Optimal. Leaf size=447 \[ \frac {2 a^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d} \]
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Rubi [A] time = 1.42, antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2895, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (-141 a^2 b^2+8 a^4+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 \left (-149 a^4 b^2-516 a^2 b^4+8 a^6-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (-147 a^2 b^2+8 a^4+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2805
Rule 2807
Rule 2895
Rule 3002
Rule 3049
Rule 3059
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac {4 \int \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (-\frac {99 b^2}{4}+\frac {5}{2} a b \sin (c+d x)-\frac {1}{4} \left (8 a^2-117 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{99 b^2}\\ &=-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac {8 \int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (-\frac {693 a b^2}{8}+\frac {3}{4} b \left (5 a^2-18 b^2\right ) \sin (c+d x)-\frac {5}{8} a \left (8 a^2-131 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{693 b^2}\\ &=-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac {16 \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-\frac {3465}{16} a^2 b^2+\frac {15}{8} a b \left (a^2-68 b^2\right ) \sin (c+d x)-\frac {15}{16} \left (8 a^4-141 a^2 b^2+36 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{3465 b^2}\\ &=-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac {32 \int \frac {\csc (c+d x) \left (-\frac {10395}{32} a^3 b^2-\frac {15}{16} b \left (a^4+480 a^2 b^2+18 b^4\right ) \sin (c+d x)-\frac {15}{32} a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{10395 b^2}\\ &=-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac {32 \int \frac {\csc (c+d x) \left (\frac {10395 a^3 b^3}{32}-\frac {15}{32} \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{10395 b^3}+\frac {\left (a \left (8 a^4-147 a^2 b^2+444 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{693 b^3}\\ &=-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+a^3 \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{693 b^3}+\frac {\left (a \left (8 a^4-147 a^2 b^2+444 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}{693 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 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)}}{693 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (a^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{\sqrt {a+b \sin (c+d x)}}-\frac {\left (\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\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}{693 b^3 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 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)}}{693 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\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}}}{693 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {2 a^3 \Pi \left (2;\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}}}{d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 4.02, size = 521, normalized size = 1.17 \[ \frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a^3 b \sin (c+d x)+2660 a^2 b^2+4 \left (113 a^2 b^2-54 b^4\right ) \cos (2 (c+d x))+1954 a b^3 \sin (c+d x)+322 a b^3 \sin (3 (c+d x))-63 b^4 \cos (4 (c+d x))-9 b^4\right )-2 \left (\frac {8 b \left (a^4+480 a^2 b^2+18 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (8 a^4+1239 a^2 b^2+444 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+\frac {2 i \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sec (c+d x) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {\frac {b (\sin (c+d x)+1)}{b-a}} \left (b \left (b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )}{b^2 \sqrt {-\frac {1}{a+b}}}\right )}{2772 b^2 d} \]
Antiderivative was successfully verified.
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fricas [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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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 = 2.04, size = 1573, normalized size = 3.52 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right )\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,\mathrm {cot}\left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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