x)^2/cos(a + b*x),x)

[Out]

(2*atanh(tan(a/2 + (b*x)/2)))/b - sin(a + b*x)/b