Optimal. Leaf size=190 \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (7 a B+7 A b+5 b C)}{21 d}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (5 a A+3 a C+3 b B)}{5 d}+\frac {2 \sin (c+d x) (7 a B+7 A b+5 b C)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) (5 a A+3 a C+3 b B)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 (a C+b B) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b C \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 0.32, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4112, 3031, 3021, 2748, 2636, 2641, 2639} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (7 a B+7 A b+5 b C)}{21 d}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (5 a A+3 a C+3 b B)}{5 d}+\frac {2 \sin (c+d x) (7 a B+7 A b+5 b C)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) (5 a A+3 a C+3 b B)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 (a C+b B) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b C \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 3021
Rule 3031
Rule 4112
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {(b+a \cos (c+d x)) \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 b C \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int \frac {-\frac {7}{2} (b B+a C)-\frac {1}{2} (7 A b+7 a B+5 b C) \cos (c+d x)-\frac {7}{2} a A \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 b C \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (b B+a C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {4}{35} \int \frac {-\frac {5}{4} (7 A b+7 a B+5 b C)-\frac {7}{4} (5 a A+3 b B+3 a C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 b C \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (b B+a C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {1}{5} (-5 a A-3 b B-3 a C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx-\frac {1}{7} (-7 A b-7 a B-5 b C) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 b C \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (b B+a C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A b+7 a B+5 b C) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 a A+3 b B+3 a C) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}-\frac {1}{5} (5 a A+3 b B+3 a C) \int \sqrt {\cos (c+d x)} \, dx-\frac {1}{21} (-7 A b-7 a B-5 b C) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 (5 a A+3 b B+3 a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (7 A b+7 a B+5 b C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b C \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (b B+a C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A b+7 a B+5 b C) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 a A+3 b B+3 a C) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.05, size = 173, normalized size = 0.91 \[ \frac {10 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (7 a B+7 A b+5 b C)-42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (5 a A+3 a C+3 b B)+\frac {\sin (c+d x) (21 \cos (c+d x) (15 a A+13 a C+13 b B)+10 \cos (2 (c+d x)) (7 a B+7 A b+5 b C)+105 a A \cos (3 (c+d x))+70 a B+63 a C \cos (3 (c+d x))+70 A b+63 b B \cos (3 (c+d x))+110 b C)}{2 \cos ^{\frac {7}{2}}(c+d x)}}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b \sec \left (d x + c\right )^{3} + {\left (C a + B b\right )} \sec \left (d x + c\right )^{2} + A a + {\left (B a + A b\right )} \sec \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 18.05, size = 851, normalized size = 4.48 \[ \text {result too large to display} \]
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 = 7.80, size = 223, normalized size = 1.17 \[ \frac {6\,C\,a\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+30\,A\,a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+10\,B\,a\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{15\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {30\,C\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+70\,A\,b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+42\,B\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{105\,d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}} \]
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 \[ \int \frac {\left (a + b \sec {\left (c + d x \right )}\right ) \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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