Integrand size = 25, antiderivative size = 104 \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=\frac {64 c d^5 \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}}-\frac {16 c d^3 (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} \sqrt {c \sec (a+b x)}}{7 b} \] Output:
64/21*c*d^5*(c*sec(b*x+a))^(1/2)/b/(d*csc(b*x+a))^(1/2)-16/21*c*d^3*(d*csc (b*x+a))^(3/2)*(c*sec(b*x+a))^(1/2)/b-2/7*c*d*(d*csc(b*x+a))^(7/2)*(c*sec( b*x+a))^(1/2)/b
Time = 0.72 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.55 \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=-\frac {2 c d^5 \left (-32+8 \csc ^2(a+b x)+3 \csc ^4(a+b x)\right ) \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}} \] Input:
Integrate[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(3/2),x]
Output:
(-2*c*d^5*(-32 + 8*Csc[a + b*x]^2 + 3*Csc[a + b*x]^4)*Sqrt[c*Sec[a + b*x]] )/(21*b*Sqrt[d*Csc[a + b*x]])
Time = 0.46 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3105, 3042, 3105, 3042, 3099}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{9/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{9/2}dx\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {8}{7} d^2 \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}dx-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{7} d^2 \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}dx-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b}\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {8}{7} d^2 \left (\frac {4}{3} d^2 \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}dx-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{7} d^2 \left (\frac {4}{3} d^2 \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}dx-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b}\) |
\(\Big \downarrow \) 3099 |
\(\displaystyle \frac {8}{7} d^2 \left (\frac {8 c d^3 \sqrt {c \sec (a+b x)}}{3 b \sqrt {d \csc (a+b x)}}-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b}\) |
Input:
Int[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(3/2),x]
Output:
(-2*c*d*(d*Csc[a + b*x])^(7/2)*Sqrt[c*Sec[a + b*x]])/(7*b) + (8*d^2*((8*c* d^3*Sqrt[c*Sec[a + b*x]])/(3*b*Sqrt[d*Csc[a + b*x]]) - (2*c*d*(d*Csc[a + b *x])^(3/2)*Sqrt[c*Sec[a + b*x]])/(3*b)))/7
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1 )/(f*(n - 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 2, 0] && NeQ[n, 1]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Simp[(-a)*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(m - 1))), x] + Simp[a^2*((m + n - 2)/(m - 1)) Int[(a*Csc[e + f* x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[ m, 1] && IntegersQ[2*m, 2*n] && !GtQ[n, m]
Time = 1.55 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {2 \left (32 \cos \left (b x +a \right )^{4}-56 \cos \left (b x +a \right )^{2}+21\right ) \sqrt {c \sec \left (b x +a \right )}\, c \,d^{4} \sqrt {d \csc \left (b x +a \right )}\, \csc \left (b x +a \right )^{3}}{21 b}\) | \(60\) |
Input:
int((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
2/21/b*(32*cos(b*x+a)^4-56*cos(b*x+a)^2+21)*(c*sec(b*x+a))^(1/2)*c*d^4*(d* csc(b*x+a))^(1/2)*csc(b*x+a)^3
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.82 \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=-\frac {2 \, {\left (32 \, c d^{4} \cos \left (b x + a\right )^{4} - 56 \, c d^{4} \cos \left (b x + a\right )^{2} + 21 \, c d^{4}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{21 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \] Input:
integrate((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="fricas")
Output:
-2/21*(32*c*d^4*cos(b*x + a)^4 - 56*c*d^4*cos(b*x + a)^2 + 21*c*d^4)*sqrt( c/cos(b*x + a))*sqrt(d/sin(b*x + a))/((b*cos(b*x + a)^2 - b)*sin(b*x + a))
Timed out. \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate((d*csc(b*x+a))**(9/2)*(c*sec(b*x+a))**(3/2),x)
Output:
Timed out
\[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=\int { \left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="maxima")
Output:
integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(3/2), x)
\[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=\int { \left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="giac")
Output:
integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(3/2), x)
Time = 11.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.06 \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=-\frac {16\,c\,d^4\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (41\,\sin \left (a+b\,x\right )-29\,\sin \left (3\,a+3\,b\,x\right )+12\,\sin \left (5\,a+5\,b\,x\right )-2\,\sin \left (7\,a+7\,b\,x\right )\right )}{21\,b\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \] Input:
int((c/cos(a + b*x))^(3/2)*(d/sin(a + b*x))^(9/2),x)
Output:
-(16*c*d^4*(c/cos(a + b*x))^(1/2)*(d/sin(a + b*x))^(1/2)*(41*sin(a + b*x) - 29*sin(3*a + 3*b*x) + 12*sin(5*a + 5*b*x) - 2*sin(7*a + 7*b*x)))/(21*b*( 15*cos(2*a + 2*b*x) - 6*cos(4*a + 4*b*x) + cos(6*a + 6*b*x) - 10))
\[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx=\sqrt {d}\, \sqrt {c}\, \left (\int \sqrt {\sec \left (b x +a \right )}\, \sqrt {\csc \left (b x +a \right )}\, \csc \left (b x +a \right )^{4} \sec \left (b x +a \right )d x \right ) c \,d^{4} \] Input:
int((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(3/2),x)
Output:
sqrt(d)*sqrt(c)*int(sqrt(sec(a + b*x))*sqrt(csc(a + b*x))*csc(a + b*x)**4* sec(a + b*x),x)*c*d**4