Optimal. Leaf size=100 \[ \frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^4 d}+\frac {10 \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{7 b d (b \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3769, 3771, 2641} \[ \frac {10 \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^4 d}+\frac {2 \sin (c+d x)}{7 b d (b \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(b \sec (c+d x))^{7/2}} \, dx &=\frac {2 \sin (c+d x)}{7 b d (b \sec (c+d x))^{5/2}}+\frac {5 \int \frac {1}{(b \sec (c+d x))^{3/2}} \, dx}{7 b^2}\\ &=\frac {2 \sin (c+d x)}{7 b d (b \sec (c+d x))^{5/2}}+\frac {10 \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {5 \int \sqrt {b \sec (c+d x)} \, dx}{21 b^4}\\ &=\frac {2 \sin (c+d x)}{7 b d (b \sec (c+d x))^{5/2}}+\frac {10 \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\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^4}\\ &=\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^4 d}+\frac {2 \sin (c+d x)}{7 b d (b \sec (c+d x))^{5/2}}+\frac {10 \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 0.66 \[ \frac {\sqrt {b \sec (c+d x)} \left (26 \sin (2 (c+d x))+3 \sin (4 (c+d x))+40 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{84 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right )}}{b^{4} \sec \left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {1}{\left (b \sec \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.90, size = 153, normalized size = 1.53 \[ -\frac {2 \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (5 i \sin \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-3 \left (\cos ^{4}\left (d x +c \right )\right )+3 \left (\cos ^{3}\left (d x +c \right )\right )-5 \left (\cos ^{2}\left (d x +c \right )\right )+5 \cos \left (d x +c \right )\right )}{21 d \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{3}} \]
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 \[ \int \frac {1}{\left (b \sec \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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