3.1.3 \(\int \tan ^3(c+d x) \, dx\) [3]

Optimal. Leaf size=27 \[ \frac {\log (\cos (c+d x))}{d}+\frac {\tan ^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \begin {gather*} \frac {\tan ^2(c+d x)}{2 d}+\frac {\log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^3,x]

[Out]

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

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]

Rule 3556

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

Rubi steps

\begin {align*} \int \tan ^3(c+d x) \, dx &=\frac {\tan ^2(c+d x)}{2 d}-\int \tan (c+d x) \, dx\\ &=\frac {\log (\cos (c+d x))}{d}+\frac {\tan ^2(c+d x)}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 25, normalized size = 0.93 \begin {gather*} \frac {2 \log (\cos (c+d x))+\tan ^2(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^3,x]

[Out]

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

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Maple [A]
time = 0.02, size = 29, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(29\)
default	\(\frac {\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(29\)
norman	\(\frac {\tan ^{2}\left (d x +c \right )}{2 d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(31\)
risch	\(-i x -\frac {2 i c}{d}+\frac {2 \,{\mathrm e}^{2 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 31, normalized size = 1.15 \begin {gather*} -\frac {\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3,x, algorithm="maxima")

[Out]

-1/2*(1/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)^2 - 1))/d

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Fricas [A]
time = 0.37, size = 27, normalized size = 1.00 \begin {gather*} \frac {\tan \left (d x + c\right )^{2} + \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.07, size = 32, normalized size = 1.19 \begin {gather*} \begin {cases} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan ^{3}{\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)**3,x)

[Out]

Piecewise((-log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*tan(c)**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (25) = 50\).
time = 0.68, size = 246, normalized size = 9.11 \begin {gather*} \frac {\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} + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \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 ) + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + \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 ) + 1}{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)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.51, size = 30, normalized size = 1.11 \begin {gather*} \frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}-\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^3,x)

[Out]

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

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