Optimal. Leaf size=180 \[ \frac {4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a^4 \cos (c+d x)}{d}+\frac {6 a^2 b^2 \cos (c+d x)}{d}-\frac {b^4 \cos (c+d x)}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}-\frac {2 b^4 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}-\frac {4 a^3 b \sin (c+d x)}{d}+\frac {6 a b^3 \sin (c+d x)}{d}+\frac {2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d} \]
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Rubi [A]
time = 0.12, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3598, 2718,
2672, 327, 212, 2670, 14, 294, 276} \begin {gather*} -\frac {a^4 \cos (c+d x)}{d}-\frac {4 a^3 b \sin (c+d x)}{d}+\frac {4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {6 a^2 b^2 \cos (c+d x)}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}+\frac {6 a b^3 \sin (c+d x)}{d}+\frac {2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d}-\frac {6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {b^4 \cos (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}-\frac {2 b^4 \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 212
Rule 276
Rule 294
Rule 327
Rule 2670
Rule 2672
Rule 2718
Rule 3598
Rubi steps
\begin {align*} \int \sin (c+d x) (a+b \tan (c+d x))^4 \, dx &=\int \left (a^4 \sin (c+d x)+4 a^3 b \sin (c+d x) \tan (c+d x)+6 a^2 b^2 \sin (c+d x) \tan ^2(c+d x)+4 a b^3 \sin (c+d x) \tan ^3(c+d x)+b^4 \sin (c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a^4 \int \sin (c+d x) \, dx+\left (4 a^3 b\right ) \int \sin (c+d x) \tan (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sin (c+d x) \tan ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sin (c+d x) \tan ^3(c+d x) \, dx+b^4 \int \sin (c+d x) \tan ^4(c+d x) \, dx\\ &=-\frac {a^4 \cos (c+d x)}{d}+\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^4 \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^4 \cos (c+d x)}{d}-\frac {4 a^3 b \sin (c+d x)}{d}+\frac {2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d}+\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (6 a b^3\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^4 \text {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a^4 \cos (c+d x)}{d}+\frac {6 a^2 b^2 \cos (c+d x)}{d}-\frac {b^4 \cos (c+d x)}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}-\frac {2 b^4 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}-\frac {4 a^3 b \sin (c+d x)}{d}+\frac {6 a b^3 \sin (c+d x)}{d}+\frac {2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d}-\frac {\left (6 a b^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a^4 \cos (c+d x)}{d}+\frac {6 a^2 b^2 \cos (c+d x)}{d}-\frac {b^4 \cos (c+d x)}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}-\frac {2 b^4 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}-\frac {4 a^3 b \sin (c+d x)}{d}+\frac {6 a b^3 \sin (c+d x)}{d}+\frac {2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(383\) vs. \(2(180)=360\).
time = 5.13, size = 383, normalized size = 2.13 \begin {gather*} \frac {72 a^2 b^2-22 b^4-12 \left (a^4-6 a^2 b^2+b^4\right ) \cos (c+d x)-24 a b \left (2 a^2-3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 a b \left (2 a^2-3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^3 (12 a+b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^4 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 b^2 \left (36 a^2-11 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {2 b^4 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {b^3 (-12 a+b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^2 \left (-36 a^2+11 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}-48 a b \left (a^2-b^2\right ) \sin (c+d x)}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 217, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {-a^{4} \cos \left (d x +c \right )+4 a^{3} b \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+6 a^{2} b^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+4 a \,b^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+b^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(217\) |
default | \(\frac {-a^{4} \cos \left (d x +c \right )+4 a^{3} b \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+6 a^{2} b^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+4 a \,b^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+b^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(217\) |
risch | \(\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{3} b}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a \,b^{3}}{d}-\frac {{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 d}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a^{2} b^{2}}{d}-\frac {{\mathrm e}^{i \left (d x +c \right )} b^{4}}{2 d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{3} b}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a \,b^{3}}{d}-\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 d}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2} b^{2}}{d}-\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{2 d}-\frac {4 b^{2} \left (-9 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+3 i a b \,{\mathrm e}^{5 i \left (d x +c \right )}-18 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+4 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-9 a^{2} {\mathrm e}^{i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{i \left (d x +c \right )}-3 i a b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 166, normalized size = 0.92 \begin {gather*} -\frac {3 \, a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} - 18 \, a^{2} b^{2} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + b^{4} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - 6 \, a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 3 \, a^{4} \cos \left (d x + c\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 176, normalized size = 0.98 \begin {gather*} -\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} - 3 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - b^{4} - 6 \, {\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (a b^{3} \cos \left (d x + c\right ) - 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end {gather*}
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 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \sin {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22530 vs.
\(2 (178) = 356\).
time = 8.20, size = 22530, normalized size = 125.17 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.29, size = 268, normalized size = 1.49 \begin {gather*} -\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (6\,a^4-48\,a^2\,b^2+\frac {32\,b^4}{3}\right )+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a\,b^3-8\,a^3\,b\right )-2\,a^4-\frac {16\,b^4}{3}+24\,a^2\,b^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (6\,a^4-24\,a^2\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (12\,a\,b^3-8\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (20\,a\,b^3-24\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (20\,a\,b^3-24\,a^3\,b\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (12\,a\,b^3-8\,a^3\,b\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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