3.167 \(\int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx\)

Optimal. Leaf size=140 \[ -\frac {5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {b \cot ^6(c+d x)}{6 d}-\frac {3 b \cot ^4(c+d x)}{4 d}-\frac {3 b \cot ^2(c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \]

[Out]

-5/16*a*arctanh(cos(d*x+c))/d-3/2*b*cot(d*x+c)^2/d-3/4*b*cot(d*x+c)^4/d-1/6*b*cot(d*x+c)^6/d-5/16*a*cot(d*x+c)
*csc(d*x+c)/d-5/24*a*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a*cot(d*x+c)*csc(d*x+c)^5/d+b*ln(tan(d*x+c))/d

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3872, 2834, 2620, 266, 43, 3768, 3770} \[ -\frac {5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {b \cot ^6(c+d x)}{6 d}-\frac {3 b \cot ^4(c+d x)}{4 d}-\frac {3 b \cot ^2(c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (3*b*Cot[c + d*x]^2)/(2*d) - (3*b*Cot[c + d*x]^4)/(4*d) - (b*Cot[c + d*x
]^6)/(6*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a*Cot[c + d*
x]*Csc[c + d*x]^5)/(6*d) + (b*Log[Tan[c + d*x]])/d

Rule 43

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2834

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx &=-\int (-b-a \cos (c+d x)) \csc ^7(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^7(c+d x) \, dx+b \int \csc ^7(c+d x) \sec (c+d x) \, dx\\ &=-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{6} (5 a) \int \csc ^5(c+d x) \, dx+\frac {b \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^7} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{8} (5 a) \int \csc ^3(c+d x) \, dx+\frac {b \operatorname {Subst}\left (\int \frac {(1+x)^3}{x^4} \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{16} (5 a) \int \csc (c+d x) \, dx+\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{x^4}+\frac {3}{x^3}+\frac {3}{x^2}+\frac {1}{x}\right ) \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=-\frac {5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {3 b \cot ^2(c+d x)}{2 d}-\frac {3 b \cot ^4(c+d x)}{4 d}-\frac {b \cot ^6(c+d x)}{6 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {b \log (\tan (c+d x))}{d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.62, size = 216, normalized size = 1.54 \[ -\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {b \left (2 \csc ^6(c+d x)+3 \csc ^4(c+d x)+6 \csc ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*a*Csc[(c + d*x)/2]^2)/(64*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (5*a*Log[C
os[(c + d*x)/2]])/(16*d) + (5*a*Log[Sin[(c + d*x)/2]])/(16*d) - (b*(6*Csc[c + d*x]^2 + 3*Csc[c + d*x]^4 + 2*Cs
c[c + d*x]^6 + 12*Log[Cos[c + d*x]] - 12*Log[Sin[c + d*x]]))/(12*d) + (5*a*Sec[(c + d*x)/2]^2)/(64*d) + (a*Sec
[(c + d*x)/2]^4)/(64*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 284, normalized size = 2.03 \[ \frac {30 \, a \cos \left (d x + c\right )^{5} + 48 \, b \cos \left (d x + c\right )^{4} - 80 \, a \cos \left (d x + c\right )^{3} - 120 \, b \cos \left (d x + c\right )^{2} + 66 \, a \cos \left (d x + c\right ) - 96 \, {\left (b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} + 3 \, b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\cos \left (d x + c\right )\right ) - 3 \, {\left ({\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a + 16 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a - 16 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, b}{96 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(30*a*cos(d*x + c)^5 + 48*b*cos(d*x + c)^4 - 80*a*cos(d*x + c)^3 - 120*b*cos(d*x + c)^2 + 66*a*cos(d*x +
c) - 96*(b*cos(d*x + c)^6 - 3*b*cos(d*x + c)^4 + 3*b*cos(d*x + c)^2 - b)*log(-cos(d*x + c)) - 3*((5*a - 16*b)*
cos(d*x + c)^6 - 3*(5*a - 16*b)*cos(d*x + c)^4 + 3*(5*a - 16*b)*cos(d*x + c)^2 - 5*a + 16*b)*log(1/2*cos(d*x +
c) + 1/2) + 3*((5*a + 16*b)*cos(d*x + c)^6 - 3*(5*a + 16*b)*cos(d*x + c)^4 + 3*(5*a + 16*b)*cos(d*x + c)^2 -
5*a - 16*b)*log(-1/2*cos(d*x + c) + 1/2) + 88*b)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 -
d)

________________________________________________________________________________________

giac [B]  time = 0.45, size = 357, normalized size = 2.55 \[ \frac {12 \, {\left (5 \, a + 16 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (a + b - \frac {9 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {12 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {87 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {110 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {352 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} - \frac {45 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {87 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(12*(5*a + 16*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 384*b*log(abs(-(cos(d*x + c) - 1)/(
cos(d*x + c) + 1) - 1)) + (a + b - 9*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 12*b*(cos(d*x + c) - 1)/(cos(d*
x + c) + 1) + 45*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 87*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
- 110*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 352*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(cos(d*x
+ c) + 1)^3/(cos(d*x + c) - 1)^3 - 45*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 87*b*(cos(d*x + c) - 1)/(cos(d
*x + c) + 1) + 9*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 12*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
- a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)/d

________________________________________________________________________________________

maple [A]  time = 0.75, size = 136, normalized size = 0.97 \[ -\frac {a \cot \left (d x +c \right ) \left (\csc ^{5}\left (d x +c \right )\right )}{6 d}-\frac {5 a \cot \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{24 d}-\frac {5 a \cot \left (d x +c \right ) \csc \left (d x +c \right )}{16 d}+\frac {5 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {b}{6 d \sin \left (d x +c \right )^{6}}-\frac {b}{4 d \sin \left (d x +c \right )^{4}}-\frac {b}{2 d \sin \left (d x +c \right )^{2}}+\frac {b \ln \left (\tan \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a*cot(d*x+c)*csc(d*x+c)^5/d-5/24*a*cot(d*x+c)*csc(d*x+c)^3/d-5/16*a*cot(d*x+c)*csc(d*x+c)/d+5/16/d*a*ln(c
sc(d*x+c)-cot(d*x+c))-1/6/d*b/sin(d*x+c)^6-1/4/d*b/sin(d*x+c)^4-1/2/d*b/sin(d*x+c)^2+b*ln(tan(d*x+c))/d

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 143, normalized size = 1.02 \[ -\frac {3 \, {\left (5 \, a - 16 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, a + 16 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 96 \, b \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{5} + 24 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, b \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) + 44 \, b\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(3*(5*a - 16*b)*log(cos(d*x + c) + 1) - 3*(5*a + 16*b)*log(cos(d*x + c) - 1) + 96*b*log(cos(d*x + c)) -
2*(15*a*cos(d*x + c)^5 + 24*b*cos(d*x + c)^4 - 40*a*cos(d*x + c)^3 - 60*b*cos(d*x + c)^2 + 33*a*cos(d*x + c) +
44*b)/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1))/d

________________________________________________________________________________________

mupad [B]  time = 1.02, size = 148, normalized size = 1.06 \[ \frac {\frac {5\,a\,{\cos \left (c+d\,x\right )}^5}{16}+\frac {b\,{\cos \left (c+d\,x\right )}^4}{2}-\frac {5\,a\,{\cos \left (c+d\,x\right )}^3}{6}-\frac {5\,b\,{\cos \left (c+d\,x\right )}^2}{4}+\frac {11\,a\,\cos \left (c+d\,x\right )}{16}+\frac {11\,b}{12}}{d\,\left ({\cos \left (c+d\,x\right )}^6-3\,{\cos \left (c+d\,x\right )}^4+3\,{\cos \left (c+d\,x\right )}^2-1\right )}+\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {5\,a}{32}+\frac {b}{2}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {5\,a}{32}-\frac {b}{2}\right )}{d}-\frac {b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((11*b)/12 + (11*a*cos(c + d*x))/16 - (5*a*cos(c + d*x)^3)/6 + (5*a*cos(c + d*x)^5)/16 - (5*b*cos(c + d*x)^2)/
4 + (b*cos(c + d*x)^4)/2)/(d*(3*cos(c + d*x)^2 - 3*cos(c + d*x)^4 + cos(c + d*x)^6 - 1)) + (log(cos(c + d*x) -
1)*((5*a)/32 + b/2))/d - (log(cos(c + d*x) + 1)*((5*a)/32 - b/2))/d - (b*log(cos(c + d*x)))/d

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________