\(\int \csc ^7(a+b x) \, dx\) [7]

Optimal result

Integrand size = 8, antiderivative size = 76 \[ \int \csc ^7(a+b x) \, dx=-\frac {5 \text {arctanh}(\cos (a+b x))}{16 b}-\frac {5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b} \]

[Out]

-5/16*arctanh(cos(b*x+a))/b-5/16*cot(b*x+a)*csc(b*x+a)/b-5/24*cot(b*x+a)*csc(b*x+a)^3/b-1/6*cot(b*x+a)*csc(b*x
+a)^5/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3855} \[ \int \csc ^7(a+b x) \, dx=-\frac {5 \text {arctanh}(\cos (a+b x))}{16 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {5 \cot (a+b x) \csc (a+b x)}{16 b} \]

[In]

Int[Csc[a + b*x]^7,x]

[Out]

(-5*ArcTanh[Cos[a + b*x]])/(16*b) - (5*Cot[a + b*x]*Csc[a + b*x])/(16*b) - (5*Cot[a + b*x]*Csc[a + b*x]^3)/(24
*b) - (Cot[a + b*x]*Csc[a + b*x]^5)/(6*b)

Rule 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac {5}{6} \int \csc ^5(a+b x) \, dx \\ & = -\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac {5}{8} \int \csc ^3(a+b x) \, dx \\ & = -\frac {5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac {5}{16} \int \csc (a+b x) \, dx \\ & = -\frac {5 \text {arctanh}(\cos (a+b x))}{16 b}-\frac {5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.99 \[ \int \csc ^7(a+b x) \, dx=-\frac {5 \csc ^2\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {\csc ^6\left (\frac {1}{2} (a+b x)\right )}{384 b}-\frac {5 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{16 b}+\frac {5 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{16 b}+\frac {5 \sec ^2\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\sec ^6\left (\frac {1}{2} (a+b x)\right )}{384 b} \]

[In]

Integrate[Csc[a + b*x]^7,x]

[Out]

(-5*Csc[(a + b*x)/2]^2)/(64*b) - Csc[(a + b*x)/2]^4/(64*b) - Csc[(a + b*x)/2]^6/(384*b) - (5*Log[Cos[(a + b*x)
/2]])/(16*b) + (5*Log[Sin[(a + b*x)/2]])/(16*b) + (5*Sec[(a + b*x)/2]^2)/(64*b) + Sec[(a + b*x)/2]^4/(64*b) +
Sec[(a + b*x)/2]^6/(384*b)

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\left (-\frac {\csc \left (x b +a \right )^{5}}{6}-\frac {5 \csc \left (x b +a \right )^{3}}{24}-\frac {5 \csc \left (x b +a \right )}{16}\right ) \cot \left (x b +a \right )+\frac {5 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{16}}{b}\)	\(60\)
default	\(\frac {\left (-\frac {\csc \left (x b +a \right )^{5}}{6}-\frac {5 \csc \left (x b +a \right )^{3}}{24}-\frac {5 \csc \left (x b +a \right )}{16}\right ) \cot \left (x b +a \right )+\frac {5 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{16}}{b}\)	\(60\)
parallelrisch	\(\frac {-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}-9 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+9 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}-45 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+45 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+120 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{384 b}\)	\(95\)
norman	\(\frac {-\frac {1}{384 b}-\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{128 b}-\frac {15 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{128 b}+\frac {15 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{128 b}+\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{10}}{128 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{12}}{384 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}+\frac {5 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{16 b}\)	\(115\)
risch	\(\frac {15 \,{\mathrm e}^{11 i \left (x b +a \right )}-85 \,{\mathrm e}^{9 i \left (x b +a \right )}+198 \,{\mathrm e}^{7 i \left (x b +a \right )}+198 \,{\mathrm e}^{5 i \left (x b +a \right )}-85 \,{\mathrm e}^{3 i \left (x b +a \right )}+15 \,{\mathrm e}^{i \left (x b +a \right )}}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{16 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{16 b}\)	\(121\)

[In]

int(csc(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/b*((-1/6*csc(b*x+a)^5-5/24*csc(b*x+a)^3-5/16*csc(b*x+a))*cot(b*x+a)+5/16*ln(csc(b*x+a)-cot(b*x+a)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (68) = 136\).

Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.04 \[ \int \csc ^7(a+b x) \, dx=\frac {30 \, \cos \left (b x + a\right )^{5} - 80 \, \cos \left (b x + a\right )^{3} - 15 \, {\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (b x + a\right )}{96 \, {\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, b \cos \left (b x + a\right )^{2} - b\right )}} \]

[In]

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

[Out]

1/96*(30*cos(b*x + a)^5 - 80*cos(b*x + a)^3 - 15*(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b*x + a)^2 - 1)*lo
g(1/2*cos(b*x + a) + 1/2) + 15*(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b*x + a)^2 - 1)*log(-1/2*cos(b*x + a
) + 1/2) + 66*cos(b*x + a))/(b*cos(b*x + a)^6 - 3*b*cos(b*x + a)^4 + 3*b*cos(b*x + a)^2 - b)

Sympy [F]

\[ \int \csc ^7(a+b x) \, dx=\int \csc ^{7}{\left (a + b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)**7,x)

[Out]

Integral(csc(a + b*x)**7, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.20 \[ \int \csc ^7(a+b x) \, dx=\frac {\frac {2 \, {\left (15 \, \cos \left (b x + a\right )^{5} - 40 \, \cos \left (b x + a\right )^{3} + 33 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1} - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{96 \, b} \]

[In]

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

[Out]

1/96*(2*(15*cos(b*x + a)^5 - 40*cos(b*x + a)^3 + 33*cos(b*x + a))/(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b
*x + a)^2 - 1) - 15*log(cos(b*x + a) + 1) + 15*log(cos(b*x + a) - 1))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (68) = 136\).

Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.39 \[ \int \csc ^7(a+b x) \, dx=-\frac {\frac {{\left (\frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {45 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}} + \frac {45 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 60 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{384 \, b} \]

[In]

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

[Out]

-1/384*((9*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 45*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 110*(cos(b*x
+ a) - 1)^3/(cos(b*x + a) + 1)^3 - 1)*(cos(b*x + a) + 1)^3/(cos(b*x + a) - 1)^3 + 45*(cos(b*x + a) - 1)/(cos(
b*x + a) + 1) - 9*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + (cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 - 60*l
og(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)))/b

Mupad [B] (verification not implemented)

Time = 21.40 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \csc ^7(a+b x) \, dx=\frac {\frac {5\,{\cos \left (a+b\,x\right )}^5}{16}-\frac {5\,{\cos \left (a+b\,x\right )}^3}{6}+\frac {11\,\cos \left (a+b\,x\right )}{16}}{b\,\left ({\cos \left (a+b\,x\right )}^6-3\,{\cos \left (a+b\,x\right )}^4+3\,{\cos \left (a+b\,x\right )}^2-1\right )}-\frac {5\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{16\,b} \]

[In]

int(1/sin(a + b*x)^7,x)

[Out]

((11*cos(a + b*x))/16 - (5*cos(a + b*x)^3)/6 + (5*cos(a + b*x)^5)/16)/(b*(3*cos(a + b*x)^2 - 3*cos(a + b*x)^4
+ cos(a + b*x)^6 - 1)) - (5*atanh(cos(a + b*x)))/(16*b)