Optimal. Leaf size=172 \[ \frac {2 a (A+B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 a (3 A+5 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a (A+B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 a (3 A+5 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d} \]
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Rubi [A] time = 0.22, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2960, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac {2 a (A+B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 a (3 A+5 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a (A+B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 a (3 A+5 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2960
Rule 3768
Rule 3771
Rule 3787
Rule 3997
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x) \, dx &=\int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) (B+A \sec (c+d x)) \, dx\\ &=\frac {2 a A \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2}{5} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a (3 A+5 B)+\frac {5}{2} a (A+B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a A \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+(a (A+B)) \int \sec ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{5} (a (3 A+5 B)) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 a (3 A+5 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (A+B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {1}{3} (a (A+B)) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{5} (a (3 A+5 B)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a (3 A+5 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (A+B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {1}{3} \left (a (A+B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{5} \left (a (3 A+5 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 a (3 A+5 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a (A+B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 a (3 A+5 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (A+B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [C] time = 1.93, size = 292, normalized size = 1.70 \[ \frac {a e^{-i c} \left (-1+e^{2 i c}\right ) \csc (c) (\cos (c+d x)+1) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \left ((3 A+5 B) e^{i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-5 i (A+B) \left (1+e^{2 i (c+d x)}\right )^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-3 A e^{i (c+d x)}-24 A e^{3 i (c+d x)}-5 A e^{4 i (c+d x)}-9 A e^{5 i (c+d x)}+5 A-15 B e^{i (c+d x)}-30 B e^{3 i (c+d x)}-5 B e^{4 i (c+d x)}-15 B e^{5 i (c+d x)}+5 B\right )}{30 d \left (1+e^{2 i (c+d x)}\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B a \cos \left (d x + c\right )^{2} + {\left (A + B\right )} a \cos \left (d x + c\right ) + A a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}, x\right ) \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.28, size = 661, normalized size = 3.84 \[ \text {result too large to display} \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,\left (a+a\,\cos \left (c+d\,x\right )\right ) \,d x \]
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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