Optimal. Leaf size=214 \[ \frac {2 \left (7 a^2 B+14 a A b+5 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (5 a^2 A+6 a b B+3 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (7 a^2 B+14 a A b+5 b^2 B\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (5 a^2 A+6 a b B+3 A b^2\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b (2 a B+A b) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 0.40, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2954, 2988, 3021, 2748, 2636, 2641, 2639} \[ \frac {2 \left (7 a^2 B+14 a A b+5 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (5 a^2 A+6 a b B+3 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (7 a^2 B+14 a A b+5 b^2 B\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (5 a^2 A+6 a b B+3 A b^2\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b (2 a B+A b) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 2954
Rule 2988
Rule 3021
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {(b+a \cos (c+d x))^2 (B+A \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int \frac {-\frac {7}{2} b (A b+2 a B)-\frac {1}{2} \left (14 a A b+7 a^2 B+5 b^2 B\right ) \cos (c+d x)-\frac {7}{2} a^2 A \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b (A b+2 a B) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {4}{35} \int \frac {-\frac {5}{4} \left (14 a A b+7 a^2 B+5 b^2 B\right )-\frac {7}{4} \left (5 a^2 A+3 A b^2+6 a b B\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b (A b+2 a B) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {1}{5} \left (-5 a^2 A-3 A b^2-6 a b B\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx-\frac {1}{7} \left (-14 a A b-7 a^2 B-5 b^2 B\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b (A b+2 a B) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}-\frac {1}{5} \left (5 a^2 A+3 A b^2+6 a b B\right ) \int \sqrt {\cos (c+d x)} \, dx-\frac {1}{21} \left (-14 a A b-7 a^2 B-5 b^2 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b (A b+2 a B) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (5 a^2 A+3 A b^2+6 a b B\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 5.26, size = 191, normalized size = 0.89 \[ \frac {2 \left (5 \left (7 a^2 B+14 a A b+5 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-21 \left (5 a^2 A+6 a b B+3 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {5 \left (7 a^2 B+14 a A b+5 b^2 B\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {21 \left (5 a^2 A+6 a b B+3 A b^2\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}+\frac {21 b (2 a B+A b) \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {15 b^2 B \sin (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B b^{2} \sec \left (d x + c\right )^{3} + A a^{2} + {\left (2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 16.10, size = 859, normalized size = 4.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [B] time = 4.98, size = 233, normalized size = 1.09 \[ \frac {6\,A\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+30\,A\,a^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+20\,A\,a\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{15\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {30\,B\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+70\,B\,a^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+84\,B\,a\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{105\,d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}} \]
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 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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