Optimal. Leaf size=330 \[ \frac {\left (32 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{60 d \sqrt {a+b \sin (c+d x)}}-\frac {a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{60 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{60 d \left (a^2-b^2\right )^2}+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d} \]
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Rubi [A] time = 0.70, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2691, 2866, 2752, 2663, 2661, 2655, 2653} \[ -\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (-13 a^2 b^2+8 a^4+5 b^4\right )-a \left (-61 a^2 b^2+32 a^4+29 b^4\right ) \sin (c+d x)\right )}{60 d \left (a^2-b^2\right )^2}+\frac {\left (32 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{60 d \sqrt {a+b \sin (c+d x)}}-\frac {a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{60 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2691
Rule 2752
Rule 2866
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {1}{5} \int \frac {\sec ^4(c+d x) \left (-4 a^2+\frac {b^2}{2}-\frac {7}{2} a b \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 d}+\frac {\int \frac {\sec ^2(c+d x) \left (\frac {1}{4} \left (32 a^4-37 a^2 b^2+5 b^4\right )+6 a b \left (a^2-b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{15 \left (a^2-b^2\right )}\\ &=-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{60 \left (a^2-b^2\right )^2 d}-\frac {\int \frac {\frac {1}{8} b^2 \left (8 a^4-13 a^2 b^2+5 b^4\right )+\frac {1}{8} a b \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{15 \left (a^2-b^2\right )^2}\\ &=-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{60 \left (a^2-b^2\right )^2 d}-\frac {\left (a \left (32 a^2-29 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{120 \left (a^2-b^2\right )}-\frac {1}{120} \left (-32 a^2+5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{60 \left (a^2-b^2\right )^2 d}-\frac {\left (a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{120 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (\left (-32 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{120 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {a \left (32 a^2-29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{60 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (32 a^2-5 b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{60 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{60 \left (a^2-b^2\right )^2 d}\\ \end {align*}
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Mathematica [A] time = 6.28, size = 364, normalized size = 1.10 \[ \frac {\sqrt {a+b \sin (c+d x)} \left (\frac {\sec (c+d x) \left (32 a^3 \sin (c+d x)-8 a^2 b-29 a b^2 \sin (c+d x)+5 b^3\right )}{60 \left (a^2-b^2\right )}+\frac {1}{5} \sec ^5(c+d x) (a \sin (c+d x)+b)+\frac {1}{30} \sec ^3(c+d x) (8 a \sin (c+d x)-b)\right )}{d}-\frac {b \left (-\frac {\left (32 a^3-29 a b^2\right ) \left (\frac {2 (a+b) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}\right )}{b}-\frac {2 \left (8 a^2 b-5 b^3\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}\right )}{120 d (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a \sec \left (d x + c\right )^{6}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 1.04, size = 1519, normalized size = 4.60 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^6} \,d x \]
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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