Optimal. Leaf size=359 \[ -\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{6 d \left (a^2-b^2\right )^3}-\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{6 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.61, antiderivative size = 359, 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 = {2694, 2866, 2752, 2663, 2661, 2655, 2653} \[ -\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (-15 a^2 b^2+4 a^4-21 b^4\right ) \sin (c+d x)\right )}{6 d \left (a^2-b^2\right )^3}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}-\frac {\left (-15 a^2 b^2+4 a^4-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{6 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2694
Rule 2752
Rule 2866
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {2 b \sec ^3(c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {2 \int \frac {\sec ^4(c+d x) \left (-\frac {a}{2}+\frac {7}{2} b \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^2-b^2}\\ &=\frac {2 b \sec ^3(c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {2 \int \frac {\sec ^2(c+d x) \left (a \left (a^2-3 b^2\right )+\frac {3}{4} b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=\frac {2 b \sec ^3(c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}-\frac {2 \int \frac {\frac {1}{8} a b^2 \left (a^2-33 b^2\right )+\frac {1}{8} b \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^3}\\ &=\frac {2 b \sec ^3(c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}+\frac {\left (a \left (a^2-3 b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^2}-\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{12 \left (a^2-b^2\right )^3}\\ &=\frac {2 b \sec ^3(c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}-\frac {\left (\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{12 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (a \left (a^2-3 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}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {2 b \sec ^3(c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^4-15 a^2 b^2-21 b^4\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)}}{6 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a \left (a^2-3 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}}}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}\\ \end {align*}
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Mathematica [A] time = 3.11, size = 348, normalized size = 0.97 \[ \frac {-4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\left (4 a^5+4 a^4 b-15 a^3 b^2-15 a^2 b^3-21 a b^4-21 b^5\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\frac {1}{8} \sec ^3(c+d x) \left (24 a^5 \sin (c+d x)+8 a^5 \sin (3 (c+d x))-24 a^4 b-64 a^3 b^2 \sin (c+d x)-32 a^3 b^2 \sin (3 (c+d x))+101 a^2 b^3+\left (-12 a^4 b+84 a^2 b^3+56 b^5\right ) \cos (2 (c+d x))+\left (-4 a^4 b+15 a^2 b^3+21 b^5\right ) \cos (4 (c+d x))+40 a b^4 \sin (c+d x)+24 a b^4 \sin (3 (c+d x))+19 b^5\right )}{6 d (a-b)^3 (a+b)^3 \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.73, size = 1646, normalized size = 4.58 \[ \text {result too large to display} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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