Optimal. Leaf size=128 \[ -\frac {4 d^4 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {4 c d^3 \sqrt {d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2625, 2630, 2572, 2639} \[ -\frac {4 c d^3 \sqrt {d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac {4 d^4 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2572
Rule 2625
Rule 2630
Rule 2639
Rubi steps
\begin {align*} \int \frac {(d \csc (a+b x))^{7/2}}{\sqrt {c \sec (a+b x)}} \, dx &=-\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}+\frac {1}{5} \left (2 d^2\right ) \int \frac {(d \csc (a+b x))^{3/2}}{\sqrt {c \sec (a+b x)}} \, dx\\ &=-\frac {4 c d^3 \sqrt {d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac {1}{5} \left (4 d^4\right ) \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}} \, dx\\ &=-\frac {4 c d^3 \sqrt {d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac {\left (4 d^4\right ) \int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)} \, dx}{5 \sqrt {c \cos (a+b x)} \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)}}\\ &=-\frac {4 c d^3 \sqrt {d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac {\left (4 d^4\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{5 \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}\\ &=-\frac {4 c d^3 \sqrt {d \csc (a+b x)}}{5 b (c \sec (a+b x))^{3/2}}-\frac {2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{3/2}}-\frac {4 d^4 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{5 b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [C] time = 1.08, size = 104, normalized size = 0.81 \[ -\frac {2 d^2 \tan ^2(a+b x) (d \csc (a+b x))^{3/2} \left (\sin (2 (a+b x)) \sqrt [4]{-\cot ^2(a+b x)} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\csc ^2(a+b x)\right )-(\cos (2 (a+b x))-2) \cot ^3(a+b x)\right )}{5 b \sqrt {c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )} d^{3} \csc \left (b x + a\right )^{3}}{c \sec \left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.23, size = 976, normalized size = 7.62 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{7/2}}{\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}} \,d x \]
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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