Optimal. Leaf size=104 \[ \frac {64 c d^5 \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}}-\frac {16 c d^3 \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{21 b}-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2625, 2619} \[ \frac {64 c d^5 \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}}-\frac {16 c d^3 \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{21 b}-\frac {2 c d \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2619
Rule 2625
Rubi steps
\begin {align*} \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx &=-\frac {2 c d (d \csc (a+b x))^{7/2} \sqrt {c \sec (a+b x)}}{7 b}+\frac {1}{7} \left (8 d^2\right ) \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2} \, dx\\ &=-\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}+\frac {1}{21} \left (32 d^4\right ) \int \sqrt {d \csc (a+b x)} (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}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.29, size = 57, normalized size = 0.55 \[ -\frac {2 c d^5 \left (3 \csc ^4(a+b x)+8 \csc ^2(a+b x)-32\right ) \sqrt {c \sec (a+b x)}}{21 b \sqrt {d \csc (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.51, size = 85, normalized size = 0.82 \[ -\frac {2 \, {\left (32 \, c d^{4} \cos \left (b x + a\right )^{4} - 56 \, c d^{4} \cos \left (b x + a\right )^{2} + 21 \, c d^{4}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{21 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.08, size = 64, normalized size = 0.62 \[ \frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )-56 \left (\cos ^{2}\left (b x +a \right )\right )+21\right ) \cos \left (b x +a \right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {9}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{21 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.17, size = 110, normalized size = 1.06 \[ -\frac {16\,c\,d^4\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (41\,\sin \left (a+b\,x\right )-29\,\sin \left (3\,a+3\,b\,x\right )+12\,\sin \left (5\,a+5\,b\,x\right )-2\,\sin \left (7\,a+7\,b\,x\right )\right )}{21\,b\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________