3.1.5 \(\int \tan ^5(c+d x) \, dx\) [5]

Optimal. Leaf size=43 \[ -\frac {\log (\cos (c+d x))}{d}-\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d} \]

[Out]

-ln(cos(d*x+c))/d-1/2*tan(d*x+c)^2/d+1/4*tan(d*x+c)^4/d

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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, 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 ^4(c+d x)}{4 d}-\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]^5,x]

[Out]

-(Log[Cos[c + d*x]]/d) - Tan[c + d*x]^2/(2*d) + Tan[c + d*x]^4/(4*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 ^5(c+d x) \, dx &=\frac {\tan ^4(c+d x)}{4 d}-\int \tan ^3(c+d x) \, dx\\ &=-\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d}+\int \tan (c+d x) \, dx\\ &=-\frac {\log (\cos (c+d x))}{d}-\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 37, normalized size = 0.86 \begin {gather*} -\frac {4 \log (\cos (c+d x))+2 \tan ^2(c+d x)-\tan ^4(c+d x)}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5,x]

[Out]

-1/4*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4)/d

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Maple [A]
time = 0.03, size = 39, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(39\)
default	\(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(39\)
norman	\(-\frac {\tan ^{2}\left (d x +c \right )}{2 d}+\frac {\tan ^{4}\left (d x +c \right )}{4 d}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(44\)
risch	\(i x +\frac {2 i c}{d}-\frac {4 \left ({\mathrm e}^{6 i \left (d x +c \right )}+{\mathrm e}^{4 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 54, normalized size = 1.26 \begin {gather*} \frac {\frac {4 \, \sin \left (d x + c\right )^{2} - 3}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5,x, algorithm="maxima")

[Out]

1/4*((4*sin(d*x + c)^2 - 3)/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 2*log(sin(d*x + c)^2 - 1))/d

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Fricas [A]
time = 0.37, size = 39, normalized size = 0.91 \begin {gather*} \frac {\tan \left (d x + c\right )^{4} - 2 \, \tan \left (d x + c\right )^{2} - 2 \, \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.10, size = 44, normalized size = 1.02 \begin {gather*} \begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {\tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan ^{5}{\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)**5,x)

[Out]

Piecewise((log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**4/(4*d) - tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*tan(c
)**5, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (39) = 78\).
time = 1.50, size = 512, normalized size = 11.91 \begin {gather*} -\frac {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 )^{4} \tan \left (c\right )^{4} + 3 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 8 \, \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 )^{3} \tan \left (c\right )^{3} + 2 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 8 \, \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 2 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{4} + 12 \, \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 )^{4} - 8 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + 4 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 8 \, \tan \left (d x\right ) \tan \left (c\right )^{3} - \tan \left (c\right )^{4} - 8 \, \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 \, \tan \left (d x\right )^{2} - 8 \, \tan \left (d x\right ) \tan \left (c\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 ) + 3}{4 \, {\left (d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 4 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 6 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 4 \, 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)^5,x, algorithm="giac")

[Out]

-1/4*(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)^4*tan(c)^4 + 3*tan(d*x)^4*tan(c)^4 - 8*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)^3*tan(c)^3 + 2*
tan(d*x)^4*tan(c)^2 - 8*tan(d*x)^3*tan(c)^3 + 2*tan(d*x)^2*tan(c)^4 + 12*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)^4 - 8*tan(d*x)^3*tan(c) + 4*tan(d*x)^2*tan(c)^2 - 8*tan(d*x)*tan(c)^3 - tan(c)^4 - 8*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))*t
an(d*x)*tan(c) + 2*tan(d*x)^2 - 8*tan(d*x)*tan(c) + 2*tan(c)^2 + 2*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*t
an(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1)) + 3)/(d*tan(d*x)^4*tan(c)^4
- 4*d*tan(d*x)^3*tan(c)^3 + 6*d*tan(d*x)^2*tan(c)^2 - 4*d*tan(d*x)*tan(c) + d)

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Mupad [B]
time = 2.50, size = 38, normalized size = 0.88 \begin {gather*} \frac {\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{4}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^5,x)

[Out]

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

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