Optimal. Leaf size=159 \[ \frac {2 a \left (5 a^2+21 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b \left (9 a^2+5 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {32 a^2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{35 d}+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d} \]
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Rubi [A] time = 0.26, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4264, 3841, 4047, 3769, 3771, 2641, 4045, 2639} \[ \frac {2 a \left (5 a^2+21 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b \left (9 a^2+5 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {32 a^2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{35 d}+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 3841
Rule 4045
Rule 4047
Rule 4264
Rubi steps
\begin {align*} \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^3}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {8 a^2 b+\frac {1}{2} a \left (5 a^2+21 b^2\right ) \sec (c+d x)+\frac {1}{2} b \left (3 a^2+7 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {8 a^2 b+\frac {1}{2} b \left (3 a^2+7 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+\frac {1}{7} \left (a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {32 a^2 b \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{5} \left (b \left (9 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {32 a^2 b \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{5} \left (b \left (9 a^2+5 b^2\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (a \left (5 a^2+21 b^2\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (9 a^2+5 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {32 a^2 b \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 110, normalized size = 0.69 \[ \frac {10 \left (5 a^3+21 a b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+42 \left (9 a^2 b+5 b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+a \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^2 \cos (2 (c+d x))+65 a^2+126 a b \cos (c+d x)+210 b^2\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a^{3} \cos \left (d x + c\right )^{3}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.12, size = 421, normalized size = 2.65 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (240 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-360 a^{3}-504 a^{2} b \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (280 a^{3}+504 a^{2} b +420 b^{2} a \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-80 a^{3}-126 a^{2} b -210 b^{2} a \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+25 a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+105 b^{2} a \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-189 a^{2} b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-105 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{3}\right )}{105 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 146, normalized size = 0.92 \[ \frac {2\,\left (b^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+a\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+a\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}-\frac {2\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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