Optimal. Leaf size=97 \[ \frac {2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 97, 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, 3769, 3771, 2639} \[ \frac {2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx &=b^3 \int \frac {1}{(b \sec (c+d x))^{9/2}} \, dx\\ &=\frac {2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {1}{9} (7 b) \int \frac {1}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac {2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {7 \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx}{15 b}\\ &=\frac {2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {7 \int \sqrt {\cos (c+d x)} \, dx}{15 b \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\\ &=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 73, normalized size = 0.75 \[ \frac {84 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+(33 \sin (c+d x)+5 \sin (3 (c+d x))) \cos ^{\frac {3}{2}}(c+d x)}{90 d \cos ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right )} \cos \left (d x + c\right )^{3}}{b^{2} \sec \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{3}}{\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.98, size = 333, normalized size = 3.43 \[ \frac {\frac {14 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}}{15}-\frac {14 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{15}-\frac {2 \left (\cos ^{6}\left (d x +c \right )\right )}{9}+\frac {14 i \sin \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{15}-\frac {14 i \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{15}-\frac {4 \left (\cos ^{4}\left (d x +c \right )\right )}{45}-\frac {28 \left (\cos ^{2}\left (d x +c \right )\right )}{45}+\frac {14 \cos \left (d x +c \right )}{15}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{3}}{\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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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