Optimal. Leaf size=291 \[ -\frac {\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 d \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-5 b^2\right )-4 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{6 d \left (a^2-b^2\right )^2}+\frac {\left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{6 d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (a^2-2 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.44, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2696, 2866, 2752, 2663, 2661, 2655, 2653} \[ -\frac {\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 d \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-5 b^2\right )-4 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{6 d \left (a^2-b^2\right )^2}+\frac {\left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{6 d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (a^2-2 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2696
Rule 2752
Rule 2866
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=-\frac {\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right ) d}-\frac {\int \frac {\sec ^2(c+d x) \left (-2 a^2+\frac {5 b^2}{2}-\frac {3}{2} a b \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac {\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-5 b^2\right )-4 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}+\frac {\int \frac {-\frac {1}{4} b^2 \left (a^2-5 b^2\right )-a b \left (a^2-2 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=-\frac {\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-5 b^2\right )-4 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}-\frac {\left (a \left (a^2-2 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{3 \left (a^2-b^2\right )^2}+\frac {\left (4 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{12 \left (a^2-b^2\right )}\\ &=-\frac {\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-5 b^2\right )-4 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}-\frac {\left (a \left (a^2-2 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}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (\left (4 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}{12 \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right ) d}-\frac {2 a \left (a^2-2 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)}}{3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 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}}}{6 \left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-5 b^2\right )-4 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}\\ \end {align*}
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Mathematica [A] time = 4.14, size = 306, normalized size = 1.05 \[ \frac {-4 \left (4 a^4-9 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+16 a \left (a^3+a^2 b-2 a b^2-2 b^3\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\sec ^3(c+d x) \left (12 a^4 \sin (c+d x)+4 a^4 \sin (3 (c+d x))+\left (14 a b^3-6 a^3 b\right ) \cos (2 (c+d x))+\left (4 a b^3-2 a^3 b\right ) \cos (4 (c+d x))-4 a^3 b-25 a^2 b^2 \sin (c+d x)-9 a^2 b^2 \sin (3 (c+d x))+10 a b^3+13 b^4 \sin (c+d x)+5 b^4 \sin (3 (c+d x))\right )}{24 d (a-b)^2 (a+b)^2 \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (d x + c\right )^{4}}{\sqrt {b \sin \left (d x + c\right ) + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{4}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.26, size = 1314, normalized size = 4.52 \[ \text {result too large to display} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{4}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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