3.2.73 \(\int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx\) [173]

Optimal. Leaf size=101 \[ \frac {b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \csc (c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d} \]

[Out]

b*arctanh(sin(d*x+c))/d-a*cot(d*x+c)/d-2/3*a*cot(d*x+c)^3/d-1/5*a*cot(d*x+c)^5/d-b*csc(d*x+c)/d-1/3*b*csc(d*x+
c)^3/d-1/5*b*csc(d*x+c)^5/d

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Rubi [A]
time = 0.08, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3957, 2917, 2701, 308, 213, 3852} \begin {gather*} -\frac {a \cot ^5(c+d x)}{5 d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-\frac {b \csc ^5(c+d x)}{5 d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc (c+d x)}{d}+\frac {b \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^6*(a + b*Sec[c + d*x]),x]

[Out]

(b*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (2*a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) - (b*
Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d)

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2701

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

Rule 2917

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

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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 ^6(c+d x) (a+b \sec (c+d x)) \, dx &=-\int (-b-a \cos (c+d x)) \csc ^6(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^6(c+d x) \, dx+b \int \csc ^6(c+d x) \sec (c+d x) \, dx\\ &=-\frac {a \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \csc (c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d}-\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \csc (c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 91, normalized size = 0.90 \begin {gather*} -\frac {8 a \cot (c+d x)}{15 d}-\frac {4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac {b \csc ^5(c+d x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\sin ^2(c+d x)\right )}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^6*(a + b*Sec[c + d*x]),x]

[Out]

(-8*a*Cot[c + d*x])/(15*d) - (4*a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d)
- (b*Csc[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, Sin[c + d*x]^2])/(5*d)

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Maple [A]
time = 0.11, size = 83, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{15}\right ) \cot \left (d x +c \right )}{d}\)	\(83\)
default	\(\frac {b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{15}\right ) \cot \left (d x +c \right )}{d}\)	\(83\)
risch	\(-\frac {2 i \left (15 b \,{\mathrm e}^{9 i \left (d x +c \right )}-80 b \,{\mathrm e}^{7 i \left (d x +c \right )}+178 b \,{\mathrm e}^{5 i \left (d x +c \right )}+80 a \,{\mathrm e}^{4 i \left (d x +c \right )}-80 b \,{\mathrm e}^{3 i \left (d x +c \right )}-40 a \,{\mathrm e}^{2 i \left (d x +c \right )}+15 b \,{\mathrm e}^{i \left (d x +c \right )}+8 a \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\)	\(146\)
norman	\(\frac {-\frac {a +b}{160 d}+\frac {\left (a -b \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {\left (5 a -11 b \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {\left (5 a -7 b \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {\left (5 a +7 b \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {\left (5 a +11 b \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^6*(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(b*(-1/5/sin(d*x+c)^5-1/3/sin(d*x+c)^3-1/sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))+a*(-8/15-1/5*csc(d*x+c)^4-4
/15*csc(d*x+c)^2)*cot(d*x+c))

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Maxima [A]
time = 0.27, size = 96, normalized size = 0.95 \begin {gather*} -\frac {b {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a}{\tan \left (d x + c\right )^{5}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

-1/30*(b*(2*(15*sin(d*x + c)^4 + 5*sin(d*x + c)^2 + 3)/sin(d*x + c)^5 - 15*log(sin(d*x + c) + 1) + 15*log(sin(
d*x + c) - 1)) + 2*(15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 + 3)*a/tan(d*x + c)^5)/d

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Fricas [A]
time = 3.43, size = 174, normalized size = 1.72 \begin {gather*} -\frac {16 \, a \cos \left (d x + c\right )^{5} + 30 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 70 \, b \cos \left (d x + c\right )^{2} - 15 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 15 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 30 \, a \cos \left (d x + c\right ) + 46 \, b}{30 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/30*(16*a*cos(d*x + c)^5 + 30*b*cos(d*x + c)^4 - 40*a*cos(d*x + c)^3 - 70*b*cos(d*x + c)^2 - 15*(b*cos(d*x +
c)^4 - 2*b*cos(d*x + c)^2 + b)*log(sin(d*x + c) + 1)*sin(d*x + c) + 15*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2
+ b)*log(-sin(d*x + c) + 1)*sin(d*x + c) + 30*a*cos(d*x + c) + 46*b)/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2
+ d)*sin(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sec {\left (c + d x \right )}\right ) \csc ^{6}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**6*(a+b*sec(d*x+c)),x)

[Out]

Integral((a + b*sec(c + d*x))*csc(c + d*x)**6, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (93) = 186\).
time = 0.46, size = 194, normalized size = 1.92 \begin {gather*} \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 480 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 480 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 330 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 330 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a + 3 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

1/480*(3*a*tan(1/2*d*x + 1/2*c)^5 - 3*b*tan(1/2*d*x + 1/2*c)^5 + 25*a*tan(1/2*d*x + 1/2*c)^3 - 35*b*tan(1/2*d*
x + 1/2*c)^3 + 480*b*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 480*b*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 150*a*tan
(1/2*d*x + 1/2*c) - 330*b*tan(1/2*d*x + 1/2*c) - (150*a*tan(1/2*d*x + 1/2*c)^4 + 330*b*tan(1/2*d*x + 1/2*c)^4
+ 25*a*tan(1/2*d*x + 1/2*c)^2 + 35*b*tan(1/2*d*x + 1/2*c)^2 + 3*a + 3*b)/tan(1/2*d*x + 1/2*c)^5)/d

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Mupad [B]
time = 1.09, size = 142, normalized size = 1.41 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a}{96}-\frac {7\,b}{96}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\left (10\,a+22\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {5\,a}{3}+\frac {7\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{5}+\frac {b}{5}\right )}{32\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a}{160}-\frac {b}{160}\right )}{d}+\frac {2\,b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,a}{16}-\frac {11\,b}{16}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x))/sin(c + d*x)^6,x)

[Out]

(tan(c/2 + (d*x)/2)^3*((5*a)/96 - (7*b)/96))/d - (cot(c/2 + (d*x)/2)^5*(a/5 + b/5 + tan(c/2 + (d*x)/2)^2*((5*a
)/3 + (7*b)/3) + tan(c/2 + (d*x)/2)^4*(10*a + 22*b)))/(32*d) + (tan(c/2 + (d*x)/2)^5*(a/160 - b/160))/d + (2*b
*atanh(tan(c/2 + (d*x)/2)))/d + (tan(c/2 + (d*x)/2)*((5*a)/16 - (11*b)/16))/d

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