3.5.15 \(\int \tan ^2(c+d x) (a+b \tan (c+d x)) \, dx\) [415]

Optimal. Leaf size=44 \[ -a x+\frac {b \log (\cos (c+d x))}{d}+\frac {a \tan (c+d x)}{d}+\frac {b \tan ^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3609, 3606, 3556} \begin {gather*} \frac {a \tan (c+d x)}{d}-a x+\frac {b \tan ^2(c+d x)}{2 d}+\frac {b \log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x) + (b*Log[Cos[c + d*x]])/d + (a*Tan[c + d*x])/d + (b*Tan[c + d*x]^2)/(2*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]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \tan ^2(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \tan ^2(c+d x)}{2 d}+\int \tan (c+d x) (-b+a \tan (c+d x)) \, dx\\ &=-a x+\frac {a \tan (c+d x)}{d}+\frac {b \tan ^2(c+d x)}{2 d}-b \int \tan (c+d x) \, dx\\ &=-a x+\frac {b \log (\cos (c+d x))}{d}+\frac {a \tan (c+d x)}{d}+\frac {b \tan ^2(c+d x)}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 51, normalized size = 1.16 \begin {gather*} -\frac {a \text {ArcTan}(\tan (c+d x))}{d}+\frac {a \tan (c+d x)}{d}+\frac {b \left (2 \log (\cos (c+d x))+\tan ^2(c+d x)\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*ArcTan[Tan[c + d*x]])/d) + (a*Tan[c + d*x])/d + (b*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)

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Maple [A]
time = 0.02, size = 49, normalized size = 1.11

method	result	size
norman	\(\frac {a \tan \left (d x +c \right )}{d}-a x +\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(48\)
derivativedivides	\(\frac {\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+a \tan \left (d x +c \right )-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(49\)
default	\(\frac {\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+a \tan \left (d x +c \right )-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(49\)
risch	\(-i b x -a x -\frac {2 i b c}{d}+\frac {2 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 i a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 47, normalized size = 1.07 \begin {gather*} \frac {b \tan \left (d x + c\right )^{2} - 2 \, {\left (d x + c\right )} a - b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \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)^2*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.34, size = 47, normalized size = 1.07 \begin {gather*} -\frac {2 \, a d x - b \tan \left (d x + c\right )^{2} - b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, a \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)^2*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.08, size = 56, normalized size = 1.27 \begin {gather*} \begin {cases} - a x + \frac {a \tan {\left (c + d x \right )}}{d} - \frac {b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \tan ^{2}{\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)**2*(a+b*tan(d*x+c)),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (42) = 84\).
time = 0.72, size = 327, normalized size = 7.43 \begin {gather*} -\frac {2 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - b \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 )^{2} \tan \left (c\right )^{2} - 4 \, a d x \tan \left (d x\right ) \tan \left (c\right ) - b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 2 \, b \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 \, a \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, a \tan \left (d x\right ) \tan \left (c\right )^{2} + 2 \, a d x - b \tan \left (d x\right )^{2} - b \tan \left (c\right )^{2} - b \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 \, a \tan \left (d x\right ) - 2 \, a \tan \left (c\right ) - b}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, 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)^2*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.05, size = 43, normalized size = 0.98 \begin {gather*} \frac {a\,\mathrm {tan}\left (c+d\,x\right )-\frac {b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-a\,d\,x}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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