Optimal. Leaf size=106 \[ -\frac {64 c^3 d^3 \sqrt {d \csc (a+b x)}}{15 b \sqrt {c \sec (a+b x)}}+\frac {16 c d^3 (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}{15 b}-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{5/2}}{5 b} \]
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Rubi [A] time = 0.16, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2625, 2626, 2619} \[ -\frac {64 c^3 d^3 \sqrt {d \csc (a+b x)}}{15 b \sqrt {c \sec (a+b x)}}+\frac {16 c d^3 (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}}{15 b}-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 2619
Rule 2625
Rule 2626
Rubi steps
\begin {align*} \int (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{5/2} \, dx &=-\frac {2 c d (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}}{5 b}+\frac {1}{5} \left (8 d^2\right ) \int (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2} \, dx\\ &=\frac {16 c d^3 \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}{15 b}-\frac {2 c d (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}}{5 b}+\frac {1}{15} \left (32 c^2 d^2\right ) \int (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)} \, dx\\ &=-\frac {64 c^3 d^3 \sqrt {d \csc (a+b x)}}{15 b \sqrt {c \sec (a+b x)}}+\frac {16 c d^3 \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}{15 b}-\frac {2 c d (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2}}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 57, normalized size = 0.54 \[ -\frac {2 c d^3 (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)} \left (32 \cos ^2(a+b x)+3 \cot ^2(a+b x)-5\right )}{15 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 89, normalized size = 0.84 \[ -\frac {2 \, {\left (32 \, c^{2} d^{3} \cos \left (b x + a\right )^{4} - 40 \, c^{2} d^{3} \cos \left (b x + a\right )^{2} + 5 \, c^{2} d^{3}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{15 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {7}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.12, size = 64, normalized size = 0.60 \[ \frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )-40 \left (\cos ^{2}\left (b x +a \right )\right )+5\right ) \cos \left (b x +a \right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {7}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \sin \left (b x +a \right )}{15 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {7}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 112, normalized size = 1.06 \[ \frac {16\,c^2\,d^3\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (5\,\cos \left (a+b\,x\right )-3\,\cos \left (3\,a+3\,b\,x\right )-4\,\cos \left (5\,a+5\,b\,x\right )+2\,\cos \left (7\,a+7\,b\,x\right )\right )}{15\,b\,\left (\cos \left (2\,a+2\,b\,x\right )+2\,\cos \left (4\,a+4\,b\,x\right )-\cos \left (6\,a+6\,b\,x\right )-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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