3.39 \(\int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx\)

Optimal. Leaf size=138 \[ \frac{a^3 \sec ^7(c+d x)}{7 d}+\frac{a^3 \sec ^6(c+d x)}{2 d}+\frac{a^3 \sec ^5(c+d x)}{5 d}-\frac{5 a^3 \sec ^4(c+d x)}{4 d}-\frac{5 a^3 \sec ^3(c+d x)}{3 d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d} \]

[Out]

-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d) - (5*a^3*Sec[c + d*x]^3)/(3
*d) - (5*a^3*Sec[c + d*x]^4)/(4*d) + (a^3*Sec[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]^6)/(2*d) + (a^3*Sec[c + d*
x]^7)/(7*d)

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Rubi [A]  time = 0.0772046, antiderivative size = 138, normalized size of antiderivative = 1., 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 = {3879, 88} \[ \frac{a^3 \sec ^7(c+d x)}{7 d}+\frac{a^3 \sec ^6(c+d x)}{2 d}+\frac{a^3 \sec ^5(c+d x)}{5 d}-\frac{5 a^3 \sec ^4(c+d x)}{4 d}-\frac{5 a^3 \sec ^3(c+d x)}{3 d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^5,x]

[Out]

-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d) - (5*a^3*Sec[c + d*x]^3)/(3
*d) - (5*a^3*Sec[c + d*x]^4)/(4*d) + (a^3*Sec[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]^6)/(2*d) + (a^3*Sec[c + d*
x]^7)/(7*d)

Rule 3879

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]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^2 (a+a x)^5}{x^8} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^7}{x^8}+\frac{3 a^7}{x^7}+\frac{a^7}{x^6}-\frac{5 a^7}{x^5}-\frac{5 a^7}{x^4}+\frac{a^7}{x^3}+\frac{3 a^7}{x^2}+\frac{a^7}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac{a^3 \log (\cos (c+d x))}{d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}-\frac{5 a^3 \sec ^3(c+d x)}{3 d}-\frac{5 a^3 \sec ^4(c+d x)}{4 d}+\frac{a^3 \sec ^5(c+d x)}{5 d}+\frac{a^3 \sec ^6(c+d x)}{2 d}+\frac{a^3 \sec ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.419514, size = 140, normalized size = 1.01 \[ -\frac{a^3 \sec ^7(c+d x) (-4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))-2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))-630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (35 \log (\cos (c+d x))+8)-3732)}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^5,x]

[Out]

-(a^3*(-3732 - 4522*Cos[2*(c + d*x)] + 1050*Cos[3*(c + d*x)] - 2380*Cos[4*(c + d*x)] - 210*Cos[5*(c + d*x)] -
630*Cos[6*(c + d*x)] + 2205*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 735*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 105*
Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[c + d*x]*(8 + 35*Log[Cos[c + d*x]]))*Sec[c + d*x]^7)/(6720*d)

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Maple [A]  time = 0.051, size = 227, normalized size = 1.6 \begin{align*}{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{22\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{22\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{105\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{22\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{35\,d\cos \left ( dx+c \right ) }}+{\frac{176\,{a}^{3}\cos \left ( dx+c \right ) }{105\,d}}+{\frac{22\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d}}+{\frac{88\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{105\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{7\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x)

[Out]

1/4/d*a^3*tan(d*x+c)^4-1/2/d*a^3*tan(d*x+c)^2-a^3*ln(cos(d*x+c))/d+22/35/d*a^3*sin(d*x+c)^6/cos(d*x+c)^5-22/10
5/d*a^3*sin(d*x+c)^6/cos(d*x+c)^3+22/35/d*a^3*sin(d*x+c)^6/cos(d*x+c)+176/105*a^3*cos(d*x+c)/d+22/35/d*a^3*cos
(d*x+c)*sin(d*x+c)^4+88/105/d*a^3*cos(d*x+c)*sin(d*x+c)^2+1/2/d*a^3*sin(d*x+c)^6/cos(d*x+c)^6+1/7/d*a^3*sin(d*
x+c)^6/cos(d*x+c)^7

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Maxima [A]  time = 1.67266, size = 149, normalized size = 1.08 \begin{align*} -\frac{420 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac{1260 \, a^{3} \cos \left (d x + c\right )^{6} + 210 \, a^{3} \cos \left (d x + c\right )^{5} - 700 \, a^{3} \cos \left (d x + c\right )^{4} - 525 \, a^{3} \cos \left (d x + c\right )^{3} + 84 \, a^{3} \cos \left (d x + c\right )^{2} + 210 \, a^{3} \cos \left (d x + c\right ) + 60 \, a^{3}}{\cos \left (d x + c\right )^{7}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x, algorithm="maxima")

[Out]

-1/420*(420*a^3*log(cos(d*x + c)) - (1260*a^3*cos(d*x + c)^6 + 210*a^3*cos(d*x + c)^5 - 700*a^3*cos(d*x + c)^4
- 525*a^3*cos(d*x + c)^3 + 84*a^3*cos(d*x + c)^2 + 210*a^3*cos(d*x + c) + 60*a^3)/cos(d*x + c)^7)/d

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Fricas [A]  time = 1.02049, size = 308, normalized size = 2.23 \begin{align*} -\frac{420 \, a^{3} \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 1260 \, a^{3} \cos \left (d x + c\right )^{6} - 210 \, a^{3} \cos \left (d x + c\right )^{5} + 700 \, a^{3} \cos \left (d x + c\right )^{4} + 525 \, a^{3} \cos \left (d x + c\right )^{3} - 84 \, a^{3} \cos \left (d x + c\right )^{2} - 210 \, a^{3} \cos \left (d x + c\right ) - 60 \, a^{3}}{420 \, d \cos \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x, algorithm="fricas")

[Out]

-1/420*(420*a^3*cos(d*x + c)^7*log(-cos(d*x + c)) - 1260*a^3*cos(d*x + c)^6 - 210*a^3*cos(d*x + c)^5 + 700*a^3
*cos(d*x + c)^4 + 525*a^3*cos(d*x + c)^3 - 84*a^3*cos(d*x + c)^2 - 210*a^3*cos(d*x + c) - 60*a^3)/(d*cos(d*x +
c)^7)

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Sympy [A]  time = 31.5841, size = 255, normalized size = 1.85 \begin{align*} \begin{cases} \frac{a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{7 d} + \frac{a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac{3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{5 d} + \frac{a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{4 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{35 d} - \frac{a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{2 d} - \frac{4 a^{3} \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{5 d} - \frac{a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{8 a^{3} \sec ^{3}{\left (c + d x \right )}}{105 d} + \frac{a^{3} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac{8 a^{3} \sec{\left (c + d x \right )}}{5 d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right )^{3} \tan ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**5,x)

[Out]

Piecewise((a**3*log(tan(c + d*x)**2 + 1)/(2*d) + a**3*tan(c + d*x)**4*sec(c + d*x)**3/(7*d) + a**3*tan(c + d*x
)**4*sec(c + d*x)**2/(2*d) + 3*a**3*tan(c + d*x)**4*sec(c + d*x)/(5*d) + a**3*tan(c + d*x)**4/(4*d) - 4*a**3*t
an(c + d*x)**2*sec(c + d*x)**3/(35*d) - a**3*tan(c + d*x)**2*sec(c + d*x)**2/(2*d) - 4*a**3*tan(c + d*x)**2*se
c(c + d*x)/(5*d) - a**3*tan(c + d*x)**2/(2*d) + 8*a**3*sec(c + d*x)**3/(105*d) + a**3*sec(c + d*x)**2/(2*d) +
8*a**3*sec(c + d*x)/(5*d), Ne(d, 0)), (x*(a*sec(c) + a)**3*tan(c)**5, True))

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Giac [B]  time = 3.89641, size = 360, normalized size = 2.61 \begin{align*} \frac{420 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{2497 \, a^{3} + \frac{18319 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{58317 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{69475 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{56035 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{28749 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{8463 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{1089 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x, algorithm="giac")

[Out]

1/420*(420*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 420*a^3*log(abs(-(cos(d*x + c) - 1)/(cos
(d*x + c) + 1) - 1)) + (2497*a^3 + 18319*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 58317*a^3*(cos(d*x + c) -
1)^2/(cos(d*x + c) + 1)^2 + 69475*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 56035*a^3*(cos(d*x + c) - 1
)^4/(cos(d*x + c) + 1)^4 + 28749*a^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 8463*a^3*(cos(d*x + c) - 1)^6
/(cos(d*x + c) + 1)^6 + 1089*a^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/((cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) + 1)^7)/d