\(\int x \csc ^7(a+b x^2) \, dx\) [15]

Optimal result

Integrand size = 12, antiderivative size = 90 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {5 \text {arctanh}\left (\cos \left (a+b x^2\right )\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b} \]

[Out]

-5/32*arctanh(cos(b*x^2+a))/b-5/32*cot(b*x^2+a)*csc(b*x^2+a)/b-5/48*cot(b*x^2+a)*csc(b*x^2+a)^3/b-1/12*cot(b*x
^2+a)*csc(b*x^2+a)^5/b

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4290, 3853, 3855} \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {5 \text {arctanh}\left (\cos \left (a+b x^2\right )\right )}{32 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b} \]

[In]

Int[x*Csc[a + b*x^2]^7,x]

[Out]

(-5*ArcTanh[Cos[a + b*x^2]])/(32*b) - (5*Cot[a + b*x^2]*Csc[a + b*x^2])/(32*b) - (5*Cot[a + b*x^2]*Csc[a + b*x
^2]^3)/(48*b) - (Cot[a + b*x^2]*Csc[a + b*x^2]^5)/(12*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]

Rule 4290

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \csc ^7(a+b x) \, dx,x,x^2\right ) \\ & = -\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}+\frac {5}{12} \text {Subst}\left (\int \csc ^5(a+b x) \, dx,x,x^2\right ) \\ & = -\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}+\frac {5}{16} \text {Subst}\left (\int \csc ^3(a+b x) \, dx,x,x^2\right ) \\ & = -\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}+\frac {5}{32} \text {Subst}\left (\int \csc (a+b x) \, dx,x,x^2\right ) \\ & = -\frac {5 \text {arctanh}\left (\cos \left (a+b x^2\right )\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.86 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {5 \csc ^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\csc ^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\csc ^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b}-\frac {5 \log \left (\cos \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}+\frac {5 \log \left (\sin \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}+\frac {5 \sec ^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\sec ^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\sec ^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b} \]

[In]

Integrate[x*Csc[a + b*x^2]^7,x]

[Out]

(-5*Csc[(a + b*x^2)/2]^2)/(128*b) - Csc[(a + b*x^2)/2]^4/(128*b) - Csc[(a + b*x^2)/2]^6/(768*b) - (5*Log[Cos[(
a + b*x^2)/2]])/(32*b) + (5*Log[Sin[(a + b*x^2)/2]])/(32*b) + (5*Sec[(a + b*x^2)/2]^2)/(128*b) + Sec[(a + b*x^
2)/2]^4/(128*b) + Sec[(a + b*x^2)/2]^6/(768*b)

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {\left (-\frac {\csc \left (b \,x^{2}+a \right )^{5}}{6}-\frac {5 \csc \left (b \,x^{2}+a \right )^{3}}{24}-\frac {5 \csc \left (b \,x^{2}+a \right )}{16}\right ) \cot \left (b \,x^{2}+a \right )+\frac {5 \ln \left (\csc \left (b \,x^{2}+a \right )-\cot \left (b \,x^{2}+a \right )\right )}{16}}{2 b}\)	\(73\)
default	\(\frac {\left (-\frac {\csc \left (b \,x^{2}+a \right )^{5}}{6}-\frac {5 \csc \left (b \,x^{2}+a \right )^{3}}{24}-\frac {5 \csc \left (b \,x^{2}+a \right )}{16}\right ) \cot \left (b \,x^{2}+a \right )+\frac {5 \ln \left (\csc \left (b \,x^{2}+a \right )-\cot \left (b \,x^{2}+a \right )\right )}{16}}{2 b}\)	\(73\)
parallelrisch	\(\frac {\tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{6}-\cot \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{6}+9 \tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{4}-9 \cot \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{4}+45 \tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{2}-45 \cot \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )^{2}+120 \ln \left (\tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )}{768 b}\)	\(109\)
risch	\(\frac {15 \,{\mathrm e}^{11 i \left (b \,x^{2}+a \right )}-85 \,{\mathrm e}^{9 i \left (b \,x^{2}+a \right )}+198 \,{\mathrm e}^{7 i \left (b \,x^{2}+a \right )}+198 \,{\mathrm e}^{5 i \left (b \,x^{2}+a \right )}-85 \,{\mathrm e}^{3 i \left (b \,x^{2}+a \right )}+15 \,{\mathrm e}^{i \left (b \,x^{2}+a \right )}}{48 b \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i \left (b \,x^{2}+a \right )}+1\right )}{32 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (b \,x^{2}+a \right )}-1\right )}{32 b}\)	\(139\)

[In]

int(x*csc(b*x^2+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/2/b*((-1/6*csc(b*x^2+a)^5-5/24*csc(b*x^2+a)^3-5/16*csc(b*x^2+a))*cot(b*x^2+a)+5/16*ln(csc(b*x^2+a)-cot(b*x^2
+a)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (82) = 164\).

Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.03 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=\frac {30 \, \cos \left (b x^{2} + a\right )^{5} - 80 \, \cos \left (b x^{2} + a\right )^{3} - 15 \, {\left (\cos \left (b x^{2} + a\right )^{6} - 3 \, \cos \left (b x^{2} + a\right )^{4} + 3 \, \cos \left (b x^{2} + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x^{2} + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x^{2} + a\right )^{6} - 3 \, \cos \left (b x^{2} + a\right )^{4} + 3 \, \cos \left (b x^{2} + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x^{2} + a\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (b x^{2} + a\right )}{192 \, {\left (b \cos \left (b x^{2} + a\right )^{6} - 3 \, b \cos \left (b x^{2} + a\right )^{4} + 3 \, b \cos \left (b x^{2} + a\right )^{2} - b\right )}} \]

[In]

integrate(x*csc(b*x^2+a)^7,x, algorithm="fricas")

[Out]

1/192*(30*cos(b*x^2 + a)^5 - 80*cos(b*x^2 + a)^3 - 15*(cos(b*x^2 + a)^6 - 3*cos(b*x^2 + a)^4 + 3*cos(b*x^2 + a
)^2 - 1)*log(1/2*cos(b*x^2 + a) + 1/2) + 15*(cos(b*x^2 + a)^6 - 3*cos(b*x^2 + a)^4 + 3*cos(b*x^2 + a)^2 - 1)*l
og(-1/2*cos(b*x^2 + a) + 1/2) + 66*cos(b*x^2 + a))/(b*cos(b*x^2 + a)^6 - 3*b*cos(b*x^2 + a)^4 + 3*b*cos(b*x^2
+ a)^2 - b)

Sympy [F]

\[ \int x \csc ^7\left (a+b x^2\right ) \, dx=\int x \csc ^{7}{\left (a + b x^{2} \right )}\, dx \]

[In]

integrate(x*csc(b*x**2+a)**7,x)

[Out]

Integral(x*csc(a + b*x**2)**7, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3543 vs. \(2 (82) = 164\).

Time = 0.28 (sec) , antiderivative size = 3543, normalized size of antiderivative = 39.37 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x*csc(b*x^2+a)^7,x, algorithm="maxima")

[Out]

1/192*(4*(15*cos(11*b*x^2 + 11*a) - 85*cos(9*b*x^2 + 9*a) + 198*cos(7*b*x^2 + 7*a) + 198*cos(5*b*x^2 + 5*a) -
85*cos(3*b*x^2 + 3*a) + 15*cos(b*x^2 + a))*cos(12*b*x^2 + 12*a) - 60*(6*cos(10*b*x^2 + 10*a) - 15*cos(8*b*x^2
+ 8*a) + 20*cos(6*b*x^2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) - 1)*cos(11*b*x^2 + 11*a) + 24*(
85*cos(9*b*x^2 + 9*a) - 198*cos(7*b*x^2 + 7*a) - 198*cos(5*b*x^2 + 5*a) + 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2
+ a))*cos(10*b*x^2 + 10*a) - 340*(15*cos(8*b*x^2 + 8*a) - 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) - 6*c
os(2*b*x^2 + 2*a) + 1)*cos(9*b*x^2 + 9*a) + 60*(198*cos(7*b*x^2 + 7*a) + 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x
^2 + 3*a) + 15*cos(b*x^2 + a))*cos(8*b*x^2 + 8*a) - 792*(20*cos(6*b*x^2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos
(2*b*x^2 + 2*a) - 1)*cos(7*b*x^2 + 7*a) - 80*(198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) + 15*cos(b*x^2 +
a))*cos(6*b*x^2 + 6*a) + 792*(15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x^2 + 2*a) + 1)*cos(5*b*x^2 + 5*a) - 300*(17*c
os(3*b*x^2 + 3*a) - 3*cos(b*x^2 + a))*cos(4*b*x^2 + 4*a) + 340*(6*cos(2*b*x^2 + 2*a) - 1)*cos(3*b*x^2 + 3*a) -
360*cos(2*b*x^2 + 2*a)*cos(b*x^2 + a) + 15*(2*(6*cos(10*b*x^2 + 10*a) - 15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*x^
2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) - 1)*cos(12*b*x^2 + 12*a) - cos(12*b*x^2 + 12*a)^2 + 1
2*(15*cos(8*b*x^2 + 8*a) - 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x^2 + 2*a) + 1)*cos(10*b*
x^2 + 10*a) - 36*cos(10*b*x^2 + 10*a)^2 + 30*(20*cos(6*b*x^2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 +
2*a) - 1)*cos(8*b*x^2 + 8*a) - 225*cos(8*b*x^2 + 8*a)^2 + 40*(15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x^2 + 2*a) + 1
)*cos(6*b*x^2 + 6*a) - 400*cos(6*b*x^2 + 6*a)^2 + 30*(6*cos(2*b*x^2 + 2*a) - 1)*cos(4*b*x^2 + 4*a) - 225*cos(4
*b*x^2 + 4*a)^2 - 36*cos(2*b*x^2 + 2*a)^2 + 2*(6*sin(10*b*x^2 + 10*a) - 15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x^2
+ 6*a) - 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(12*b*x^2 + 12*a) - sin(12*b*x^2 + 12*a)^2 + 12*(15
*sin(8*b*x^2 + 8*a) - 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) - 6*sin(2*b*x^2 + 2*a))*sin(10*b*x^2 + 10*
a) - 36*sin(10*b*x^2 + 10*a)^2 + 30*(20*sin(6*b*x^2 + 6*a) - 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin
(8*b*x^2 + 8*a) - 225*sin(8*b*x^2 + 8*a)^2 + 120*(5*sin(4*b*x^2 + 4*a) - 2*sin(2*b*x^2 + 2*a))*sin(6*b*x^2 + 6
*a) - 400*sin(6*b*x^2 + 6*a)^2 - 225*sin(4*b*x^2 + 4*a)^2 + 180*sin(4*b*x^2 + 4*a)*sin(2*b*x^2 + 2*a) - 36*sin
(2*b*x^2 + 2*a)^2 + 12*cos(2*b*x^2 + 2*a) - 1)*log(cos(b*x^2)^2 + 2*cos(b*x^2)*cos(a) + cos(a)^2 + sin(b*x^2)^
2 - 2*sin(b*x^2)*sin(a) + sin(a)^2) - 15*(2*(6*cos(10*b*x^2 + 10*a) - 15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 +
6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) - 1)*cos(12*b*x^2 + 12*a) - cos(12*b*x^2 + 12*a)^2 + 12*(
15*cos(8*b*x^2 + 8*a) - 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x^2 + 2*a) + 1)*cos(10*b*x^2
+ 10*a) - 36*cos(10*b*x^2 + 10*a)^2 + 30*(20*cos(6*b*x^2 + 6*a) - 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a
) - 1)*cos(8*b*x^2 + 8*a) - 225*cos(8*b*x^2 + 8*a)^2 + 40*(15*cos(4*b*x^2 + 4*a) - 6*cos(2*b*x^2 + 2*a) + 1)*c
os(6*b*x^2 + 6*a) - 400*cos(6*b*x^2 + 6*a)^2 + 30*(6*cos(2*b*x^2 + 2*a) - 1)*cos(4*b*x^2 + 4*a) - 225*cos(4*b*
x^2 + 4*a)^2 - 36*cos(2*b*x^2 + 2*a)^2 + 2*(6*sin(10*b*x^2 + 10*a) - 15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x^2 +
6*a) - 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(12*b*x^2 + 12*a) - sin(12*b*x^2 + 12*a)^2 + 12*(15*si
n(8*b*x^2 + 8*a) - 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) - 6*sin(2*b*x^2 + 2*a))*sin(10*b*x^2 + 10*a)
- 36*sin(10*b*x^2 + 10*a)^2 + 30*(20*sin(6*b*x^2 + 6*a) - 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(8*
b*x^2 + 8*a) - 225*sin(8*b*x^2 + 8*a)^2 + 120*(5*sin(4*b*x^2 + 4*a) - 2*sin(2*b*x^2 + 2*a))*sin(6*b*x^2 + 6*a)
- 400*sin(6*b*x^2 + 6*a)^2 - 225*sin(4*b*x^2 + 4*a)^2 + 180*sin(4*b*x^2 + 4*a)*sin(2*b*x^2 + 2*a) - 36*sin(2*
b*x^2 + 2*a)^2 + 12*cos(2*b*x^2 + 2*a) - 1)*log(cos(b*x^2)^2 - 2*cos(b*x^2)*cos(a) + cos(a)^2 + sin(b*x^2)^2 +
2*sin(b*x^2)*sin(a) + sin(a)^2) + 4*(15*sin(11*b*x^2 + 11*a) - 85*sin(9*b*x^2 + 9*a) + 198*sin(7*b*x^2 + 7*a)
+ 198*sin(5*b*x^2 + 5*a) - 85*sin(3*b*x^2 + 3*a) + 15*sin(b*x^2 + a))*sin(12*b*x^2 + 12*a) - 60*(6*sin(10*b*x
^2 + 10*a) - 15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x^2 + 6*a) - 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin
(11*b*x^2 + 11*a) + 24*(85*sin(9*b*x^2 + 9*a) - 198*sin(7*b*x^2 + 7*a) - 198*sin(5*b*x^2 + 5*a) + 85*sin(3*b*x
^2 + 3*a) - 15*sin(b*x^2 + a))*sin(10*b*x^2 + 10*a) - 340*(15*sin(8*b*x^2 + 8*a) - 20*sin(6*b*x^2 + 6*a) + 15*
sin(4*b*x^2 + 4*a) - 6*sin(2*b*x^2 + 2*a))*sin(9*b*x^2 + 9*a) + 60*(198*sin(7*b*x^2 + 7*a) + 198*sin(5*b*x^2 +
5*a) - 85*sin(3*b*x^2 + 3*a) + 15*sin(b*x^2 + a))*sin(8*b*x^2 + 8*a) - 792*(20*sin(6*b*x^2 + 6*a) - 15*sin(4*
b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(7*b*x^2 + 7*a) - 80*(198*sin(5*b*x^2 + 5*a) - 85*sin(3*b*x^2 + 3*a) +
15*sin(b*x^2 + a))*sin(6*b*x^2 + 6*a) + 2376*(5*sin(4*b*x^2 + 4*a) - 2*sin(2*b*x^2 + 2*a))*sin(5*b*x^2 + 5*a)
- 300*(17*sin(3*b*x^2 + 3*a) - 3*sin(b*x^2 + a))*sin(4*b*x^2 + 4*a) + 2040*sin(3*b*x^2 + 3*a)*sin(2*b*x^2 + 2
*a) - 360*sin(2*b*x^2 + 2*a)*sin(b*x^2 + a) + 60*cos(b*x^2 + a))/(b*cos(12*b*x^2 + 12*a)^2 + 36*b*cos(10*b*x^2
+ 10*a)^2 + 225*b*cos(8*b*x^2 + 8*a)^2 + 400*b*cos(6*b*x^2 + 6*a)^2 + 225*b*cos(4*b*x^2 + 4*a)^2 + 36*b*cos(2
*b*x^2 + 2*a)^2 + b*sin(12*b*x^2 + 12*a)^2 + 36*b*sin(10*b*x^2 + 10*a)^2 + 225*b*sin(8*b*x^2 + 8*a)^2 + 400*b*
sin(6*b*x^2 + 6*a)^2 + 225*b*sin(4*b*x^2 + 4*a)^2 - 180*b*sin(4*b*x^2 + 4*a)*sin(2*b*x^2 + 2*a) + 36*b*sin(2*b
*x^2 + 2*a)^2 - 2*(6*b*cos(10*b*x^2 + 10*a) - 15*b*cos(8*b*x^2 + 8*a) + 20*b*cos(6*b*x^2 + 6*a) - 15*b*cos(4*b
*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) - b)*cos(12*b*x^2 + 12*a) - 12*(15*b*cos(8*b*x^2 + 8*a) - 20*b*cos(6*b*x^
2 + 6*a) + 15*b*cos(4*b*x^2 + 4*a) - 6*b*cos(2*b*x^2 + 2*a) + b)*cos(10*b*x^2 + 10*a) - 30*(20*b*cos(6*b*x^2 +
6*a) - 15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) - b)*cos(8*b*x^2 + 8*a) - 40*(15*b*cos(4*b*x^2 + 4*a)
- 6*b*cos(2*b*x^2 + 2*a) + b)*cos(6*b*x^2 + 6*a) - 30*(6*b*cos(2*b*x^2 + 2*a) - b)*cos(4*b*x^2 + 4*a) - 12*b*
cos(2*b*x^2 + 2*a) - 2*(6*b*sin(10*b*x^2 + 10*a) - 15*b*sin(8*b*x^2 + 8*a) + 20*b*sin(6*b*x^2 + 6*a) - 15*b*si
n(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(12*b*x^2 + 12*a) - 12*(15*b*sin(8*b*x^2 + 8*a) - 20*b*sin(6*b*x
^2 + 6*a) + 15*b*sin(4*b*x^2 + 4*a) - 6*b*sin(2*b*x^2 + 2*a))*sin(10*b*x^2 + 10*a) - 30*(20*b*sin(6*b*x^2 + 6*
a) - 15*b*sin(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(8*b*x^2 + 8*a) - 120*(5*b*sin(4*b*x^2 + 4*a) - 2*b*
sin(2*b*x^2 + 2*a))*sin(6*b*x^2 + 6*a) + b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (82) = 164\).

Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.34 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {\frac {{\left (\frac {9 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}}{\cos \left (b x^{2} + a\right ) + 1} - \frac {45 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}} + \frac {45 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}}{\cos \left (b x^{2} + a\right ) + 1} - \frac {9 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}} - 60 \, \log \left (-\frac {\cos \left (b x^{2} + a\right ) - 1}{\cos \left (b x^{2} + a\right ) + 1}\right )}{768 \, b} \]

[In]

integrate(x*csc(b*x^2+a)^7,x, algorithm="giac")

[Out]

-1/768*((9*(cos(b*x^2 + a) - 1)/(cos(b*x^2 + a) + 1) - 45*(cos(b*x^2 + a) - 1)^2/(cos(b*x^2 + a) + 1)^2 + 110*
(cos(b*x^2 + a) - 1)^3/(cos(b*x^2 + a) + 1)^3 - 1)*(cos(b*x^2 + a) + 1)^3/(cos(b*x^2 + a) - 1)^3 + 45*(cos(b*x
^2 + a) - 1)/(cos(b*x^2 + a) + 1) - 9*(cos(b*x^2 + a) - 1)^2/(cos(b*x^2 + a) + 1)^2 + (cos(b*x^2 + a) - 1)^3/(
cos(b*x^2 + a) + 1)^3 - 60*log(-(cos(b*x^2 + a) - 1)/(cos(b*x^2 + a) + 1)))/b

Mupad [B] (verification not implemented)

Time = 28.55 (sec) , antiderivative size = 491, normalized size of antiderivative = 5.46 \[ \int x \csc ^7\left (a+b x^2\right ) \, dx=-\frac {5\,\ln \left (-\frac {x\,5{}\mathrm {i}}{8}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}\,5{}\mathrm {i}}{8}\right )}{32\,b}+\frac {5\,\ln \left (\frac {x\,5{}\mathrm {i}}{8}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}\,5{}\mathrm {i}}{8}\right )}{32\,b}+\frac {8\,{\mathrm {e}}^{3{}\mathrm {i}\,b\,x^2+a\,3{}\mathrm {i}}}{3\,b\,\left (5\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-10\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+10\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}-5\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}+{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}-1\right )}+\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{6\,b\,\left (3\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}-1\right )}+\frac {5\,{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{16\,b\,\left ({\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-1\right )}+\frac {16\,{\mathrm {e}}^{5{}\mathrm {i}\,b\,x^2+a\,5{}\mathrm {i}}}{3\,b\,\left (1+15\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-20\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+15\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}-6\,{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}+{\mathrm {e}}^{12{}\mathrm {i}\,b\,x^2+a\,12{}\mathrm {i}}-6\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{b\,\left (1+6\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-4\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}-4\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )}-\frac {5\,{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{24\,b\,\left (1+{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )} \]

[In]

int(x/sin(a + b*x^2)^7,x)

[Out]

(5*log((x*5i)/8 - (x*exp(a*1i)*exp(b*x^2*1i)*5i)/8))/(32*b) - (5*log(- (x*5i)/8 - (x*exp(a*1i)*exp(b*x^2*1i)*5
i)/8))/(32*b) + (8*exp(a*3i + b*x^2*3i))/(3*b*(5*exp(a*2i + b*x^2*2i) - 10*exp(a*4i + b*x^2*4i) + 10*exp(a*6i
+ b*x^2*6i) - 5*exp(a*8i + b*x^2*8i) + exp(a*10i + b*x^2*10i) - 1)) + exp(a*1i + b*x^2*1i)/(6*b*(3*exp(a*2i +
b*x^2*2i) - 3*exp(a*4i + b*x^2*4i) + exp(a*6i + b*x^2*6i) - 1)) + (5*exp(a*1i + b*x^2*1i))/(16*b*(exp(a*2i + b
*x^2*2i) - 1)) + (16*exp(a*5i + b*x^2*5i))/(3*b*(15*exp(a*4i + b*x^2*4i) - 6*exp(a*2i + b*x^2*2i) - 20*exp(a*6
i + b*x^2*6i) + 15*exp(a*8i + b*x^2*8i) - 6*exp(a*10i + b*x^2*10i) + exp(a*12i + b*x^2*12i) + 1)) + exp(a*1i +
b*x^2*1i)/(b*(6*exp(a*4i + b*x^2*4i) - 4*exp(a*2i + b*x^2*2i) - 4*exp(a*6i + b*x^2*6i) + exp(a*8i + b*x^2*8i)
+ 1)) - (5*exp(a*1i + b*x^2*1i))/(24*b*(exp(a*4i + b*x^2*4i) - 2*exp(a*2i + b*x^2*2i) + 1))