3.1.7 \(\int \tan ^7(c+d x) \, dx\) [7]

Optimal. Leaf size=57 \[ \frac {\log (\cos (c+d x))}{d}+\frac {\tan ^2(c+d x)}{2 d}-\frac {\tan ^4(c+d x)}{4 d}+\frac {\tan ^6(c+d x)}{6 d} \]

[Out]

ln(cos(d*x+c))/d+1/2*tan(d*x+c)^2/d-1/4*tan(d*x+c)^4/d+1/6*tan(d*x+c)^6/d

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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, 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 ^6(c+d x)}{6 d}-\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]^7,x]

[Out]

Log[Cos[c + d*x]]/d + Tan[c + d*x]^2/(2*d) - Tan[c + d*x]^4/(4*d) + Tan[c + d*x]^6/(6*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 ^7(c+d x) \, dx &=\frac {\tan ^6(c+d x)}{6 d}-\int \tan ^5(c+d x) \, dx\\ &=-\frac {\tan ^4(c+d x)}{4 d}+\frac {\tan ^6(c+d x)}{6 d}+\int \tan ^3(c+d x) \, dx\\ &=\frac {\tan ^2(c+d x)}{2 d}-\frac {\tan ^4(c+d x)}{4 d}+\frac {\tan ^6(c+d x)}{6 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}+\frac {\tan ^6(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 47, normalized size = 0.82 \begin {gather*} \frac {12 \log (\cos (c+d x))+6 \tan ^2(c+d x)-3 \tan ^4(c+d x)+2 \tan ^6(c+d x)}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^7,x]

[Out]

(12*Log[Cos[c + d*x]] + 6*Tan[c + d*x]^2 - 3*Tan[c + d*x]^4 + 2*Tan[c + d*x]^6)/(12*d)

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Maple [A]
time = 0.04, size = 49, normalized size = 0.86

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(49\)
default	\(\frac {\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(49\)
norman	\(\frac {\tan ^{2}\left (d x +c \right )}{2 d}-\frac {\tan ^{4}\left (d x +c \right )}{4 d}+\frac {\tan ^{6}\left (d x +c \right )}{6 d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(57\)
risch	\(-i x -\frac {2 i c}{d}+\frac {6 \,{\mathrm e}^{10 i \left (d x +c \right )}+12 \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {68 \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+12 \,{\mathrm e}^{4 i \left (d x +c \right )}+6 \,{\mathrm e}^{2 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/6*tan(d*x+c)^6-1/4*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 = 74, normalized size = 1.30 \begin {gather*} -\frac {\frac {18 \, \sin \left (d x + c\right )^{4} - 27 \, \sin \left (d x + c\right )^{2} + 11}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 6 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7,x, algorithm="maxima")

[Out]

-1/12*((18*sin(d*x + c)^4 - 27*sin(d*x + c)^2 + 11)/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1)
- 6*log(sin(d*x + c)^2 - 1))/d

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Fricas [A]
time = 0.36, size = 51, normalized size = 0.89 \begin {gather*} \frac {2 \, \tan \left (d x + c\right )^{6} - 3 \, \tan \left (d x + c\right )^{4} + 6 \, \tan \left (d x + c\right )^{2} + 6 \, \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.16, size = 56, normalized size = 0.98 \begin {gather*} \begin {cases} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{6}{\left (c + d x \right )}}{6 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 ^{7}{\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)**7,x)

[Out]

Piecewise((-log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**6/(6*d) - tan(c + d*x)**4/(4*d) + tan(c + d*x)**2/(
2*d), Ne(d, 0)), (x*tan(c)**7, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 810 vs. \(2 (51) = 102\).
time = 3.25, size = 810, normalized size = 14.21 \begin {gather*} \frac {6 \, \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 )^{6} \tan \left (c\right )^{6} + 11 \, \tan \left (d x\right )^{6} \tan \left (c\right )^{6} - 36 \, \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 )^{5} \tan \left (c\right )^{5} + 6 \, \tan \left (d x\right )^{6} \tan \left (c\right )^{4} - 54 \, \tan \left (d x\right )^{5} \tan \left (c\right )^{5} + 6 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{6} + 90 \, \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 )^{6} \tan \left (c\right )^{2} - 36 \, \tan \left (d x\right )^{5} \tan \left (c\right )^{3} + 99 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 36 \, \tan \left (d x\right )^{3} \tan \left (c\right )^{5} - 3 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{6} - 120 \, \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 )^{6} + 18 \, \tan \left (d x\right )^{5} \tan \left (c\right ) + 90 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 72 \, \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 90 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{4} + 18 \, \tan \left (d x\right ) \tan \left (c\right )^{5} + 2 \, \tan \left (c\right )^{6} + 90 \, \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} - 3 \, \tan \left (d x\right )^{4} - 36 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + 99 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 36 \, \tan \left (d x\right ) \tan \left (c\right )^{3} - 3 \, \tan \left (c\right )^{4} - 36 \, \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 ) + 6 \, \tan \left (d x\right )^{2} - 54 \, \tan \left (d x\right ) \tan \left (c\right ) + 6 \, \tan \left (c\right )^{2} + 6 \, \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 ) + 11}{12 \, {\left (d \tan \left (d x\right )^{6} \tan \left (c\right )^{6} - 6 \, d \tan \left (d x\right )^{5} \tan \left (c\right )^{5} + 15 \, d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 20 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 15 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 6 \, 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)^7,x, algorithm="giac")

[Out]

1/12*(6*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)^6*tan(c)^6 + 11*tan(d*x)^6*tan(c)^6 - 36*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)^5*tan(c)^5 +
6*tan(d*x)^6*tan(c)^4 - 54*tan(d*x)^5*tan(c)^5 + 6*tan(d*x)^4*tan(c)^6 + 90*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)^6*tan(c)^2 - 36*tan(d*x)^5*tan(c)^3 + 99*tan(d*x)^4*tan(c)^4 - 36*tan(d*x)^3*tan(c)^5 - 3*tan(d*
x)^2*tan(c)^6 - 120*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*ta
n(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 + 2*tan(d*x)^6 + 18*tan(d*x)^5*tan(c) + 90*tan(d*x)^4*t
an(c)^2 - 72*tan(d*x)^3*tan(c)^3 + 90*tan(d*x)^2*tan(c)^4 + 18*tan(d*x)*tan(c)^5 + 2*tan(c)^6 + 90*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 - 3*tan(d*x)^4 - 36*tan(d*x)^3*tan(c) + 99*tan(d*x)^2*tan(c)^2 - 36*tan(d*x)*tan(c)^3
- 3*tan(c)^4 - 36*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) + 6*tan(d*x)^2 - 54*tan(d*x)*tan(c) + 6*tan(c)^2 + 6*log(4*(t
an(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)) + 11)/(d*tan(d*x)^6*tan(c)^6 - 6*d*tan(d*x)^5*tan(c)^5 + 15*d*tan(d*x)^4*tan(c)^4 - 20*d*tan(d*x)^3*tan
(c)^3 + 15*d*tan(d*x)^2*tan(c)^2 - 6*d*tan(d*x)*tan(c) + d)

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Mupad [B]
time = 2.49, size = 49, normalized size = 0.86 \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}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{6}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^7,x)

[Out]

-(log(tan(c + d*x)^2 + 1)/2 - tan(c + d*x)^2/2 + tan(c + d*x)^4/4 - tan(c + d*x)^6/6)/d

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