Optimal. Leaf size=313 \[ -\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}+\frac {16 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{63 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {16 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{63 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.54, antiderivative size = 313, 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 = {2693, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}+\frac {16 a \left (-65 a^2 b^2+32 a^4+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{63 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {16 \left (-57 a^2 b^2+32 a^4+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{63 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}} \]
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 ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}-\frac {10 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{b}\\ &=-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {40 \int \frac {\cos ^2(c+d x) \left (-\frac {a b}{2}-\frac {1}{2} \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{21 b^3}\\ &=-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac {32 \int \frac {a b \left (2 a^2-3 b^2\right )+\frac {1}{4} \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{63 b^5}\\ &=-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac {\left (8 \left (32 a^4-57 a^2 b^2+21 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{63 b^6}+\frac {\left (8 a \left (32 a^4-65 a^2 b^2+33 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{63 b^6}\\ &=-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}-\frac {\left (8 \left (32 a^4-57 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}{63 b^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a \left (32 a^4-65 a^2 b^2+33 b^4\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}{63 b^6 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {2 \cos ^5(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^3 d}-\frac {16 \left (32 a^4-57 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)}}{63 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {16 a \left (32 a^4-65 a^2 b^2+33 b^4\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}}}{63 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 b^5 d}\\ \end {align*}
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Mathematica [A] time = 1.51, size = 273, normalized size = 0.87 \[ \frac {-64 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+b \cos (c+d x) \left (-1024 a^4-256 a^3 b \sin (c+d x)+1760 a^2 b^2+\left (84 b^4-64 a^2 b^2\right ) \cos (2 (c+d x))+404 a b^3 \sin (c+d x)+20 a b^3 \sin (3 (c+d x))+7 b^4 \cos (4 (c+d x))-595 b^4\right )+64 \left (32 a^5+32 a^4 b-57 a^3 b^2-57 a^2 b^3+21 a b^4+21 b^5\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{252 b^6 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{6}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{6}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.82, size = 1195, normalized size = 3.82 \[ \text {result too large to display} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{6}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^6}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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