Optimal. Leaf size=194 \[ -\frac {2 a \left (5 a^2+9 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (5 a^2+9 b^2\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A]
time = 0.22, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4349, 3927,
4132, 3853, 3856, 2720, 4131, 2719} \begin {gather*} \frac {2 b \left (21 a^2+5 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 a \left (5 a^2+9 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (5 a^2+9 b^2\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{7 d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3927
Rule 4131
Rule 4132
Rule 4349
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx\\ &=\frac {2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (7 a^2+3 b^2\right )+\frac {1}{2} b \left (21 a^2+5 b^2\right ) \sec (c+d x)+8 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (7 a^2+3 b^2\right )+8 a b^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} \left (b \left (21 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{21} \left (b \left (21 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{5} \left (a \left (5 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (5 a^2+9 b^2\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{21} \left (b \left (21 a^2+5 b^2\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{5} \left (a \left (5 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 b \left (21 a^2+5 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (5 a^2+9 b^2\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {1}{5} \left (a \left (5 a^2+9 b^2\right )\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 a \left (5 a^2+9 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (5 a^2+9 b^2\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^2 (a+b \sec (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac {5}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 177, normalized size = 0.91 \begin {gather*} \frac {-42 a \left (5 a^2+9 b^2\right ) \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 b \left (21 a^2+5 b^2\right ) \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+126 a b^2 \sin (c+d x)+210 a^3 \cos ^2(c+d x) \sin (c+d x)+378 a b^2 \cos ^2(c+d x) \sin (c+d x)+105 a^2 b \sin (2 (c+d x))+25 b^3 \sin (2 (c+d x))+30 b^3 \tan (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(819\) vs.
\(2(226)=452\).
time = 0.48, size = 820, normalized size = 4.23
method | result | size |
default | \(\text {Expression too large to display}\) | \(820\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.68, size = 270, normalized size = 1.39 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (21 i \, a^{2} b + 5 i \, b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-21 i \, a^{2} b - 5 i \, b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, a^{3} + 9 i \, a b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, a^{3} - 9 i \, a b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (63 \, a b^{2} \cos \left (d x + c\right ) + 21 \, {\left (5 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, b^{3} + 5 \, {\left (21 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{3}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.24, size = 147, normalized size = 0.76 \begin {gather*} \frac {\frac {2\,b^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7}+2\,a^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+\frac {6\,a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5}+2\,a^2\,b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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