Optimal. Leaf size=484 \[ -\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d}+\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (384 a^4+116 a^2 b^2-35 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{1920 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (48 a^4-24 a^2 b^2+7 b^4\right ) \Pi \left (2;\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^4 d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A]
time = 1.01, antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2972, 3126,
3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} \frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d}-\frac {\left (384 a^4+332 a^2 b^2-105 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 )}{1920 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (48 a^4-24 a^2 b^2+7 b^4\right ) \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 )}{128 a^4 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (384 a^4+116 a^2 b^2-35 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 )}{1920 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 2972
Rule 3081
Rule 3126
Rule 3134
Rule 3138
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx &=\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}-\frac {\int \csc ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {1}{4} \left (96 a^2-35 b^2\right )+\frac {1}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2}\\ &=\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}-\frac {\int \frac {\csc ^3(c+d x) \left (\frac {1}{8} b \left (108 a^2-35 b^2\right )-\frac {1}{4} a \left (48 a^2+b^2\right ) \sin (c+d x)-\frac {3}{8} b \left (64 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{60 a^2}\\ &=\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (\frac {1}{16} \left (-384 a^4-332 a^2 b^2+105 b^4\right )-\frac {1}{8} a b \left (276 a^2-7 b^2\right ) \sin (c+d x)+\frac {1}{16} b^2 \left (108 a^2-35 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^3}\\ &=-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d}+\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {15}{32} b \left (48 a^4-24 a^2 b^2+7 b^4\right )+\frac {1}{16} a b^2 \left (108 a^2-35 b^2\right ) \sin (c+d x)+\frac {1}{32} b \left (384 a^4+332 a^2 b^2-105 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^4}\\ &=-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d}+\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {15}{32} b^2 \left (48 a^4-24 a^2 b^2+7 b^4\right )+\frac {1}{32} a b \left (384 a^4+116 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^4 b}-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{3840 a^4}\\ &=-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d}+\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}+\frac {\left (384 a^4+116 a^2 b^2-35 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{3840 a^3}+\frac {\left (b \left (48 a^4-24 a^2 b^2+7 b^4\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{256 a^4}-\frac {\left (\left (384 a^4+332 a^2 b^2-105 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}{3840 a^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d}+\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}-\frac {\left (384 a^4+332 a^2 b^2-105 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)}}{1920 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (\left (384 a^4+116 a^2 b^2-35 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}{3840 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (b \left (48 a^4-24 a^2 b^2+7 b^4\right ) \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}{256 a^4 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {\left (384 a^4+332 a^2 b^2-105 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1920 a^4 d}+\frac {b \left (108 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{960 a^3 d}+\frac {\left (96 a^2-35 b^2\right ) \cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{240 a^2 d}+\frac {7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}-\frac {\left (384 a^4+332 a^2 b^2-105 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)}}{1920 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (384 a^4+116 a^2 b^2-35 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}}}{1920 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (48 a^4-24 a^2 b^2+7 b^4\right ) \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}}}{128 a^4 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 16.05, size = 545, normalized size = 1.13 \begin {gather*} \frac {-4 \cot (c+d x) \left (384 a^4+332 a^2 b^2-105 b^4+\left (-216 a^3 b+70 a b^3\right ) \csc (c+d x)-8 a^2 \left (96 a^2+7 b^2\right ) \csc ^2(c+d x)+48 a^3 b \csc ^3(c+d x)+384 a^4 \csc ^4(c+d x)\right ) \sqrt {a+b \sin (c+d x)}+b \left (-\frac {2 i \left (384 a^4+332 a^2 b^2-105 b^4\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (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 )+b \left (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 )-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 )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}+\frac {8 a b \left (108 a^2-35 b^2\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (1056 a^4-1052 a^2 b^2+315 b^4\right ) \Pi \left (2;\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )}{7680 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2074\) vs.
\(2(545)=1090\).
time = 14.38, size = 2075, normalized size = 4.29
method | result | size |
default | \(\text {Expression too large to display}\) | \(2075\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2\,\sqrt {a+b\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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