3.1.88 \(\int \frac {\tan ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [88]

Optimal. Leaf size=65 \[ \frac {\log (\cos (c+d x))}{a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {3 \sec ^2(c+d x)}{2 a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d} \]

[Out]

ln(cos(d*x+c))/a^3/d+3*sec(d*x+c)/a^3/d-3/2*sec(d*x+c)^2/a^3/d+1/3*sec(d*x+c)^3/a^3/d

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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3964, 45} \begin {gather*} \frac {\sec ^3(c+d x)}{3 a^3 d}-\frac {3 \sec ^2(c+d x)}{2 a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}+\frac {\log (\cos (c+d x))}{a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]

[Out]

Log[Cos[c + d*x]]/(a^3*d) + (3*Sec[c + d*x])/(a^3*d) - (3*Sec[c + d*x]^2)/(2*a^3*d) + Sec[c + d*x]^3/(3*a^3*d)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 3964

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {\tan ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-a x)^3}{x^4} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {3 a^3}{x^3}+\frac {3 a^3}{x^2}-\frac {a^3}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac {\log (\cos (c+d x))}{a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {3 \sec ^2(c+d x)}{2 a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 64, normalized size = 0.98 \begin {gather*} \frac {(22+18 \cos (2 (c+d x))+9 \cos (c+d x) (-2+\log (\cos (c+d x)))+3 \cos (3 (c+d x)) \log (\cos (c+d x))) \sec ^3(c+d x)}{12 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]

[Out]

((22 + 18*Cos[2*(c + d*x)] + 9*Cos[c + d*x]*(-2 + Log[Cos[c + d*x]]) + 3*Cos[3*(c + d*x)]*Log[Cos[c + d*x]])*S
ec[c + d*x]^3)/(12*a^3*d)

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Maple [A]
time = 0.10, size = 45, normalized size = 0.69

method	result	size
derivativedivides	\(-\frac {-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sec ^{2}\left (d x +c \right )\right )}{2}-3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}\)	\(45\)
default	\(-\frac {-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sec ^{2}\left (d x +c \right )\right )}{2}-3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}\)	\(45\)
risch	\(-\frac {i x}{a^{3}}-\frac {2 i c}{a^{3} d}+\frac {6 \,{\mathrm e}^{5 i \left (d x +c \right )}-6 \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {44 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{3} d}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^7/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/d/a^3*(-1/3*sec(d*x+c)^3+3/2*sec(d*x+c)^2-3*sec(d*x+c)+ln(sec(d*x+c)))

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Maxima [A]
time = 0.27, size = 50, normalized size = 0.77 \begin {gather*} \frac {\frac {6 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} + \frac {18 \, \cos \left (d x + c\right )^{2} - 9 \, \cos \left (d x + c\right ) + 2}{a^{3} \cos \left (d x + c\right )^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/6*(6*log(cos(d*x + c))/a^3 + (18*cos(d*x + c)^2 - 9*cos(d*x + c) + 2)/(a^3*cos(d*x + c)^3))/d

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Fricas [A]
time = 3.67, size = 55, normalized size = 0.85 \begin {gather*} \frac {6 \, \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) + 18 \, \cos \left (d x + c\right )^{2} - 9 \, \cos \left (d x + c\right ) + 2}{6 \, a^{3} d \cos \left (d x + c\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/6*(6*cos(d*x + c)^3*log(-cos(d*x + c)) + 18*cos(d*x + c)^2 - 9*cos(d*x + c) + 2)/(a^3*d*cos(d*x + c)^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\tan ^{7}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**7/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(tan(c + d*x)**7/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (61) = 122\).
time = 3.72, size = 158, normalized size = 2.43 \begin {gather*} -\frac {\frac {6 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} - \frac {6 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3}} - \frac {\frac {75 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {51 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {11 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 29}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^7/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/6*(6*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^3 - 6*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) - 1))/a^3 - (75*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 51*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1
1*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 29)/(a^3*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^3))/d

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Mupad [B]
time = 2.03, size = 109, normalized size = 1.68 \begin {gather*} -\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {20}{3}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )}-\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^7/(a + a/cos(c + d*x))^3,x)

[Out]

- (14*tan(c/2 + (d*x)/2)^4 - 18*tan(c/2 + (d*x)/2)^2 + 20/3)/(d*(3*a^3*tan(c/2 + (d*x)/2)^2 - 3*a^3*tan(c/2 +
(d*x)/2)^4 + a^3*tan(c/2 + (d*x)/2)^6 - a^3)) - (2*atanh(tan(c/2 + (d*x)/2)^2))/(a^3*d)

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