Optimal. Leaf size=100 \[ \frac {2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b^6 d}+\frac {10 \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 b^4 d}+\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3768, 3771, 2641} \[ \frac {2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b^6 d}+\frac {10 \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 b^4 d}+\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^3 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2641
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sec ^7(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=\frac {\int (b \sec (c+d x))^{9/2} \, dx}{b^7}\\ &=\frac {2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}+\frac {5 \int (b \sec (c+d x))^{5/2} \, dx}{7 b^5}\\ &=\frac {10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^4 d}+\frac {2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}+\frac {5 \int \sqrt {b \sec (c+d x)} \, dx}{21 b^3}\\ &=\frac {10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^4 d}+\frac {2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}+\frac {\left (5 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 b^3}\\ &=\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^3 d}+\frac {10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^4 d}+\frac {2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 64, normalized size = 0.64 \[ \frac {(b \sec (c+d x))^{5/2} \left (5 \sin (2 (c+d x))+6 \tan (c+d x)+10 \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{21 b^5 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right )} \sec \left (d x + c\right )^{4}}{b^{3}}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{7}}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.90, size = 152, normalized size = 1.52 \[ -\frac {2 \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (5 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-5 \left (\cos ^{3}\left (d x +c \right )\right )+5 \left (\cos ^{2}\left (d x +c \right )\right )-3 \cos \left (d x +c \right )+3\right )}{21 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{7}}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^7\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{7}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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