Optimal. Leaf size=229 \[ -\frac {32 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{5 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{5 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.33, antiderivative size = 229, 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 = {2693, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac {32 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{5 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{5 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2693
Rule 2752
Rule 2865
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}-\frac {6 \int \frac {\cos ^2(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{b}\\ &=-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}-\frac {8 \int \frac {-\frac {a b}{2}-\frac {1}{2} \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{5 b^3}\\ &=-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {\left (4 \left (4 a^2-3 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{5 b^4}-\frac {\left (16 a \left (a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{5 b^4}\\ &=-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {\left (4 \left (4 a^2-3 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}{5 b^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a \left (a^2-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}{5 b^4 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {8 \left (4 a^2-3 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)}}{5 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {32 a \left (a^2-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}}}{5 b^4 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.12, size = 187, normalized size = 0.82 \[ \frac {b \cos (c+d x) \left (16 a^2+4 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-11 b^2\right )+32 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )-8 \left (4 a^3+4 a^2 b-3 a b^2-3 b^3\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{5 b^4 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.70, size = 797, normalized size = 3.48 \[ \frac {\frac {32 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b}{5}-\frac {24 a^{2} \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{2}}{5}-\frac {32 a \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{3}}{5}+\frac {24 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{4}}{5}-\frac {32 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4}}{5}+\frac {56 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{2}}{5}-\frac {24 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{4}}{5}+\frac {2 b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{5}-\frac {4 a \,b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{5}-\frac {16 a^{2} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{5}+\frac {8 b^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{5}+\frac {4 a \,b^{3} \sin \left (d x +c \right )}{5}+\frac {16 a^{2} b^{2}}{5}-2 b^{4}}{b^{5} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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