3.2.46 \(\int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx\) [146]

Optimal. Leaf size=38 \[ -a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]

[Out]

-a*x+b*cos(d*x+c)/d+b*sec(d*x+c)/d+a*tan(d*x+c)/d

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2801, 3554, 8, 2670, 14} \begin {gather*} \frac {a \tan (c+d x)}{d}-a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x) + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d

Rule 8

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2670

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 2801

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Int[Expan
dIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2
, 0] && IGtQ[m, 0]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

\begin {align*} \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx &=\int \left (a \tan ^2(c+d x)+b \sin (c+d x) \tan ^2(c+d x)\right ) \, dx\\ &=a \int \tan ^2(c+d x) \, dx+b \int \sin (c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {a \tan (c+d x)}{d}-a \int 1 \, dx-\frac {b \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a x+\frac {a \tan (c+d x)}{d}-\frac {b \text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 47, normalized size = 1.24 \begin {gather*} -\frac {a \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*ArcTan[Tan[c + d*x]])/d) + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 59, normalized size = 1.55

method	result	size
derivativedivides	\(\frac {a \left (\tan \left (d x +c \right )-d x -c \right )+b \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\)	\(59\)
default	\(\frac {a \left (\tan \left (d x +c \right )-d x -c \right )+b \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\)	\(59\)
risch	\(-a x +\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i a +2 b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a*(tan(d*x+c)-d*x-c)+b*(sin(d*x+c)^4/cos(d*x+c)+(2+sin(d*x+c)^2)*cos(d*x+c)))

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 39, normalized size = 1.03 \begin {gather*} -\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} a - b {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((d*x + c - tan(d*x + c))*a - b*(1/cos(d*x + c) + cos(d*x + c)))/d

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 47, normalized size = 1.24 \begin {gather*} -\frac {a d x \cos \left (d x + c\right ) - b \cos \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - b}{d \cos \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right ) \tan ^{2}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (38) = 76\).
time = 10.58, size = 1008, normalized size = 26.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) - a*d*x*tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*a*d*x*tan(d*x)*tan
(1/2*d*x)^3*tan(1/2*c)^3*tan(c) - 2*b*tan(d*x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + a*tan(d*x)*tan(1/2*d*x)^4*
tan(1/2*c)^4 + a*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) + 4*a*d*x*tan(1/2*d*x)^3*tan(1/2*c)^3 + 2*b*tan(1/2*d*x)^4
*tan(1/2*c)^4 - a*d*x*tan(d*x)*tan(1/2*d*x)^4*tan(c) - 4*a*d*x*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 4*a
*d*x*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) + 8*b*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) - a*d*x*tan(d
*x)*tan(1/2*c)^4*tan(c) - 4*a*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)^3 - 4*a*tan(1/2*d*x)^3*tan(1/2*c)^3*tan(c) +
a*d*x*tan(1/2*d*x)^4 + 4*a*d*x*tan(1/2*d*x)^3*tan(1/2*c) + 4*a*d*x*tan(1/2*d*x)*tan(1/2*c)^3 - 8*b*tan(1/2*d*x
)^3*tan(1/2*c)^3 + a*d*x*tan(1/2*c)^4 - 2*b*tan(d*x)*tan(1/2*d*x)^4*tan(c) - 4*a*d*x*tan(d*x)*tan(1/2*d*x)*tan
(1/2*c)*tan(c) - 8*b*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 24*b*tan(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan
(c) - 8*b*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - 2*b*tan(d*x)*tan(1/2*c)^4*tan(c) - a*tan(d*x)*tan(1/2*d*
x)^4 - 4*a*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c) - 4*a*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3 - a*tan(d*x)*tan(1/2*c)
^4 - a*tan(1/2*d*x)^4*tan(c) - 4*a*tan(1/2*d*x)^3*tan(1/2*c)*tan(c) - 4*a*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - a
*tan(1/2*c)^4*tan(c) + 2*b*tan(1/2*d*x)^4 + 4*a*d*x*tan(1/2*d*x)*tan(1/2*c) + 8*b*tan(1/2*d*x)^3*tan(1/2*c) +
24*b*tan(1/2*d*x)^2*tan(1/2*c)^2 + 8*b*tan(1/2*d*x)*tan(1/2*c)^3 + 2*b*tan(1/2*c)^4 + a*d*x*tan(d*x)*tan(c) +
8*b*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)*tan(c) - 4*a*tan(d*x)*tan(1/2*d*x)*tan(1/2*c) - 4*a*tan(1/2*d*x)*tan(1/2*
c)*tan(c) - a*d*x - 8*b*tan(1/2*d*x)*tan(1/2*c) - 2*b*tan(d*x)*tan(c) + a*tan(d*x) + a*tan(c) + 2*b)/(d*tan(d*
x)*tan(1/2*d*x)^4*tan(1/2*c)^4*tan(c) - d*tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*d*tan(d*x)*tan(1/2*d*x)^3*tan(1/2*c)
^3*tan(c) + 4*d*tan(1/2*d*x)^3*tan(1/2*c)^3 - d*tan(d*x)*tan(1/2*d*x)^4*tan(c) - 4*d*tan(d*x)*tan(1/2*d*x)^3*t
an(1/2*c)*tan(c) - 4*d*tan(d*x)*tan(1/2*d*x)*tan(1/2*c)^3*tan(c) - d*tan(d*x)*tan(1/2*c)^4*tan(c) + d*tan(1/2*
d*x)^4 + 4*d*tan(1/2*d*x)^3*tan(1/2*c) + 4*d*tan(1/2*d*x)*tan(1/2*c)^3 + d*tan(1/2*c)^4 - 4*d*tan(d*x)*tan(1/2
*d*x)*tan(1/2*c)*tan(c) + 4*d*tan(1/2*d*x)*tan(1/2*c) + d*tan(d*x)*tan(c) - d)

________________________________________________________________________________________

Mupad [B]
time = 6.60, size = 55, normalized size = 1.45 \begin {gather*} -a\,x-\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,b}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- a*x - (4*b + 2*a*tan(c/2 + (d*x)/2) + 2*a*tan(c/2 + (d*x)/2)^3)/(d*(tan(c/2 + (d*x)/2)^4 - 1))

________________________________________________________________________________________