3.5.16 \(\int \tan (c+d x) (a+b \tan (c+d x)) \, dx\) [416]

Optimal. Leaf size=29 \[ -b x-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3606, 3556} \begin {gather*} -\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d}-b x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3556

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

Rule 3606

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[b*d*(Tan[e + f*x]/f), x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rubi steps

\begin {align*} \int \tan (c+d x) (a+b \tan (c+d x)) \, dx &=-b x+\frac {b \tan (c+d x)}{d}+a \int \tan (c+d x) \, dx\\ &=-b x-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 38, normalized size = 1.31 \begin {gather*} -\frac {b \text {ArcTan}(\tan (c+d x))}{d}-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 38, normalized size = 1.31

method	result	size
norman	\(\frac {b \tan \left (d x +c \right )}{d}-b x +\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(34\)
derivativedivides	\(\frac {b \tan \left (d x +c \right )+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(38\)
default	\(\frac {b \tan \left (d x +c \right )+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(38\)
risch	\(-b x +i a x +\frac {2 i a c}{d}+\frac {2 i b}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b*tan(d*x+c)+1/2*a*ln(1+tan(d*x+c)^2)-b*arctan(tan(d*x+c)))

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Maxima [A]
time = 0.52, size = 37, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (d x + c\right )} b - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, b \tan \left (d x + c\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.51, size = 35, normalized size = 1.21 \begin {gather*} -\frac {2 \, b d x + a \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, b \tan \left (d x + c\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 41, normalized size = 1.41 \begin {gather*} \begin {cases} \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - b x + \frac {b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \tan {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*log(tan(c + d*x)**2 + 1)/(2*d) - b*x + b*tan(c + d*x)/d, Ne(d, 0)), (x*(a + b*tan(c))*tan(c), Tru
e))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (29) = 58\).
time = 0.72, size = 174, normalized size = 6.00 \begin {gather*} -\frac {2 \, b d x \tan \left (d x\right ) \tan \left (c\right ) + a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) - 2 \, b d x - a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) + 2 \, b \tan \left (d x\right ) + 2 \, b \tan \left (c\right )}{2 \, {\left (d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.06, size = 32, normalized size = 1.10 \begin {gather*} \frac {b\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-b\,d\,x}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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