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*ln(cos(d*x+c))/d-16/35*b*sec(d*x+c)/d+1/70*(35*a+16*b*sec(d*x+c))*tan(d*x+c)^2/d-1/140*(35*a+24*b*sec(d*x+c)
)*tan(d*x+c)^4/d+1/42*(7*a+6*b*sec(d*x+c))*tan(d*x+c)^6/d

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Rubi [A]  time = 0.16, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3475

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

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]

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*}

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Mathematica [A]  time = 0.47, 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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fricas [A]  time = 1.29, size = 101, normalized size = 0.91 \[ \frac {420 \, a \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 420 \, b \cos \left (d x + c\right )^{6} + 630 \, a \cos \left (d x + c\right )^{5} + 420 \, b \cos \left (d x + c\right )^{4} - 315 \, a \cos \left (d x + c\right )^{3} - 252 \, b \cos \left (d x + c\right )^{2} + 70 \, a \cos \left (d x + c\right ) + 60 \, b}{420 \, d \cos \left (d x + c\right )^{7}} \]

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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giac [B]  time = 11.58, size = 317, normalized size = 2.86 \[ -\frac {420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {1089 \, a + 384 \, b + \frac {8463 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2688 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {28749 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8064 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {56035 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {13440 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {28749 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1089 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \]

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

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maple [B]  time = 0.62, size = 216, normalized size = 1.95 \[ \frac {\left (\tan ^{6}\left (d x +c \right )\right ) a}{6 d}-\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {b \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )^{7}}-\frac {b \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{5}}+\frac {b \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{3}}-\frac {b \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )}-\frac {16 b \cos \left (d x +c \right )}{35 d}-\frac {b \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{7 d}-\frac {6 b \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}-\frac {8 b \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d} \]

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*tan(d*x+c)^6*a-1/4*a*tan(d*x+c)^4/d+1/2*a*tan(d*x+c)^2/d+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*b*cos(d*x+c)/d-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 = 0.32, size = 94, normalized size = 0.85 \[ \frac {420 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac {420 \, b \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{5} - 420 \, b \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{3} + 252 \, b \cos \left (d x + c\right )^{2} - 70 \, a \cos \left (d x + c\right ) - 60 \, b}{\cos \left (d x + c\right )^{7}}}{420 \, d} \]

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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mupad [B]  time = 5.24, size = 221, normalized size = 1.99 \[ \frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {128\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\left (-\frac {128\,a}{3}-32\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (14\,a+\frac {96\,b}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-2\,a-\frac {32\,b}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {32\,b}{35}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((32*b)/35 - tan(c/2 + (d*x)/2)^2*(2*a + (32*b)/5) + tan(c/2 + (d*x)/2)^4*(14*a + (96*b)/5) - tan(c/2 + (d*x)/
2)^6*((128*a)/3 + 32*b) + (128*a*tan(c/2 + (d*x)/2)^8)/3 - 14*a*tan(c/2 + (d*x)/2)^10 + 2*a*tan(c/2 + (d*x)/2)
^12)/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8
+ 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1)) - (2*a*atanh(tan(c/2 + (d*x
)/2)^2))/d

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sympy [A]  time = 8.48, size = 148, normalized size = 1.33 \[ \begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {a \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {b \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{7 d} - \frac {6 b \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {8 b \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} - \frac {16 b \sec {\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sec {\relax (c )}\right ) \tan ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]

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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