3.256 \(\int (a+b \sec (c+d x)) \tan ^7(c+d x) \, dx\)

Optimal. Leaf size=111 \[ \frac{\tan ^6(c+d x) (7 a+6 b \sec (c+d x))}{42 d}-\frac{\tan ^4(c+d x) (35 a+24 b \sec (c+d x))}{140 d}+\frac{\tan ^2(c+d x) (35 a+16 b \sec (c+d x))}{70 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{16 b \sec (c+d x)}{35 d} \]

[Out]

(a*Log[Cos[c + d*x]])/d - (16*b*Sec[c + d*x])/(35*d) + ((35*a + 16*b*Sec[c + d*x])*Tan[c + d*x]^2)/(70*d) - ((
35*a + 24*b*Sec[c + d*x])*Tan[c + d*x]^4)/(140*d) + ((7*a + 6*b*Sec[c + d*x])*Tan[c + d*x]^6)/(42*d)

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Rubi [A]  time = 0.156044, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3881, 3884, 3475, 2606, 8} \[ \frac{\tan ^6(c+d x) (7 a+6 b \sec (c+d x))}{42 d}-\frac{\tan ^4(c+d x) (35 a+24 b \sec (c+d x))}{140 d}+\frac{\tan ^2(c+d x) (35 a+16 b \sec (c+d x))}{70 d}+\frac{a \log (\cos (c+d x))}{d}-\frac{16 b \sec (c+d x)}{35 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Log[Cos[c + d*x]])/d - (16*b*Sec[c + d*x])/(35*d) + ((35*a + 16*b*Sec[c + d*x])*Tan[c + d*x]^2)/(70*d) - ((
35*a + 24*b*Sec[c + d*x])*Tan[c + d*x]^4)/(140*d) + ((7*a + 6*b*Sec[c + d*x])*Tan[c + d*x]^6)/(42*d)

Rule 3881

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(e*(e*Cot[
c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x]))/(d*m*(m - 1)), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m - 2)
*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]

Rule 3884

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(e*
Cot[c + d*x])^m, x], x] + Dist[b, Int[(e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]

Rule 3475

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

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (a+b \sec (c+d x)) \tan ^7(c+d x) \, dx &=\frac{(7 a+6 b \sec (c+d x)) \tan ^6(c+d x)}{42 d}-\frac{1}{7} \int (7 a+6 b \sec (c+d x)) \tan ^5(c+d x) \, dx\\ &=-\frac{(35 a+24 b \sec (c+d x)) \tan ^4(c+d x)}{140 d}+\frac{(7 a+6 b \sec (c+d x)) \tan ^6(c+d x)}{42 d}+\frac{1}{35} \int (35 a+24 b \sec (c+d x)) \tan ^3(c+d x) \, dx\\ &=\frac{(35 a+16 b \sec (c+d x)) \tan ^2(c+d x)}{70 d}-\frac{(35 a+24 b \sec (c+d x)) \tan ^4(c+d x)}{140 d}+\frac{(7 a+6 b \sec (c+d x)) \tan ^6(c+d x)}{42 d}-\frac{1}{105} \int (105 a+48 b \sec (c+d x)) \tan (c+d x) \, dx\\ &=\frac{(35 a+16 b \sec (c+d x)) \tan ^2(c+d x)}{70 d}-\frac{(35 a+24 b \sec (c+d x)) \tan ^4(c+d x)}{140 d}+\frac{(7 a+6 b \sec (c+d x)) \tan ^6(c+d x)}{42 d}-a \int \tan (c+d x) \, dx-\frac{1}{35} (16 b) \int \sec (c+d x) \tan (c+d x) \, dx\\ &=\frac{a \log (\cos (c+d x))}{d}+\frac{(35 a+16 b \sec (c+d x)) \tan ^2(c+d x)}{70 d}-\frac{(35 a+24 b \sec (c+d x)) \tan ^4(c+d x)}{140 d}+\frac{(7 a+6 b \sec (c+d x)) \tan ^6(c+d x)}{42 d}-\frac{(16 b) \operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{35 d}\\ &=\frac{a \log (\cos (c+d x))}{d}-\frac{16 b \sec (c+d x)}{35 d}+\frac{(35 a+16 b \sec (c+d x)) \tan ^2(c+d x)}{70 d}-\frac{(35 a+24 b \sec (c+d x)) \tan ^4(c+d x)}{140 d}+\frac{(7 a+6 b \sec (c+d x)) \tan ^6(c+d x)}{42 d}\\ \end{align*}

Mathematica [A]  time = 0.447335, size = 106, normalized size = 0.95 \[ \frac{a \left (2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))\right )}{12 d}+\frac{b \sec ^7(c+d x)}{7 d}-\frac{3 b \sec ^5(c+d x)}{5 d}+\frac{b \sec ^3(c+d x)}{d}-\frac{b \sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*Sec[c + d*x])/d) + (b*Sec[c + d*x]^3)/d - (3*b*Sec[c + d*x]^5)/(5*d) + (b*Sec[c + d*x]^7)/(7*d) + (a*(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 [B]  time = 0.045, size = 216, normalized size = 2. \begin{align*}{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}a}{2\,d}}+{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{7\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}-{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{7\,d\cos \left ( dx+c \right ) }}-{\frac{16\,b\cos \left ( dx+c \right ) }{35\,d}}-{\frac{b\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{7\,d}}-{\frac{6\,b\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d}}-{\frac{8\,b\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{35\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))*tan(d*x+c)^7,x)

[Out]

1/6/d*a*tan(d*x+c)^6-1/4/d*a*tan(d*x+c)^4+1/2/d*a*tan(d*x+c)^2+a*ln(cos(d*x+c))/d+1/7/d*b*sin(d*x+c)^8/cos(d*x
+c)^7-1/35/d*b*sin(d*x+c)^8/cos(d*x+c)^5+1/35/d*b*sin(d*x+c)^8/cos(d*x+c)^3-1/7/d*b*sin(d*x+c)^8/cos(d*x+c)-16
/35/d*b*cos(d*x+c)-1/7/d*b*cos(d*x+c)*sin(d*x+c)^6-6/35/d*b*cos(d*x+c)*sin(d*x+c)^4-8/35/d*b*cos(d*x+c)*sin(d*
x+c)^2

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Maxima [A]  time = 1.03822, size = 127, normalized size = 1.14 \begin{align*} \frac{420 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac{420 \, b \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{5} - 420 \, b \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{3} + 252 \, b \cos \left (d x + c\right )^{2} - 70 \, a \cos \left (d x + c\right ) - 60 \, b}{\cos \left (d x + c\right )^{7}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(420*a*log(cos(d*x + c)) - (420*b*cos(d*x + c)^6 - 630*a*cos(d*x + c)^5 - 420*b*cos(d*x + c)^4 + 315*a*c
os(d*x + c)^3 + 252*b*cos(d*x + c)^2 - 70*a*cos(d*x + c) - 60*b)/cos(d*x + c)^7)/d

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Fricas [A]  time = 1.49646, size = 284, normalized size = 2.56 \begin{align*} \frac{420 \, a \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 420 \, b \cos \left (d x + c\right )^{6} + 630 \, a \cos \left (d x + c\right )^{5} + 420 \, b \cos \left (d x + c\right )^{4} - 315 \, a \cos \left (d x + c\right )^{3} - 252 \, b \cos \left (d x + c\right )^{2} + 70 \, a \cos \left (d x + c\right ) + 60 \, b}{420 \, d \cos \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(420*a*cos(d*x + c)^7*log(-cos(d*x + c)) - 420*b*cos(d*x + c)^6 + 630*a*cos(d*x + c)^5 + 420*b*cos(d*x +
c)^4 - 315*a*cos(d*x + c)^3 - 252*b*cos(d*x + c)^2 + 70*a*cos(d*x + c) + 60*b)/(d*cos(d*x + c)^7)

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Sympy [A]  time = 19.3387, size = 148, normalized size = 1.33 \begin{align*} \begin{cases} - \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac{a \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac{a \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{b \tan ^{6}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{7 d} - \frac{6 b \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{35 d} + \frac{8 b \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{35 d} - \frac{16 b \sec{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a + b \sec{\left (c \right )}\right ) \tan ^{7}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*log(tan(c + d*x)**2 + 1)/(2*d) + a*tan(c + d*x)**6/(6*d) - a*tan(c + d*x)**4/(4*d) + a*tan(c + d
*x)**2/(2*d) + b*tan(c + d*x)**6*sec(c + d*x)/(7*d) - 6*b*tan(c + d*x)**4*sec(c + d*x)/(35*d) + 8*b*tan(c + d*
x)**2*sec(c + d*x)/(35*d) - 16*b*sec(c + d*x)/(35*d), Ne(d, 0)), (x*(a + b*sec(c))*tan(c)**7, True))

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Giac [B]  time = 8.00617, size = 428, normalized size = 3.86 \begin{align*} -\frac{420 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{1089 \, a + 384 \, b + \frac{8463 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{2688 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{28749 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{8064 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{56035 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{13440 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{56035 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{28749 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{8463 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{1089 \, a{\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+b*sec(d*x+c))*tan(d*x+c)^7,x, algorithm="giac")

[Out]

-1/420*(420*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 420*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*
x + c) + 1) - 1)) + (1089*a + 384*b + 8463*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2688*b*(cos(d*x + c) - 1)
/(cos(d*x + c) + 1) + 28749*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 8064*b*(cos(d*x + c) - 1)^2/(cos(d*x
+ c) + 1)^2 + 56035*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 13440*b*(cos(d*x + c) - 1)^3/(cos(d*x + c)
+ 1)^3 + 56035*a*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 28749*a*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5
+ 8463*a*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1089*a*(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