Optimal. Leaf size=112 \[ a^4 x-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \cot (c+d x)}{d}-\frac {b^3 (4 a+3 b) \cot ^5(c+d x)}{5 d}-\frac {b^4 \cot ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4128, 390, 203} \[ -\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \cot (c+d x)}{d}+a^4 x-\frac {b^3 (4 a+3 b) \cot ^5(c+d x)}{5 d}-\frac {b^4 \cot ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 390
Rule 4128
Rubi steps
\begin {align*} \int \left (a+b \csc ^2(c+d x)\right )^4 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (b (2 a+b) \left (2 a^2+2 a b+b^2\right )+b^2 \left (6 a^2+8 a b+3 b^2\right ) x^2+b^3 (4 a+3 b) x^4+b^4 x^6+\frac {a^4}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \cot (c+d x)}{d}-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {b^3 (4 a+3 b) \cot ^5(c+d x)}{5 d}-\frac {b^4 \cot ^7(c+d x)}{7 d}-\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=a^4 x-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \cot (c+d x)}{d}-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {b^3 (4 a+3 b) \cot ^5(c+d x)}{5 d}-\frac {b^4 \cot ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 3.57, size = 148, normalized size = 1.32 \[ -\frac {16 \sin ^8(c+d x) \left (a+b \csc ^2(c+d x)\right )^4 \left (b \cot (c+d x) \left (420 a^3+2 b \left (105 a^2+56 a b+12 b^2\right ) \csc ^2(c+d x)+420 a^2 b+6 b^2 (14 a+3 b) \csc ^4(c+d x)+224 a b^2+15 b^3 \csc ^6(c+d x)+48 b^3\right )-105 a^4 (c+d x)\right )}{105 d (a (-\cos (2 (c+d x)))+a+2 b)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 249, normalized size = 2.22 \[ -\frac {4 \, {\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cos \left (d x + c\right )^{7} - 14 \, {\left (90 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + 70 \, {\left (18 \, a^{3} b + 24 \, a^{2} b^{2} + 14 \, a b^{3} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \cos \left (d x + c\right ) - 105 \, {\left (a^{4} d x \cos \left (d x + c\right )^{6} - 3 \, a^{4} d x \cos \left (d x + c\right )^{4} + 3 \, a^{4} d x \cos \left (d x + c\right )^{2} - a^{4} d x\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 351, normalized size = 3.13 \[ \frac {15 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 336 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13440 \, {\left (d x + c\right )} a^{4} + 26880 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 30240 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3675 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {26880 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 30240 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 16800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3675 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 735 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 147 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, b^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.59, size = 129, normalized size = 1.15 \[ \frac {a^{4} \left (d x +c \right )-4 a^{3} b \cot \left (d x +c \right )+6 a^{2} b^{2} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )+4 a \,b^{3} \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{15}\right ) \cot \left (d x +c \right )+b^{4} \left (-\frac {16}{35}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\csc ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\csc ^{2}\left (d x +c \right )\right )}{35}\right ) \cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 141, normalized size = 1.26 \[ a^{4} x - \frac {4 \, a^{3} b}{d \tan \left (d x + c\right )} - \frac {2 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a^{2} b^{2}}{d \tan \left (d x + c\right )^{3}} - \frac {4 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a b^{3}}{15 \, d \tan \left (d x + c\right )^{5}} - \frac {{\left (35 \, \tan \left (d x + c\right )^{6} + 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} + 5\right )} b^{4}}{35 \, d \tan \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.74, size = 106, normalized size = 0.95 \[ a^4\,x-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {3\,b^4}{5}+\frac {4\,a\,b^3}{5}\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (2\,a^2\,b^2+\frac {8\,a\,b^3}{3}+b^4\right )+\frac {b^4}{7}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^7} \]
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 \[ \int \left (a + b \csc ^{2}{\left (c + d x \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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