Optimal. Leaf size=281 \[ \frac {a^2 (c+d x)^{1+m}}{d (1+m)}-\frac {b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {2^{-3-m} b^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {a b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}+\frac {a b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}-\frac {2^{-3-m} b^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f} \]
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Rubi [A]
time = 0.26, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3398, 3389,
2212, 3393, 3388} \begin {gather*} \frac {a b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {f (c+d x)}{d}\right )}{f}+\frac {a b e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {f (c+d x)}{d}\right )}{f}+\frac {b^2 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}-\frac {b^2 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {a^2 (c+d x)^{m+1}}{d (m+1)}-\frac {b^2 (c+d x)^{m+1}}{2 d (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3389
Rule 3393
Rule 3398
Rubi steps
\begin {align*} \int (c+d x)^m (a+b \sinh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^m+2 a b (c+d x)^m \sinh (e+f x)+b^2 (c+d x)^m \sinh ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}+(2 a b) \int (c+d x)^m \sinh (e+f x) \, dx+b^2 \int (c+d x)^m \sinh ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}+(a b) \int e^{-i (i e+i f x)} (c+d x)^m \, dx-(a b) \int e^{i (i e+i f x)} (c+d x)^m \, dx-b^2 \int \left (\frac {1}{2} (c+d x)^m-\frac {1}{2} (c+d x)^m \cosh (2 e+2 f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}-\frac {b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {a b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}+\frac {a b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}+\frac {1}{2} b^2 \int (c+d x)^m \cosh (2 e+2 f x) \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}-\frac {b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {a b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}+\frac {a b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}+\frac {1}{4} b^2 \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac {1}{4} b^2 \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}-\frac {b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {2^{-3-m} b^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {a b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}+\frac {a b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}-\frac {2^{-3-m} b^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}\\ \end {align*}
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Mathematica [A]
time = 6.48, size = 241, normalized size = 0.86 \begin {gather*} \frac {(c+d x)^m \left (\frac {8 a^2 f (c+d x)}{d (1+m)}-\frac {4 b^2 f (c+d x)}{d (1+m)}+8 a b e^{-e+\frac {c f}{d}} \left (f \left (\frac {c}{d}+x\right )\right )^{-m} \Gamma \left (1+m,f \left (\frac {c}{d}+x\right )\right )+2^{-m} b^2 e^{2 e-\frac {2 c f}{d}} \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )+8 a b e^{e-\frac {c f}{d}} \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )-b^2 e^{-2 e+\frac {2 c f}{d}} \left (\frac {2 c f}{d}+2 f x\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )\right )}{8 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \left (a +b \sinh \left (f x +e \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.08, size = 212, normalized size = 0.75 \begin {gather*} {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (\frac {c f}{d} - e\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-\frac {c f}{d} + e\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a b - \frac {1}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-\frac {2 \, c f}{d} + 2 \, e\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} b^{2} + \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 563 vs.
\(2 (279) = 558\).
time = 0.15, size = 563, normalized size = 2.00 \begin {gather*} -\frac {{\left (b^{2} d m + b^{2} d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) - 2 \, c f + 2 \, d \cosh \left (1\right ) + 2 \, d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - 8 \, {\left (a b d m + a b d\right )} \cosh \left (\frac {d m \log \left (\frac {f}{d}\right ) - c f + d \cosh \left (1\right ) + d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) - 8 \, {\left (a b d m + a b d\right )} \cosh \left (\frac {d m \log \left (-\frac {f}{d}\right ) + c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) - {\left (b^{2} d m + b^{2} d\right )} \cosh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) + 2 \, c f - 2 \, d \cosh \left (1\right ) - 2 \, d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) - 2 \, c f + 2 \, d \cosh \left (1\right ) + 2 \, d \sinh \left (1\right )}{d}\right ) + 8 \, {\left (a b d m + a b d\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {f}{d}\right ) - c f + d \cosh \left (1\right ) + d \sinh \left (1\right )}{d}\right ) + 8 \, {\left (a b d m + a b d\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {f}{d}\right ) + c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right ) + {\left (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) + 2 \, c f - 2 \, d \cosh \left (1\right ) - 2 \, d \sinh \left (1\right )}{d}\right ) - 4 \, {\left ({\left (2 \, a^{2} - b^{2}\right )} d f x + {\left (2 \, a^{2} - b^{2}\right )} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 4 \, {\left ({\left (2 \, a^{2} - b^{2}\right )} d f x + {\left (2 \, a^{2} - b^{2}\right )} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{8 \, {\left (d f m + d f\right )}} \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: TypeError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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